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Metamath Proof Explorer

Theorem psgnunilem4

Description: Lemma for psgnuni . An odd-length representation of the identity is impossible, as it could be repeatedly shortened to a length of 1, but a length 1 permutation must be a transposition. (Contributed by Stefan O'Rear, 25-Aug-2015)

Ref Expression
Hypotheses psgnunilem4.g 
|- G = ( SymGrp ` D )
psgnunilem4.t 
|- T = ran ( pmTrsp ` D )
psgnunilem4.d 
|- ( ph -> D e. V )
psgnunilem4.w1 
|- ( ph -> W e. Word T )
psgnunilem4.w2 
|- ( ph -> ( G gsum W ) = ( _I |` D ) )
Assertion psgnunilem4 
|- ( ph -> ( -u 1 ^ ( # ` W ) ) = 1 )

Proof

Step Hyp Ref Expression
1 psgnunilem4.g 
 |-  G = ( SymGrp ` D )
2 psgnunilem4.t 
 |-  T = ran ( pmTrsp ` D )
3 psgnunilem4.d 
 |-  ( ph -> D e. V )
4 psgnunilem4.w1 
 |-  ( ph -> W e. Word T )
5 psgnunilem4.w2 
 |-  ( ph -> ( G gsum W ) = ( _I |` D ) )
6 wrdfin 
 |-  ( W e. Word T -> W e. Fin )
7 hashcl 
 |-  ( W e. Fin -> ( # ` W ) e. NN0 )
8 4 6 7 3syl 
 |-  ( ph -> ( # ` W ) e. NN0 )
9 nn0uz 
 |-  NN0 = ( ZZ>= ` 0 )
10 8 9 eleqtrdi 
 |-  ( ph -> ( # ` W ) e. ( ZZ>= ` 0 ) )
11 fveq2 
 |-  ( w = (/) -> ( # ` w ) = ( # ` (/) ) )
12 hash0 
 |-  ( # ` (/) ) = 0
13 11 12 eqtrdi 
 |-  ( w = (/) -> ( # ` w ) = 0 )
14 13 oveq2d 
 |-  ( w = (/) -> ( -u 1 ^ ( # ` w ) ) = ( -u 1 ^ 0 ) )
15 neg1cn 
 |-  -u 1 e. CC
16 exp0 
 |-  ( -u 1 e. CC -> ( -u 1 ^ 0 ) = 1 )
17 15 16 ax-mp 
 |-  ( -u 1 ^ 0 ) = 1
18 14 17 eqtrdi 
 |-  ( w = (/) -> ( -u 1 ^ ( # ` w ) ) = 1 )
19 18 2a1d 
 |-  ( w = (/) -> ( ( ph /\ A. x ( ( # ` x ) e. ( 0 ..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsum x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) ) -> ( ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) ) )
20 simpl1 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ -. E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) -> ph )
21 20 3 syl 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ -. E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) -> D e. V )
22 simpl3l 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ -. E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) -> w e. Word T )
23 eqidd 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ -. E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) -> ( # ` w ) = ( # ` w ) )
24 wrdfin 
 |-  ( w e. Word T -> w e. Fin )
25 22 24 syl 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ -. E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) -> w e. Fin )
26 simpl2 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ -. E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) -> w =/= (/) )
27 hashnncl 
 |-  ( w e. Fin -> ( ( # ` w ) e. NN <-> w =/= (/) ) )
28 27 biimpar 
 |-  ( ( w e. Fin /\ w =/= (/) ) -> ( # ` w ) e. NN )
29 25 26 28 syl2anc 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ -. E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) -> ( # ` w ) e. NN )
30 simpl3r 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ -. E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) -> ( G gsum w ) = ( _I |` D ) )
31 fveqeq2 
 |-  ( x = y -> ( ( # ` x ) = ( ( # ` w ) - 2 ) <-> ( # ` y ) = ( ( # ` w ) - 2 ) ) )
32 oveq2 
 |-  ( x = y -> ( G gsum x ) = ( G gsum y ) )
33 32 eqeq1d 
 |-  ( x = y -> ( ( G gsum x ) = ( _I |` D ) <-> ( G gsum y ) = ( _I |` D ) ) )
34 31 33 anbi12d 
 |-  ( x = y -> ( ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) <-> ( ( # ` y ) = ( ( # ` w ) - 2 ) /\ ( G gsum y ) = ( _I |` D ) ) ) )
35 34 cbvrexvw 
 |-  ( E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) <-> E. y e. Word T ( ( # ` y ) = ( ( # ` w ) - 2 ) /\ ( G gsum y ) = ( _I |` D ) ) )
36 35 notbii 
 |-  ( -. E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) <-> -. E. y e. Word T ( ( # ` y ) = ( ( # ` w ) - 2 ) /\ ( G gsum y ) = ( _I |` D ) ) )
37 36 bilani 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ -. E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) -> -. E. y e. Word T ( ( # ` y ) = ( ( # ` w ) - 2 ) /\ ( G gsum y ) = ( _I |` D ) ) )
38 1 2 21 22 23 29 30 37 psgnunilem3 
 |-  -. ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ -. E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) )
39 iman 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) -> E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) <-> -. ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ -. E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) )
40 38 39 mpbir 
 |-  ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) -> E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) )
41 df-rex 
 |-  ( E. x e. Word T ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) <-> E. x ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) )
42 40 41 sylib 
 |-  ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) -> E. x ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) )
43 simprl 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) ) -> x e. Word T )
44 simprrr 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) ) -> ( G gsum x ) = ( _I |` D ) )
45 43 44 jca 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) ) -> ( x e. Word T /\ ( G gsum x ) = ( _I |` D ) ) )
46 wrdfin 
 |-  ( x e. Word T -> x e. Fin )
47 hashcl 
 |-  ( x e. Fin -> ( # ` x ) e. NN0 )
48 43 46 47 3syl 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) ) -> ( # ` x ) e. NN0 )
49 simp3l 
 |-  ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) -> w e. Word T )
50 49 24 syl 
 |-  ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) -> w e. Fin )
51 simp2 
 |-  ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) -> w =/= (/) )
52 50 51 28 syl2anc 
 |-  ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) -> ( # ` w ) e. NN )
53 52 adantr 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) ) -> ( # ` w ) e. NN )
54 simprrl 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) ) -> ( # ` x ) = ( ( # ` w ) - 2 ) )
55 53 nnred 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) ) -> ( # ` w ) e. RR )
56 2rp 
 |-  2 e. RR+
57 ltsubrp 
 |-  ( ( ( # ` w ) e. RR /\ 2 e. RR+ ) -> ( ( # ` w ) - 2 ) < ( # ` w ) )
58 55 56 57 sylancl 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) ) -> ( ( # ` w ) - 2 ) < ( # ` w ) )
59 54 58 eqbrtrd 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) ) -> ( # ` x ) < ( # ` w ) )
60 elfzo0 
 |-  ( ( # ` x ) e. ( 0 ..^ ( # ` w ) ) <-> ( ( # ` x ) e. NN0 /\ ( # ` w ) e. NN /\ ( # ` x ) < ( # ` w ) ) )
61 48 53 59 60 syl3anbrc 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) ) -> ( # ` x ) e. ( 0 ..^ ( # ` w ) ) )
62 id 
 |-  ( ( ( # ` x ) e. ( 0 ..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsum x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) -> ( ( # ` x ) e. ( 0 ..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsum x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) )
63 62 com13 
 |-  ( ( x e. Word T /\ ( G gsum x ) = ( _I |` D ) ) -> ( ( # ` x ) e. ( 0 ..^ ( # ` w ) ) -> ( ( ( # ` x ) e. ( 0 ..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsum x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) )
64 45 61 63 sylc 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) ) -> ( ( ( # ` x ) e. ( 0 ..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsum x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) )
65 54 oveq2d 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) ) -> ( -u 1 ^ ( # ` x ) ) = ( -u 1 ^ ( ( # ` w ) - 2 ) ) )
66 15 a1i 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) ) -> -u 1 e. CC )
67 neg1ne0 
 |-  -u 1 =/= 0
68 67 a1i 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) ) -> -u 1 =/= 0 )
69 2z 
 |-  2 e. ZZ
70 69 a1i 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) ) -> 2 e. ZZ )
71 53 nnzd 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) ) -> ( # ` w ) e. ZZ )
72 66 68 70 71 expsubd 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) ) -> ( -u 1 ^ ( ( # ` w ) - 2 ) ) = ( ( -u 1 ^ ( # ` w ) ) / ( -u 1 ^ 2 ) ) )
73 neg1sqe1 
 |-  ( -u 1 ^ 2 ) = 1
74 73 oveq2i 
 |-  ( ( -u 1 ^ ( # ` w ) ) / ( -u 1 ^ 2 ) ) = ( ( -u 1 ^ ( # ` w ) ) / 1 )
75 m1expcl 
 |-  ( ( # ` w ) e. ZZ -> ( -u 1 ^ ( # ` w ) ) e. ZZ )
76 75 zcnd 
 |-  ( ( # ` w ) e. ZZ -> ( -u 1 ^ ( # ` w ) ) e. CC )
77 71 76 syl 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) ) -> ( -u 1 ^ ( # ` w ) ) e. CC )
78 77 div1d 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) ) -> ( ( -u 1 ^ ( # ` w ) ) / 1 ) = ( -u 1 ^ ( # ` w ) ) )
79 74 78 eqtrid 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) ) -> ( ( -u 1 ^ ( # ` w ) ) / ( -u 1 ^ 2 ) ) = ( -u 1 ^ ( # ` w ) ) )
80 65 72 79 3eqtrd 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) ) -> ( -u 1 ^ ( # ` x ) ) = ( -u 1 ^ ( # ` w ) ) )
81 80 eqeq1d 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) ) -> ( ( -u 1 ^ ( # ` x ) ) = 1 <-> ( -u 1 ^ ( # ` w ) ) = 1 ) )
82 64 81 sylibd 
 |-  ( ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) /\ ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) ) -> ( ( ( # ` x ) e. ( 0 ..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsum x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) )
83 82 ex 
 |-  ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) -> ( ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) -> ( ( ( # ` x ) e. ( 0 ..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsum x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) ) )
84 83 com23 
 |-  ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) -> ( ( ( # ` x ) e. ( 0 ..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsum x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) -> ( ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) ) )
85 84 alimdv 
 |-  ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) -> ( A. x ( ( # ` x ) e. ( 0 ..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsum x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) -> A. x ( ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) ) )
86 19.23v 
 |-  ( A. x ( ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) <-> ( E. x ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) )
87 85 86 imbitrdi 
 |-  ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) -> ( A. x ( ( # ` x ) e. ( 0 ..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsum x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) -> ( E. x ( x e. Word T /\ ( ( # ` x ) = ( ( # ` w ) - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) ) )
88 42 87 mpid 
 |-  ( ( ph /\ w =/= (/) /\ ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) ) -> ( A. x ( ( # ` x ) e. ( 0 ..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsum x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) )
89 88 3exp 
 |-  ( ph -> ( w =/= (/) -> ( ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) -> ( A. x ( ( # ` x ) e. ( 0 ..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsum x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) ) ) )
90 89 com34 
 |-  ( ph -> ( w =/= (/) -> ( A. x ( ( # ` x ) e. ( 0 ..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsum x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) -> ( ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) ) ) )
91 90 com12 
 |-  ( w =/= (/) -> ( ph -> ( A. x ( ( # ` x ) e. ( 0 ..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsum x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) -> ( ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) ) ) )
92 91 impd 
 |-  ( w =/= (/) -> ( ( ph /\ A. x ( ( # ` x ) e. ( 0 ..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsum x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) ) -> ( ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) ) )
93 19 92 pm2.61ine 
 |-  ( ( ph /\ A. x ( ( # ` x ) e. ( 0 ..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsum x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) ) -> ( ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) )
94 93 3adant2 
 |-  ( ( ph /\ ( # ` w ) e. ( 0 ... ( # ` W ) ) /\ A. x ( ( # ` x ) e. ( 0 ..^ ( # ` w ) ) -> ( ( x e. Word T /\ ( G gsum x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) ) -> ( ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) )
95 eleq1 
 |-  ( w = x -> ( w e. Word T <-> x e. Word T ) )
96 oveq2 
 |-  ( w = x -> ( G gsum w ) = ( G gsum x ) )
97 96 eqeq1d 
 |-  ( w = x -> ( ( G gsum w ) = ( _I |` D ) <-> ( G gsum x ) = ( _I |` D ) ) )
98 95 97 anbi12d 
 |-  ( w = x -> ( ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) <-> ( x e. Word T /\ ( G gsum x ) = ( _I |` D ) ) ) )
99 fveq2 
 |-  ( w = x -> ( # ` w ) = ( # ` x ) )
100 99 oveq2d 
 |-  ( w = x -> ( -u 1 ^ ( # ` w ) ) = ( -u 1 ^ ( # ` x ) ) )
101 100 eqeq1d 
 |-  ( w = x -> ( ( -u 1 ^ ( # ` w ) ) = 1 <-> ( -u 1 ^ ( # ` x ) ) = 1 ) )
102 98 101 imbi12d 
 |-  ( w = x -> ( ( ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) <-> ( ( x e. Word T /\ ( G gsum x ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` x ) ) = 1 ) ) )
103 eleq1 
 |-  ( w = W -> ( w e. Word T <-> W e. Word T ) )
104 oveq2 
 |-  ( w = W -> ( G gsum w ) = ( G gsum W ) )
105 104 eqeq1d 
 |-  ( w = W -> ( ( G gsum w ) = ( _I |` D ) <-> ( G gsum W ) = ( _I |` D ) ) )
106 103 105 anbi12d 
 |-  ( w = W -> ( ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) <-> ( W e. Word T /\ ( G gsum W ) = ( _I |` D ) ) ) )
107 fveq2 
 |-  ( w = W -> ( # ` w ) = ( # ` W ) )
108 107 oveq2d 
 |-  ( w = W -> ( -u 1 ^ ( # ` w ) ) = ( -u 1 ^ ( # ` W ) ) )
109 108 eqeq1d 
 |-  ( w = W -> ( ( -u 1 ^ ( # ` w ) ) = 1 <-> ( -u 1 ^ ( # ` W ) ) = 1 ) )
110 106 109 imbi12d 
 |-  ( w = W -> ( ( ( w e. Word T /\ ( G gsum w ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` w ) ) = 1 ) <-> ( ( W e. Word T /\ ( G gsum W ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` W ) ) = 1 ) ) )
111 4 10 94 102 110 99 107 uzindi 
 |-  ( ph -> ( ( W e. Word T /\ ( G gsum W ) = ( _I |` D ) ) -> ( -u 1 ^ ( # ` W ) ) = 1 ) )
112 4 5 111 mp2and 
 |-  ( ph -> ( -u 1 ^ ( # ` W ) ) = 1 )