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Metamath Proof Explorer

Theorem pcadd

Description: An inequality for the prime count of a sum. This is the source of the ultrametric inequality for the p-adic metric. (Contributed by Mario Carneiro, 9-Sep-2014)

Ref Expression
Hypotheses pcadd.1 
|- ( ph -> P e. Prime )
pcadd.2 
|- ( ph -> A e. QQ )
pcadd.3 
|- ( ph -> B e. QQ )
pcadd.4 
|- ( ph -> ( P pCnt A ) <_ ( P pCnt B ) )
Assertion pcadd 
|- ( ph -> ( P pCnt A ) <_ ( P pCnt ( A + B ) ) )

Proof

Step Hyp Ref Expression
1 pcadd.1 
 |-  ( ph -> P e. Prime )
2 pcadd.2 
 |-  ( ph -> A e. QQ )
3 pcadd.3 
 |-  ( ph -> B e. QQ )
4 pcadd.4 
 |-  ( ph -> ( P pCnt A ) <_ ( P pCnt B ) )
5 elq 
 |-  ( A e. QQ <-> E. x e. ZZ E. y e. NN A = ( x / y ) )
6 2 5 sylib 
 |-  ( ph -> E. x e. ZZ E. y e. NN A = ( x / y ) )
7 elq 
 |-  ( B e. QQ <-> E. z e. ZZ E. w e. NN B = ( z / w ) )
8 3 7 sylib 
 |-  ( ph -> E. z e. ZZ E. w e. NN B = ( z / w ) )
9 pcxcl 
 |-  ( ( P e. Prime /\ A e. QQ ) -> ( P pCnt A ) e. RR* )
10 1 2 9 syl2anc 
 |-  ( ph -> ( P pCnt A ) e. RR* )
11 10 xrleidd 
 |-  ( ph -> ( P pCnt A ) <_ ( P pCnt A ) )
12 11 adantr 
 |-  ( ( ph /\ B = 0 ) -> ( P pCnt A ) <_ ( P pCnt A ) )
13 oveq2 
 |-  ( B = 0 -> ( A + B ) = ( A + 0 ) )
14 qcn 
 |-  ( A e. QQ -> A e. CC )
15 2 14 syl 
 |-  ( ph -> A e. CC )
16 15 addridd 
 |-  ( ph -> ( A + 0 ) = A )
17 13 16 sylan9eqr 
 |-  ( ( ph /\ B = 0 ) -> ( A + B ) = A )
18 17 oveq2d 
 |-  ( ( ph /\ B = 0 ) -> ( P pCnt ( A + B ) ) = ( P pCnt A ) )
19 12 18 breqtrrd 
 |-  ( ( ph /\ B = 0 ) -> ( P pCnt A ) <_ ( P pCnt ( A + B ) ) )
20 19 a1d 
 |-  ( ( ph /\ B = 0 ) -> ( ( E. x e. ZZ E. y e. NN A = ( x / y ) /\ E. z e. ZZ E. w e. NN B = ( z / w ) ) -> ( P pCnt A ) <_ ( P pCnt ( A + B ) ) ) )
21 reeanv 
 |-  ( E. x e. ZZ E. z e. ZZ ( E. y e. NN A = ( x / y ) /\ E. w e. NN B = ( z / w ) ) <-> ( E. x e. ZZ E. y e. NN A = ( x / y ) /\ E. z e. ZZ E. w e. NN B = ( z / w ) ) )
22 reeanv 
 |-  ( E. y e. NN E. w e. NN ( A = ( x / y ) /\ B = ( z / w ) ) <-> ( E. y e. NN A = ( x / y ) /\ E. w e. NN B = ( z / w ) ) )
23 1 ad3antrrr 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> P e. Prime )
24 prmnn 
 |-  ( P e. Prime -> P e. NN )
25 23 24 syl 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> P e. NN )
26 simplrl 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> x e. ZZ )
27 simprrl 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> A = ( x / y ) )
28 pc0 
 |-  ( P e. Prime -> ( P pCnt 0 ) = +oo )
29 23 28 syl 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P pCnt 0 ) = +oo )
30 3 ad3antrrr 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> B e. QQ )
31 simpllr 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> B =/= 0 )
32 pcqcl 
 |-  ( ( P e. Prime /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt B ) e. ZZ )
33 23 30 31 32 syl12anc 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P pCnt B ) e. ZZ )
34 33 zred 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P pCnt B ) e. RR )
35 ltpnf 
 |-  ( ( P pCnt B ) e. RR -> ( P pCnt B ) < +oo )
36 rexr 
 |-  ( ( P pCnt B ) e. RR -> ( P pCnt B ) e. RR* )
37 pnfxr 
 |-  +oo e. RR*
38 xrltnle 
 |-  ( ( ( P pCnt B ) e. RR* /\ +oo e. RR* ) -> ( ( P pCnt B ) < +oo <-> -. +oo <_ ( P pCnt B ) ) )
39 36 37 38 sylancl 
 |-  ( ( P pCnt B ) e. RR -> ( ( P pCnt B ) < +oo <-> -. +oo <_ ( P pCnt B ) ) )
40 35 39 mpbid 
 |-  ( ( P pCnt B ) e. RR -> -. +oo <_ ( P pCnt B ) )
41 34 40 syl 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> -. +oo <_ ( P pCnt B ) )
42 29 41 eqnbrtrd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> -. ( P pCnt 0 ) <_ ( P pCnt B ) )
43 4 ad3antrrr 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P pCnt A ) <_ ( P pCnt B ) )
44 oveq2 
 |-  ( A = 0 -> ( P pCnt A ) = ( P pCnt 0 ) )
45 44 breq1d 
 |-  ( A = 0 -> ( ( P pCnt A ) <_ ( P pCnt B ) <-> ( P pCnt 0 ) <_ ( P pCnt B ) ) )
46 43 45 syl5ibcom 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( A = 0 -> ( P pCnt 0 ) <_ ( P pCnt B ) ) )
47 46 necon3bd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( -. ( P pCnt 0 ) <_ ( P pCnt B ) -> A =/= 0 ) )
48 42 47 mpd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> A =/= 0 )
49 27 48 eqnetrrd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( x / y ) =/= 0 )
50 simprll 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> y e. NN )
51 50 nncnd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> y e. CC )
52 50 nnne0d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> y =/= 0 )
53 51 52 div0d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( 0 / y ) = 0 )
54 oveq1 
 |-  ( x = 0 -> ( x / y ) = ( 0 / y ) )
55 54 eqeq1d 
 |-  ( x = 0 -> ( ( x / y ) = 0 <-> ( 0 / y ) = 0 ) )
56 53 55 syl5ibrcom 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( x = 0 -> ( x / y ) = 0 ) )
57 56 necon3d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( ( x / y ) =/= 0 -> x =/= 0 ) )
58 49 57 mpd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> x =/= 0 )
59 pczcl 
 |-  ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) ) -> ( P pCnt x ) e. NN0 )
60 23 26 58 59 syl12anc 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P pCnt x ) e. NN0 )
61 25 60 nnexpcld 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt x ) ) e. NN )
62 61 nncnd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt x ) ) e. CC )
63 62 51 mulcomd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( ( P ^ ( P pCnt x ) ) x. y ) = ( y x. ( P ^ ( P pCnt x ) ) ) )
64 63 oveq2d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( ( x x. ( P ^ ( P pCnt y ) ) ) / ( ( P ^ ( P pCnt x ) ) x. y ) ) = ( ( x x. ( P ^ ( P pCnt y ) ) ) / ( y x. ( P ^ ( P pCnt x ) ) ) ) )
65 26 zcnd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> x e. CC )
66 23 50 pccld 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P pCnt y ) e. NN0 )
67 25 66 nnexpcld 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt y ) ) e. NN )
68 67 nncnd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt y ) ) e. CC )
69 61 nnne0d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt x ) ) =/= 0 )
70 67 nnne0d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt y ) ) =/= 0 )
71 65 62 51 68 69 70 52 divdivdivd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( ( x / ( P ^ ( P pCnt x ) ) ) / ( y / ( P ^ ( P pCnt y ) ) ) ) = ( ( x x. ( P ^ ( P pCnt y ) ) ) / ( ( P ^ ( P pCnt x ) ) x. y ) ) )
72 27 oveq2d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P pCnt A ) = ( P pCnt ( x / y ) ) )
73 pcdiv 
 |-  ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) -> ( P pCnt ( x / y ) ) = ( ( P pCnt x ) - ( P pCnt y ) ) )
74 23 26 58 50 73 syl121anc 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P pCnt ( x / y ) ) = ( ( P pCnt x ) - ( P pCnt y ) ) )
75 72 74 eqtrd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P pCnt A ) = ( ( P pCnt x ) - ( P pCnt y ) ) )
76 75 oveq2d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt A ) ) = ( P ^ ( ( P pCnt x ) - ( P pCnt y ) ) ) )
77 25 nncnd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> P e. CC )
78 25 nnne0d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> P =/= 0 )
79 66 nn0zd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P pCnt y ) e. ZZ )
80 60 nn0zd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P pCnt x ) e. ZZ )
81 77 78 79 80 expsubd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( ( P pCnt x ) - ( P pCnt y ) ) ) = ( ( P ^ ( P pCnt x ) ) / ( P ^ ( P pCnt y ) ) ) )
82 76 81 eqtrd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt A ) ) = ( ( P ^ ( P pCnt x ) ) / ( P ^ ( P pCnt y ) ) ) )
83 82 oveq2d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( A / ( P ^ ( P pCnt A ) ) ) = ( A / ( ( P ^ ( P pCnt x ) ) / ( P ^ ( P pCnt y ) ) ) ) )
84 27 oveq1d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( A / ( ( P ^ ( P pCnt x ) ) / ( P ^ ( P pCnt y ) ) ) ) = ( ( x / y ) / ( ( P ^ ( P pCnt x ) ) / ( P ^ ( P pCnt y ) ) ) ) )
85 65 51 62 68 52 70 69 divdivdivd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( ( x / y ) / ( ( P ^ ( P pCnt x ) ) / ( P ^ ( P pCnt y ) ) ) ) = ( ( x x. ( P ^ ( P pCnt y ) ) ) / ( y x. ( P ^ ( P pCnt x ) ) ) ) )
86 83 84 85 3eqtrd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( A / ( P ^ ( P pCnt A ) ) ) = ( ( x x. ( P ^ ( P pCnt y ) ) ) / ( y x. ( P ^ ( P pCnt x ) ) ) ) )
87 64 71 86 3eqtr4d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( ( x / ( P ^ ( P pCnt x ) ) ) / ( y / ( P ^ ( P pCnt y ) ) ) ) = ( A / ( P ^ ( P pCnt A ) ) ) )
88 87 oveq2d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( ( P ^ ( P pCnt A ) ) x. ( ( x / ( P ^ ( P pCnt x ) ) ) / ( y / ( P ^ ( P pCnt y ) ) ) ) ) = ( ( P ^ ( P pCnt A ) ) x. ( A / ( P ^ ( P pCnt A ) ) ) ) )
89 2 ad3antrrr 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> A e. QQ )
90 89 14 syl 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> A e. CC )
91 pcqcl 
 |-  ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( P pCnt A ) e. ZZ )
92 23 89 48 91 syl12anc 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P pCnt A ) e. ZZ )
93 77 78 92 expclzd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt A ) ) e. CC )
94 77 78 92 expne0d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt A ) ) =/= 0 )
95 90 93 94 divcan2d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( ( P ^ ( P pCnt A ) ) x. ( A / ( P ^ ( P pCnt A ) ) ) ) = A )
96 88 95 eqtr2d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> A = ( ( P ^ ( P pCnt A ) ) x. ( ( x / ( P ^ ( P pCnt x ) ) ) / ( y / ( P ^ ( P pCnt y ) ) ) ) ) )
97 simplrr 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> z e. ZZ )
98 simprrr 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> B = ( z / w ) )
99 98 31 eqnetrrd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( z / w ) =/= 0 )
100 simprlr 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> w e. NN )
101 100 nncnd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> w e. CC )
102 100 nnne0d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> w =/= 0 )
103 101 102 div0d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( 0 / w ) = 0 )
104 oveq1 
 |-  ( z = 0 -> ( z / w ) = ( 0 / w ) )
105 104 eqeq1d 
 |-  ( z = 0 -> ( ( z / w ) = 0 <-> ( 0 / w ) = 0 ) )
106 103 105 syl5ibrcom 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( z = 0 -> ( z / w ) = 0 ) )
107 106 necon3d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( ( z / w ) =/= 0 -> z =/= 0 ) )
108 99 107 mpd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> z =/= 0 )
109 pczcl 
 |-  ( ( P e. Prime /\ ( z e. ZZ /\ z =/= 0 ) ) -> ( P pCnt z ) e. NN0 )
110 23 97 108 109 syl12anc 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P pCnt z ) e. NN0 )
111 25 110 nnexpcld 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt z ) ) e. NN )
112 111 nncnd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt z ) ) e. CC )
113 112 101 mulcomd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( ( P ^ ( P pCnt z ) ) x. w ) = ( w x. ( P ^ ( P pCnt z ) ) ) )
114 113 oveq2d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( ( z x. ( P ^ ( P pCnt w ) ) ) / ( ( P ^ ( P pCnt z ) ) x. w ) ) = ( ( z x. ( P ^ ( P pCnt w ) ) ) / ( w x. ( P ^ ( P pCnt z ) ) ) ) )
115 97 zcnd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> z e. CC )
116 23 100 pccld 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P pCnt w ) e. NN0 )
117 25 116 nnexpcld 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt w ) ) e. NN )
118 117 nncnd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt w ) ) e. CC )
119 111 nnne0d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt z ) ) =/= 0 )
120 117 nnne0d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt w ) ) =/= 0 )
121 115 112 101 118 119 120 102 divdivdivd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( ( z / ( P ^ ( P pCnt z ) ) ) / ( w / ( P ^ ( P pCnt w ) ) ) ) = ( ( z x. ( P ^ ( P pCnt w ) ) ) / ( ( P ^ ( P pCnt z ) ) x. w ) ) )
122 98 oveq2d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P pCnt B ) = ( P pCnt ( z / w ) ) )
123 pcdiv 
 |-  ( ( P e. Prime /\ ( z e. ZZ /\ z =/= 0 ) /\ w e. NN ) -> ( P pCnt ( z / w ) ) = ( ( P pCnt z ) - ( P pCnt w ) ) )
124 23 97 108 100 123 syl121anc 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P pCnt ( z / w ) ) = ( ( P pCnt z ) - ( P pCnt w ) ) )
125 122 124 eqtrd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P pCnt B ) = ( ( P pCnt z ) - ( P pCnt w ) ) )
126 125 oveq2d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt B ) ) = ( P ^ ( ( P pCnt z ) - ( P pCnt w ) ) ) )
127 116 nn0zd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P pCnt w ) e. ZZ )
128 110 nn0zd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P pCnt z ) e. ZZ )
129 77 78 127 128 expsubd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( ( P pCnt z ) - ( P pCnt w ) ) ) = ( ( P ^ ( P pCnt z ) ) / ( P ^ ( P pCnt w ) ) ) )
130 126 129 eqtrd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt B ) ) = ( ( P ^ ( P pCnt z ) ) / ( P ^ ( P pCnt w ) ) ) )
131 130 oveq2d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( B / ( P ^ ( P pCnt B ) ) ) = ( B / ( ( P ^ ( P pCnt z ) ) / ( P ^ ( P pCnt w ) ) ) ) )
132 98 oveq1d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( B / ( ( P ^ ( P pCnt z ) ) / ( P ^ ( P pCnt w ) ) ) ) = ( ( z / w ) / ( ( P ^ ( P pCnt z ) ) / ( P ^ ( P pCnt w ) ) ) ) )
133 115 101 112 118 102 120 119 divdivdivd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( ( z / w ) / ( ( P ^ ( P pCnt z ) ) / ( P ^ ( P pCnt w ) ) ) ) = ( ( z x. ( P ^ ( P pCnt w ) ) ) / ( w x. ( P ^ ( P pCnt z ) ) ) ) )
134 131 132 133 3eqtrd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( B / ( P ^ ( P pCnt B ) ) ) = ( ( z x. ( P ^ ( P pCnt w ) ) ) / ( w x. ( P ^ ( P pCnt z ) ) ) ) )
135 114 121 134 3eqtr4d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( ( z / ( P ^ ( P pCnt z ) ) ) / ( w / ( P ^ ( P pCnt w ) ) ) ) = ( B / ( P ^ ( P pCnt B ) ) ) )
136 135 oveq2d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( ( P ^ ( P pCnt B ) ) x. ( ( z / ( P ^ ( P pCnt z ) ) ) / ( w / ( P ^ ( P pCnt w ) ) ) ) ) = ( ( P ^ ( P pCnt B ) ) x. ( B / ( P ^ ( P pCnt B ) ) ) ) )
137 qcn 
 |-  ( B e. QQ -> B e. CC )
138 30 137 syl 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> B e. CC )
139 77 78 33 expclzd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt B ) ) e. CC )
140 77 78 33 expne0d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt B ) ) =/= 0 )
141 138 139 140 divcan2d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( ( P ^ ( P pCnt B ) ) x. ( B / ( P ^ ( P pCnt B ) ) ) ) = B )
142 136 141 eqtr2d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> B = ( ( P ^ ( P pCnt B ) ) x. ( ( z / ( P ^ ( P pCnt z ) ) ) / ( w / ( P ^ ( P pCnt w ) ) ) ) ) )
143 eluz 
 |-  ( ( ( P pCnt A ) e. ZZ /\ ( P pCnt B ) e. ZZ ) -> ( ( P pCnt B ) e. ( ZZ>= ` ( P pCnt A ) ) <-> ( P pCnt A ) <_ ( P pCnt B ) ) )
144 92 33 143 syl2anc 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( ( P pCnt B ) e. ( ZZ>= ` ( P pCnt A ) ) <-> ( P pCnt A ) <_ ( P pCnt B ) ) )
145 43 144 mpbird 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P pCnt B ) e. ( ZZ>= ` ( P pCnt A ) ) )
146 pczdvds 
 |-  ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) ) -> ( P ^ ( P pCnt x ) ) || x )
147 23 26 58 146 syl12anc 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt x ) ) || x )
148 61 nnzd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt x ) ) e. ZZ )
149 dvdsval2 
 |-  ( ( ( P ^ ( P pCnt x ) ) e. ZZ /\ ( P ^ ( P pCnt x ) ) =/= 0 /\ x e. ZZ ) -> ( ( P ^ ( P pCnt x ) ) || x <-> ( x / ( P ^ ( P pCnt x ) ) ) e. ZZ ) )
150 148 69 26 149 syl3anc 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( ( P ^ ( P pCnt x ) ) || x <-> ( x / ( P ^ ( P pCnt x ) ) ) e. ZZ ) )
151 147 150 mpbid 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( x / ( P ^ ( P pCnt x ) ) ) e. ZZ )
152 pczndvds2 
 |-  ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) ) -> -. P || ( x / ( P ^ ( P pCnt x ) ) ) )
153 23 26 58 152 syl12anc 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> -. P || ( x / ( P ^ ( P pCnt x ) ) ) )
154 151 153 jca 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( ( x / ( P ^ ( P pCnt x ) ) ) e. ZZ /\ -. P || ( x / ( P ^ ( P pCnt x ) ) ) ) )
155 pcdvds 
 |-  ( ( P e. Prime /\ y e. NN ) -> ( P ^ ( P pCnt y ) ) || y )
156 23 50 155 syl2anc 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt y ) ) || y )
157 67 nnzd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt y ) ) e. ZZ )
158 50 nnzd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> y e. ZZ )
159 dvdsval2 
 |-  ( ( ( P ^ ( P pCnt y ) ) e. ZZ /\ ( P ^ ( P pCnt y ) ) =/= 0 /\ y e. ZZ ) -> ( ( P ^ ( P pCnt y ) ) || y <-> ( y / ( P ^ ( P pCnt y ) ) ) e. ZZ ) )
160 157 70 158 159 syl3anc 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( ( P ^ ( P pCnt y ) ) || y <-> ( y / ( P ^ ( P pCnt y ) ) ) e. ZZ ) )
161 156 160 mpbid 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( y / ( P ^ ( P pCnt y ) ) ) e. ZZ )
162 50 nnred 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> y e. RR )
163 67 nnred 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt y ) ) e. RR )
164 50 nngt0d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> 0 < y )
165 67 nngt0d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> 0 < ( P ^ ( P pCnt y ) ) )
166 162 163 164 165 divgt0d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> 0 < ( y / ( P ^ ( P pCnt y ) ) ) )
167 elnnz 
 |-  ( ( y / ( P ^ ( P pCnt y ) ) ) e. NN <-> ( ( y / ( P ^ ( P pCnt y ) ) ) e. ZZ /\ 0 < ( y / ( P ^ ( P pCnt y ) ) ) ) )
168 161 166 167 sylanbrc 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( y / ( P ^ ( P pCnt y ) ) ) e. NN )
169 pcndvds2 
 |-  ( ( P e. Prime /\ y e. NN ) -> -. P || ( y / ( P ^ ( P pCnt y ) ) ) )
170 23 50 169 syl2anc 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> -. P || ( y / ( P ^ ( P pCnt y ) ) ) )
171 168 170 jca 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( ( y / ( P ^ ( P pCnt y ) ) ) e. NN /\ -. P || ( y / ( P ^ ( P pCnt y ) ) ) ) )
172 pczdvds 
 |-  ( ( P e. Prime /\ ( z e. ZZ /\ z =/= 0 ) ) -> ( P ^ ( P pCnt z ) ) || z )
173 23 97 108 172 syl12anc 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt z ) ) || z )
174 111 nnzd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt z ) ) e. ZZ )
175 dvdsval2 
 |-  ( ( ( P ^ ( P pCnt z ) ) e. ZZ /\ ( P ^ ( P pCnt z ) ) =/= 0 /\ z e. ZZ ) -> ( ( P ^ ( P pCnt z ) ) || z <-> ( z / ( P ^ ( P pCnt z ) ) ) e. ZZ ) )
176 174 119 97 175 syl3anc 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( ( P ^ ( P pCnt z ) ) || z <-> ( z / ( P ^ ( P pCnt z ) ) ) e. ZZ ) )
177 173 176 mpbid 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( z / ( P ^ ( P pCnt z ) ) ) e. ZZ )
178 pczndvds2 
 |-  ( ( P e. Prime /\ ( z e. ZZ /\ z =/= 0 ) ) -> -. P || ( z / ( P ^ ( P pCnt z ) ) ) )
179 23 97 108 178 syl12anc 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> -. P || ( z / ( P ^ ( P pCnt z ) ) ) )
180 177 179 jca 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( ( z / ( P ^ ( P pCnt z ) ) ) e. ZZ /\ -. P || ( z / ( P ^ ( P pCnt z ) ) ) ) )
181 pcdvds 
 |-  ( ( P e. Prime /\ w e. NN ) -> ( P ^ ( P pCnt w ) ) || w )
182 23 100 181 syl2anc 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt w ) ) || w )
183 117 nnzd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt w ) ) e. ZZ )
184 100 nnzd 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> w e. ZZ )
185 dvdsval2 
 |-  ( ( ( P ^ ( P pCnt w ) ) e. ZZ /\ ( P ^ ( P pCnt w ) ) =/= 0 /\ w e. ZZ ) -> ( ( P ^ ( P pCnt w ) ) || w <-> ( w / ( P ^ ( P pCnt w ) ) ) e. ZZ ) )
186 183 120 184 185 syl3anc 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( ( P ^ ( P pCnt w ) ) || w <-> ( w / ( P ^ ( P pCnt w ) ) ) e. ZZ ) )
187 182 186 mpbid 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( w / ( P ^ ( P pCnt w ) ) ) e. ZZ )
188 100 nnred 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> w e. RR )
189 117 nnred 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P ^ ( P pCnt w ) ) e. RR )
190 100 nngt0d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> 0 < w )
191 117 nngt0d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> 0 < ( P ^ ( P pCnt w ) ) )
192 188 189 190 191 divgt0d 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> 0 < ( w / ( P ^ ( P pCnt w ) ) ) )
193 elnnz 
 |-  ( ( w / ( P ^ ( P pCnt w ) ) ) e. NN <-> ( ( w / ( P ^ ( P pCnt w ) ) ) e. ZZ /\ 0 < ( w / ( P ^ ( P pCnt w ) ) ) ) )
194 187 192 193 sylanbrc 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( w / ( P ^ ( P pCnt w ) ) ) e. NN )
195 pcndvds2 
 |-  ( ( P e. Prime /\ w e. NN ) -> -. P || ( w / ( P ^ ( P pCnt w ) ) ) )
196 23 100 195 syl2anc 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> -. P || ( w / ( P ^ ( P pCnt w ) ) ) )
197 194 196 jca 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( ( w / ( P ^ ( P pCnt w ) ) ) e. NN /\ -. P || ( w / ( P ^ ( P pCnt w ) ) ) ) )
198 23 96 142 145 154 171 180 197 pcaddlem 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( ( y e. NN /\ w e. NN ) /\ ( A = ( x / y ) /\ B = ( z / w ) ) ) ) -> ( P pCnt A ) <_ ( P pCnt ( A + B ) ) )
199 198 expr 
 |-  ( ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) /\ ( y e. NN /\ w e. NN ) ) -> ( ( A = ( x / y ) /\ B = ( z / w ) ) -> ( P pCnt A ) <_ ( P pCnt ( A + B ) ) ) )
200 199 rexlimdvva 
 |-  ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) -> ( E. y e. NN E. w e. NN ( A = ( x / y ) /\ B = ( z / w ) ) -> ( P pCnt A ) <_ ( P pCnt ( A + B ) ) ) )
201 22 200 biimtrrid 
 |-  ( ( ( ph /\ B =/= 0 ) /\ ( x e. ZZ /\ z e. ZZ ) ) -> ( ( E. y e. NN A = ( x / y ) /\ E. w e. NN B = ( z / w ) ) -> ( P pCnt A ) <_ ( P pCnt ( A + B ) ) ) )
202 201 rexlimdvva 
 |-  ( ( ph /\ B =/= 0 ) -> ( E. x e. ZZ E. z e. ZZ ( E. y e. NN A = ( x / y ) /\ E. w e. NN B = ( z / w ) ) -> ( P pCnt A ) <_ ( P pCnt ( A + B ) ) ) )
203 21 202 biimtrrid 
 |-  ( ( ph /\ B =/= 0 ) -> ( ( E. x e. ZZ E. y e. NN A = ( x / y ) /\ E. z e. ZZ E. w e. NN B = ( z / w ) ) -> ( P pCnt A ) <_ ( P pCnt ( A + B ) ) ) )
204 20 203 pm2.61dane 
 |-  ( ph -> ( ( E. x e. ZZ E. y e. NN A = ( x / y ) /\ E. z e. ZZ E. w e. NN B = ( z / w ) ) -> ( P pCnt A ) <_ ( P pCnt ( A + B ) ) ) )
205 6 8 204 mp2and 
 |-  ( ph -> ( P pCnt A ) <_ ( P pCnt ( A + B ) ) )