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Metamath Proof Explorer

Theorem 4sqlem10

Description: Lemma for 4sq . (Contributed by Mario Carneiro, 16-Jul-2014)

Ref Expression
Hypotheses 4sqlem5.2 
|- ( ph -> A e. ZZ )
4sqlem5.3 
|- ( ph -> M e. NN )
4sqlem5.4 
|- B = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )
4sqlem10.5 
|- ( ( ph /\ ps ) -> ( ( ( ( M ^ 2 ) / 2 ) / 2 ) - ( B ^ 2 ) ) = 0 )
Assertion 4sqlem10 
|- ( ( ph /\ ps ) -> ( M ^ 2 ) || ( ( A ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) )

Proof

Step Hyp Ref Expression
1 4sqlem5.2 
 |-  ( ph -> A e. ZZ )
2 4sqlem5.3 
 |-  ( ph -> M e. NN )
3 4sqlem5.4 
 |-  B = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )
4 4sqlem10.5 
 |-  ( ( ph /\ ps ) -> ( ( ( ( M ^ 2 ) / 2 ) / 2 ) - ( B ^ 2 ) ) = 0 )
5 2 adantr 
 |-  ( ( ph /\ ps ) -> M e. NN )
6 5 nnzd 
 |-  ( ( ph /\ ps ) -> M e. ZZ )
7 zsqcl 
 |-  ( M e. ZZ -> ( M ^ 2 ) e. ZZ )
8 6 7 syl 
 |-  ( ( ph /\ ps ) -> ( M ^ 2 ) e. ZZ )
9 1 adantr 
 |-  ( ( ph /\ ps ) -> A e. ZZ )
10 5 nnred 
 |-  ( ( ph /\ ps ) -> M e. RR )
11 10 rehalfcld 
 |-  ( ( ph /\ ps ) -> ( M / 2 ) e. RR )
12 11 recnd 
 |-  ( ( ph /\ ps ) -> ( M / 2 ) e. CC )
13 12 negnegd 
 |-  ( ( ph /\ ps ) -> -u -u ( M / 2 ) = ( M / 2 ) )
14 1 2 3 4sqlem5 
 |-  ( ph -> ( B e. ZZ /\ ( ( A - B ) / M ) e. ZZ ) )
15 14 adantr 
 |-  ( ( ph /\ ps ) -> ( B e. ZZ /\ ( ( A - B ) / M ) e. ZZ ) )
16 15 simpld 
 |-  ( ( ph /\ ps ) -> B e. ZZ )
17 16 zred 
 |-  ( ( ph /\ ps ) -> B e. RR )
18 1 2 3 4sqlem6 
 |-  ( ph -> ( -u ( M / 2 ) <_ B /\ B < ( M / 2 ) ) )
19 18 adantr 
 |-  ( ( ph /\ ps ) -> ( -u ( M / 2 ) <_ B /\ B < ( M / 2 ) ) )
20 19 simprd 
 |-  ( ( ph /\ ps ) -> B < ( M / 2 ) )
21 17 20 ltned 
 |-  ( ( ph /\ ps ) -> B =/= ( M / 2 ) )
22 21 neneqd 
 |-  ( ( ph /\ ps ) -> -. B = ( M / 2 ) )
23 2cnd 
 |-  ( ( ph /\ ps ) -> 2 e. CC )
24 23 sqvald 
 |-  ( ( ph /\ ps ) -> ( 2 ^ 2 ) = ( 2 x. 2 ) )
25 24 oveq2d 
 |-  ( ( ph /\ ps ) -> ( ( M ^ 2 ) / ( 2 ^ 2 ) ) = ( ( M ^ 2 ) / ( 2 x. 2 ) ) )
26 5 nncnd 
 |-  ( ( ph /\ ps ) -> M e. CC )
27 2ne0 
 |-  2 =/= 0
28 27 a1i 
 |-  ( ( ph /\ ps ) -> 2 =/= 0 )
29 26 23 28 sqdivd 
 |-  ( ( ph /\ ps ) -> ( ( M / 2 ) ^ 2 ) = ( ( M ^ 2 ) / ( 2 ^ 2 ) ) )
30 26 sqcld 
 |-  ( ( ph /\ ps ) -> ( M ^ 2 ) e. CC )
31 30 23 23 28 28 divdiv1d 
 |-  ( ( ph /\ ps ) -> ( ( ( M ^ 2 ) / 2 ) / 2 ) = ( ( M ^ 2 ) / ( 2 x. 2 ) ) )
32 25 29 31 3eqtr4d 
 |-  ( ( ph /\ ps ) -> ( ( M / 2 ) ^ 2 ) = ( ( ( M ^ 2 ) / 2 ) / 2 ) )
33 30 halfcld 
 |-  ( ( ph /\ ps ) -> ( ( M ^ 2 ) / 2 ) e. CC )
34 33 halfcld 
 |-  ( ( ph /\ ps ) -> ( ( ( M ^ 2 ) / 2 ) / 2 ) e. CC )
35 16 zcnd 
 |-  ( ( ph /\ ps ) -> B e. CC )
36 35 sqcld 
 |-  ( ( ph /\ ps ) -> ( B ^ 2 ) e. CC )
37 34 36 4 subeq0d 
 |-  ( ( ph /\ ps ) -> ( ( ( M ^ 2 ) / 2 ) / 2 ) = ( B ^ 2 ) )
38 32 37 eqtr2d 
 |-  ( ( ph /\ ps ) -> ( B ^ 2 ) = ( ( M / 2 ) ^ 2 ) )
39 sqeqor 
 |-  ( ( B e. CC /\ ( M / 2 ) e. CC ) -> ( ( B ^ 2 ) = ( ( M / 2 ) ^ 2 ) <-> ( B = ( M / 2 ) \/ B = -u ( M / 2 ) ) ) )
40 35 12 39 syl2anc 
 |-  ( ( ph /\ ps ) -> ( ( B ^ 2 ) = ( ( M / 2 ) ^ 2 ) <-> ( B = ( M / 2 ) \/ B = -u ( M / 2 ) ) ) )
41 38 40 mpbid 
 |-  ( ( ph /\ ps ) -> ( B = ( M / 2 ) \/ B = -u ( M / 2 ) ) )
42 41 ord 
 |-  ( ( ph /\ ps ) -> ( -. B = ( M / 2 ) -> B = -u ( M / 2 ) ) )
43 22 42 mpd 
 |-  ( ( ph /\ ps ) -> B = -u ( M / 2 ) )
44 43 16 eqeltrrd 
 |-  ( ( ph /\ ps ) -> -u ( M / 2 ) e. ZZ )
45 44 znegcld 
 |-  ( ( ph /\ ps ) -> -u -u ( M / 2 ) e. ZZ )
46 13 45 eqeltrrd 
 |-  ( ( ph /\ ps ) -> ( M / 2 ) e. ZZ )
47 9 46 zaddcld 
 |-  ( ( ph /\ ps ) -> ( A + ( M / 2 ) ) e. ZZ )
48 zsqcl 
 |-  ( ( A + ( M / 2 ) ) e. ZZ -> ( ( A + ( M / 2 ) ) ^ 2 ) e. ZZ )
49 47 48 syl 
 |-  ( ( ph /\ ps ) -> ( ( A + ( M / 2 ) ) ^ 2 ) e. ZZ )
50 47 6 zmulcld 
 |-  ( ( ph /\ ps ) -> ( ( A + ( M / 2 ) ) x. M ) e. ZZ )
51 47 zred 
 |-  ( ( ph /\ ps ) -> ( A + ( M / 2 ) ) e. RR )
52 5 nnrpd 
 |-  ( ( ph /\ ps ) -> M e. RR+ )
53 51 52 modcld 
 |-  ( ( ph /\ ps ) -> ( ( A + ( M / 2 ) ) mod M ) e. RR )
54 53 recnd 
 |-  ( ( ph /\ ps ) -> ( ( A + ( M / 2 ) ) mod M ) e. CC )
55 0cnd 
 |-  ( ( ph /\ ps ) -> 0 e. CC )
56 df-neg 
 |-  -u ( M / 2 ) = ( 0 - ( M / 2 ) )
57 43 3 56 3eqtr3g 
 |-  ( ( ph /\ ps ) -> ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) ) = ( 0 - ( M / 2 ) ) )
58 54 55 12 57 subcan2d 
 |-  ( ( ph /\ ps ) -> ( ( A + ( M / 2 ) ) mod M ) = 0 )
59 dvdsval3 
 |-  ( ( M e. NN /\ ( A + ( M / 2 ) ) e. ZZ ) -> ( M || ( A + ( M / 2 ) ) <-> ( ( A + ( M / 2 ) ) mod M ) = 0 ) )
60 5 47 59 syl2anc 
 |-  ( ( ph /\ ps ) -> ( M || ( A + ( M / 2 ) ) <-> ( ( A + ( M / 2 ) ) mod M ) = 0 ) )
61 58 60 mpbird 
 |-  ( ( ph /\ ps ) -> M || ( A + ( M / 2 ) ) )
62 dvdssq 
 |-  ( ( M e. ZZ /\ ( A + ( M / 2 ) ) e. ZZ ) -> ( M || ( A + ( M / 2 ) ) <-> ( M ^ 2 ) || ( ( A + ( M / 2 ) ) ^ 2 ) ) )
63 6 47 62 syl2anc 
 |-  ( ( ph /\ ps ) -> ( M || ( A + ( M / 2 ) ) <-> ( M ^ 2 ) || ( ( A + ( M / 2 ) ) ^ 2 ) ) )
64 61 63 mpbid 
 |-  ( ( ph /\ ps ) -> ( M ^ 2 ) || ( ( A + ( M / 2 ) ) ^ 2 ) )
65 26 sqvald 
 |-  ( ( ph /\ ps ) -> ( M ^ 2 ) = ( M x. M ) )
66 5 nnne0d 
 |-  ( ( ph /\ ps ) -> M =/= 0 )
67 dvdsmulcr 
 |-  ( ( M e. ZZ /\ ( A + ( M / 2 ) ) e. ZZ /\ ( M e. ZZ /\ M =/= 0 ) ) -> ( ( M x. M ) || ( ( A + ( M / 2 ) ) x. M ) <-> M || ( A + ( M / 2 ) ) ) )
68 6 47 6 66 67 syl112anc 
 |-  ( ( ph /\ ps ) -> ( ( M x. M ) || ( ( A + ( M / 2 ) ) x. M ) <-> M || ( A + ( M / 2 ) ) ) )
69 61 68 mpbird 
 |-  ( ( ph /\ ps ) -> ( M x. M ) || ( ( A + ( M / 2 ) ) x. M ) )
70 65 69 eqbrtrd 
 |-  ( ( ph /\ ps ) -> ( M ^ 2 ) || ( ( A + ( M / 2 ) ) x. M ) )
71 8 49 50 64 70 dvds2subd 
 |-  ( ( ph /\ ps ) -> ( M ^ 2 ) || ( ( ( A + ( M / 2 ) ) ^ 2 ) - ( ( A + ( M / 2 ) ) x. M ) ) )
72 47 zcnd 
 |-  ( ( ph /\ ps ) -> ( A + ( M / 2 ) ) e. CC )
73 72 sqvald 
 |-  ( ( ph /\ ps ) -> ( ( A + ( M / 2 ) ) ^ 2 ) = ( ( A + ( M / 2 ) ) x. ( A + ( M / 2 ) ) ) )
74 73 oveq1d 
 |-  ( ( ph /\ ps ) -> ( ( ( A + ( M / 2 ) ) ^ 2 ) - ( ( A + ( M / 2 ) ) x. M ) ) = ( ( ( A + ( M / 2 ) ) x. ( A + ( M / 2 ) ) ) - ( ( A + ( M / 2 ) ) x. M ) ) )
75 72 72 26 subdid 
 |-  ( ( ph /\ ps ) -> ( ( A + ( M / 2 ) ) x. ( ( A + ( M / 2 ) ) - M ) ) = ( ( ( A + ( M / 2 ) ) x. ( A + ( M / 2 ) ) ) - ( ( A + ( M / 2 ) ) x. M ) ) )
76 26 2halvesd 
 |-  ( ( ph /\ ps ) -> ( ( M / 2 ) + ( M / 2 ) ) = M )
77 76 oveq2d 
 |-  ( ( ph /\ ps ) -> ( ( A + ( M / 2 ) ) - ( ( M / 2 ) + ( M / 2 ) ) ) = ( ( A + ( M / 2 ) ) - M ) )
78 9 zcnd 
 |-  ( ( ph /\ ps ) -> A e. CC )
79 78 12 12 pnpcan2d 
 |-  ( ( ph /\ ps ) -> ( ( A + ( M / 2 ) ) - ( ( M / 2 ) + ( M / 2 ) ) ) = ( A - ( M / 2 ) ) )
80 77 79 eqtr3d 
 |-  ( ( ph /\ ps ) -> ( ( A + ( M / 2 ) ) - M ) = ( A - ( M / 2 ) ) )
81 80 oveq2d 
 |-  ( ( ph /\ ps ) -> ( ( A + ( M / 2 ) ) x. ( ( A + ( M / 2 ) ) - M ) ) = ( ( A + ( M / 2 ) ) x. ( A - ( M / 2 ) ) ) )
82 subsq 
 |-  ( ( A e. CC /\ ( M / 2 ) e. CC ) -> ( ( A ^ 2 ) - ( ( M / 2 ) ^ 2 ) ) = ( ( A + ( M / 2 ) ) x. ( A - ( M / 2 ) ) ) )
83 78 12 82 syl2anc 
 |-  ( ( ph /\ ps ) -> ( ( A ^ 2 ) - ( ( M / 2 ) ^ 2 ) ) = ( ( A + ( M / 2 ) ) x. ( A - ( M / 2 ) ) ) )
84 32 oveq2d 
 |-  ( ( ph /\ ps ) -> ( ( A ^ 2 ) - ( ( M / 2 ) ^ 2 ) ) = ( ( A ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) )
85 81 83 84 3eqtr2d 
 |-  ( ( ph /\ ps ) -> ( ( A + ( M / 2 ) ) x. ( ( A + ( M / 2 ) ) - M ) ) = ( ( A ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) )
86 74 75 85 3eqtr2d 
 |-  ( ( ph /\ ps ) -> ( ( ( A + ( M / 2 ) ) ^ 2 ) - ( ( A + ( M / 2 ) ) x. M ) ) = ( ( A ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) )
87 71 86 breqtrd 
 |-  ( ( ph /\ ps ) -> ( M ^ 2 ) || ( ( A ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) )