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Theorem psgnunilem4 16313
Description: Lemma for psgnuni 16315. 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.)
Hypotheses
Ref Expression
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  gsumg  W )  =  (  _I  |`  D )
)
Assertion
Ref Expression
psgnunilem4  |-  ( ph  ->  ( -u 1 ^ ( # `  W
) )  =  1 )

Proof of Theorem psgnunilem4
Dummy variables  x  w  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psgnunilem4.w1 . 2  |-  ( ph  ->  W  e. Word  T )
2 psgnunilem4.w2 . 2  |-  ( ph  ->  ( G  gsumg  W )  =  (  _I  |`  D )
)
3 wrdfin 12516 . . . . 5  |-  ( W  e. Word  T  ->  W  e.  Fin )
4 hashcl 12385 . . . . 5  |-  ( W  e.  Fin  ->  ( # `
 W )  e. 
NN0 )
51, 3, 43syl 20 . . . 4  |-  ( ph  ->  ( # `  W
)  e.  NN0 )
6 nn0uz 11107 . . . 4  |-  NN0  =  ( ZZ>= `  0 )
75, 6syl6eleq 2560 . . 3  |-  ( ph  ->  ( # `  W
)  e.  ( ZZ>= ` 
0 ) )
8 fveq2 5859 . . . . . . . . . 10  |-  ( w  =  (/)  ->  ( # `  w )  =  (
# `  (/) ) )
9 hash0 12394 . . . . . . . . . 10  |-  ( # `  (/) )  =  0
108, 9syl6eq 2519 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( # `  w )  =  0 )
1110oveq2d 6293 . . . . . . . 8  |-  ( w  =  (/)  ->  ( -u
1 ^ ( # `  w ) )  =  ( -u 1 ^ 0 ) )
12 neg1cn 10630 . . . . . . . . 9  |-  -u 1  e.  CC
13 exp0 12128 . . . . . . . . 9  |-  ( -u
1  e.  CC  ->  (
-u 1 ^ 0 )  =  1 )
1412, 13ax-mp 5 . . . . . . . 8  |-  ( -u
1 ^ 0 )  =  1
1511, 14syl6eq 2519 . . . . . . 7  |-  ( w  =  (/)  ->  ( -u
1 ^ ( # `  w ) )  =  1 )
1615a1d 25 . . . . . 6  |-  ( w  =  (/)  ->  ( ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
)  ->  ( -u 1 ^ ( # `  w
) )  =  1 ) )
1716a1d 25 . . . . 5  |-  ( w  =  (/)  ->  ( (
ph  /\  A. x
( ( # `  x
)  e.  ( 0..^ ( # `  w
) )  ->  (
( x  e. Word  T  /\  ( G  gsumg  x )  =  (  _I  |`  D )
)  ->  ( -u 1 ^ ( # `  x
) )  =  1 ) ) )  -> 
( ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D ) )  ->  ( -u 1 ^ ( # `  w
) )  =  1 ) ) )
18 psgnunilem4.g . . . . . . . . . . . . 13  |-  G  =  ( SymGrp `  D )
19 psgnunilem4.t . . . . . . . . . . . . 13  |-  T  =  ran  (pmTrsp `  D
)
20 simpl1 994 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  -.  E. x  e. Word  T ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) )  ->  ph )
21 psgnunilem4.d . . . . . . . . . . . . . 14  |-  ( ph  ->  D  e.  V )
2220, 21syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  -.  E. x  e. Word  T ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) )  ->  D  e.  V )
23 simpl3l 1046 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  -.  E. x  e. Word  T ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) )  ->  w  e. Word  T )
24 eqidd 2463 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  -.  E. x  e. Word  T ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) )  ->  ( # `
 w )  =  ( # `  w
) )
25 wrdfin 12516 . . . . . . . . . . . . . . 15  |-  ( w  e. Word  T  ->  w  e.  Fin )
2623, 25syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  -.  E. x  e. Word  T ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) )  ->  w  e.  Fin )
27 simpl2 995 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  -.  E. x  e. Word  T ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) )  ->  w  =/=  (/) )
28 hashnncl 12393 . . . . . . . . . . . . . . 15  |-  ( w  e.  Fin  ->  (
( # `  w )  e.  NN  <->  w  =/=  (/) ) )
2928biimpar 485 . . . . . . . . . . . . . 14  |-  ( ( w  e.  Fin  /\  w  =/=  (/) )  ->  ( # `
 w )  e.  NN )
3026, 27, 29syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  -.  E. x  e. Word  T ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) )  ->  ( # `
 w )  e.  NN )
31 simpl3r 1047 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  -.  E. x  e. Word  T ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) )  ->  ( G  gsumg  w )  =  (  _I  |`  D )
)
32 fveq2 5859 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  y  ->  ( # `
 x )  =  ( # `  y
) )
3332eqeq1d 2464 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  y  ->  (
( # `  x )  =  ( ( # `  w )  -  2 )  <->  ( # `  y
)  =  ( (
# `  w )  -  2 ) ) )
34 oveq2 6285 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  y  ->  ( G  gsumg  x )  =  ( G  gsumg  y ) )
3534eqeq1d 2464 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  y  ->  (
( G  gsumg  x )  =  (  _I  |`  D )  <->  ( G  gsumg  y )  =  (  _I  |`  D )
) )
3633, 35anbi12d 710 . . . . . . . . . . . . . . . . 17  |-  ( x  =  y  ->  (
( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
)  <->  ( ( # `  y )  =  ( ( # `  w
)  -  2 )  /\  ( G  gsumg  y )  =  (  _I  |`  D ) ) ) )
3736cbvrexv 3084 . . . . . . . . . . . . . . . 16  |-  ( E. x  e. Word  T ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
)  <->  E. y  e. Word  T
( ( # `  y
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  y )  =  (  _I  |`  D )
) )
3837notbii 296 . . . . . . . . . . . . . . 15  |-  ( -. 
E. x  e. Word  T
( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
)  <->  -.  E. y  e. Word  T ( ( # `  y )  =  ( ( # `  w
)  -  2 )  /\  ( G  gsumg  y )  =  (  _I  |`  D ) ) )
3938biimpi 194 . . . . . . . . . . . . . 14  |-  ( -. 
E. x  e. Word  T
( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
)  ->  -.  E. y  e. Word  T ( ( # `  y )  =  ( ( # `  w
)  -  2 )  /\  ( G  gsumg  y )  =  (  _I  |`  D ) ) )
4039adantl 466 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  -.  E. x  e. Word  T ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) )  ->  -.  E. y  e. Word  T ( ( # `  y
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  y )  =  (  _I  |`  D )
) )
4118, 19, 22, 23, 24, 30, 31, 40psgnunilem3 16312 . . . . . . . . . . . 12  |-  -.  (
( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  -.  E. x  e. Word  T ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) )
42 iman 424 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  ->  E. x  e. Word  T ( ( # `  x )  =  ( ( # `  w
)  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D ) ) )  <->  -.  (
( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  -.  E. x  e. Word  T ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) ) )
4341, 42mpbir 209 . . . . . . . . . . 11  |-  ( (
ph  /\  w  =/=  (/) 
/\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D ) ) )  ->  E. x  e. Word  T ( ( # `  x )  =  ( ( # `  w
)  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D ) ) )
44 df-rex 2815 . . . . . . . . . . 11  |-  ( E. x  e. Word  T ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
)  <->  E. x ( x  e. Word  T  /\  (
( # `  x )  =  ( ( # `  w )  -  2 )  /\  ( G 
gsumg  x )  =  (  _I  |`  D )
) ) )
4543, 44sylib 196 . . . . . . . . . 10  |-  ( (
ph  /\  w  =/=  (/) 
/\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D ) ) )  ->  E. x
( x  e. Word  T  /\  ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) ) )
46 simprl 755 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  (
x  e. Word  T  /\  ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) ) )  ->  x  e. Word  T )
47 simprrr 764 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  (
x  e. Word  T  /\  ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) ) )  -> 
( G  gsumg  x )  =  (  _I  |`  D )
)
4846, 47jca 532 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  (
x  e. Word  T  /\  ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) ) )  -> 
( x  e. Word  T  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) )
49 wrdfin 12516 . . . . . . . . . . . . . . . . . 18  |-  ( x  e. Word  T  ->  x  e.  Fin )
50 hashcl 12385 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  Fin  ->  ( # `
 x )  e. 
NN0 )
5146, 49, 503syl 20 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  (
x  e. Word  T  /\  ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) ) )  -> 
( # `  x )  e.  NN0 )
52 simp3l 1019 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  w  =/=  (/) 
/\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D ) ) )  ->  w  e. Word  T )
5352, 25syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  w  =/=  (/) 
/\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D ) ) )  ->  w  e.  Fin )
54 simp2 992 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  w  =/=  (/) 
/\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D ) ) )  ->  w  =/=  (/) )
5553, 54, 29syl2anc 661 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  w  =/=  (/) 
/\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D ) ) )  ->  ( # `
 w )  e.  NN )
5655adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  (
x  e. Word  T  /\  ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) ) )  -> 
( # `  w )  e.  NN )
57 simprrl 763 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  (
x  e. Word  T  /\  ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) ) )  -> 
( # `  x )  =  ( ( # `  w )  -  2 ) )
5856nnred 10542 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  (
x  e. Word  T  /\  ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) ) )  -> 
( # `  w )  e.  RR )
59 2rp 11216 . . . . . . . . . . . . . . . . . . 19  |-  2  e.  RR+
60 ltsubrp 11242 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( # `  w
)  e.  RR  /\  2  e.  RR+ )  -> 
( ( # `  w
)  -  2 )  <  ( # `  w
) )
6158, 59, 60sylancl 662 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  (
x  e. Word  T  /\  ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) ) )  -> 
( ( # `  w
)  -  2 )  <  ( # `  w
) )
6257, 61eqbrtrd 4462 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  (
x  e. Word  T  /\  ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) ) )  -> 
( # `  x )  <  ( # `  w
) )
63 elfzo0 11822 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  x )  e.  ( 0..^ ( # `  w ) )  <->  ( ( # `
 x )  e. 
NN0  /\  ( # `  w
)  e.  NN  /\  ( # `  x )  <  ( # `  w
) ) )
6451, 56, 62, 63syl3anbrc 1175 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  (
x  e. Word  T  /\  ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) ) )  -> 
( # `  x )  e.  ( 0..^ (
# `  w )
) )
65 id 22 . . . . . . . . . . . . . . . . 17  |-  ( ( ( # `  x
)  e.  ( 0..^ ( # `  w
) )  ->  (
( x  e. Word  T  /\  ( G  gsumg  x )  =  (  _I  |`  D )
)  ->  ( -u 1 ^ ( # `  x
) )  =  1 ) )  ->  (
( # `  x )  e.  ( 0..^ (
# `  w )
)  ->  ( (
x  e. Word  T  /\  ( G  gsumg  x )  =  (  _I  |`  D )
)  ->  ( -u 1 ^ ( # `  x
) )  =  1 ) ) )
6665com13 80 . . . . . . . . . . . . . . . 16  |-  ( ( x  e. Word  T  /\  ( G  gsumg  x )  =  (  _I  |`  D )
)  ->  ( ( # `
 x )  e.  ( 0..^ ( # `  w ) )  -> 
( ( ( # `  x )  e.  ( 0..^ ( # `  w
) )  ->  (
( x  e. Word  T  /\  ( G  gsumg  x )  =  (  _I  |`  D )
)  ->  ( -u 1 ^ ( # `  x
) )  =  1 ) )  ->  ( -u 1 ^ ( # `  x ) )  =  1 ) ) )
6748, 64, 66sylc 60 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  (
x  e. Word  T  /\  ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) ) )  -> 
( ( ( # `  x )  e.  ( 0..^ ( # `  w
) )  ->  (
( x  e. Word  T  /\  ( G  gsumg  x )  =  (  _I  |`  D )
)  ->  ( -u 1 ^ ( # `  x
) )  =  1 ) )  ->  ( -u 1 ^ ( # `  x ) )  =  1 ) )
6857oveq2d 6293 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  (
x  e. Word  T  /\  ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) ) )  -> 
( -u 1 ^ ( # `
 x ) )  =  ( -u 1 ^ ( ( # `  w )  -  2 ) ) )
6912a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  (
x  e. Word  T  /\  ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) ) )  ->  -u 1  e.  CC )
70 neg1ne0 10632 . . . . . . . . . . . . . . . . . . 19  |-  -u 1  =/=  0
7170a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  (
x  e. Word  T  /\  ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) ) )  ->  -u 1  =/=  0 )
72 2z 10887 . . . . . . . . . . . . . . . . . . 19  |-  2  e.  ZZ
7372a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  (
x  e. Word  T  /\  ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) ) )  -> 
2  e.  ZZ )
7456nnzd 10956 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  (
x  e. Word  T  /\  ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) ) )  -> 
( # `  w )  e.  ZZ )
7569, 71, 73, 74expsubd 12278 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  (
x  e. Word  T  /\  ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) ) )  -> 
( -u 1 ^ (
( # `  w )  -  2 ) )  =  ( ( -u
1 ^ ( # `  w ) )  / 
( -u 1 ^ 2 ) ) )
76 neg1sqe1 12220 . . . . . . . . . . . . . . . . . . 19  |-  ( -u
1 ^ 2 )  =  1
7776oveq2i 6288 . . . . . . . . . . . . . . . . . 18  |-  ( (
-u 1 ^ ( # `
 w ) )  /  ( -u 1 ^ 2 ) )  =  ( ( -u
1 ^ ( # `  w ) )  / 
1 )
78 m1expcl 12147 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  w )  e.  ZZ  ->  ( -u 1 ^ ( # `  w
) )  e.  ZZ )
7978zcnd 10958 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  w )  e.  ZZ  ->  ( -u 1 ^ ( # `  w
) )  e.  CC )
8074, 79syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  (
x  e. Word  T  /\  ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) ) )  -> 
( -u 1 ^ ( # `
 w ) )  e.  CC )
8180div1d 10303 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  (
x  e. Word  T  /\  ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) ) )  -> 
( ( -u 1 ^ ( # `  w
) )  /  1
)  =  ( -u
1 ^ ( # `  w ) ) )
8277, 81syl5eq 2515 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  (
x  e. Word  T  /\  ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) ) )  -> 
( ( -u 1 ^ ( # `  w
) )  /  ( -u 1 ^ 2 ) )  =  ( -u
1 ^ ( # `  w ) ) )
8368, 75, 823eqtrd 2507 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  (
x  e. Word  T  /\  ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) ) )  -> 
( -u 1 ^ ( # `
 x ) )  =  ( -u 1 ^ ( # `  w
) ) )
8483eqeq1d 2464 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  (
x  e. Word  T  /\  ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) ) )  -> 
( ( -u 1 ^ ( # `  x
) )  =  1  <-> 
( -u 1 ^ ( # `
 w ) )  =  1 ) )
8567, 84sylibd 214 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  w  =/=  (/)  /\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
) )  /\  (
x  e. Word  T  /\  ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) ) )  -> 
( ( ( # `  x )  e.  ( 0..^ ( # `  w
) )  ->  (
( x  e. Word  T  /\  ( G  gsumg  x )  =  (  _I  |`  D )
)  ->  ( -u 1 ^ ( # `  x
) )  =  1 ) )  ->  ( -u 1 ^ ( # `  w ) )  =  1 ) )
8685ex 434 . . . . . . . . . . . . 13  |-  ( (
ph  /\  w  =/=  (/) 
/\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D ) ) )  ->  (
( x  e. Word  T  /\  ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) )  ->  (
( ( # `  x
)  e.  ( 0..^ ( # `  w
) )  ->  (
( x  e. Word  T  /\  ( G  gsumg  x )  =  (  _I  |`  D )
)  ->  ( -u 1 ^ ( # `  x
) )  =  1 ) )  ->  ( -u 1 ^ ( # `  w ) )  =  1 ) ) )
8786com23 78 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  =/=  (/) 
/\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D ) ) )  ->  (
( ( # `  x
)  e.  ( 0..^ ( # `  w
) )  ->  (
( x  e. Word  T  /\  ( G  gsumg  x )  =  (  _I  |`  D )
)  ->  ( -u 1 ^ ( # `  x
) )  =  1 ) )  ->  (
( x  e. Word  T  /\  ( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) )  ->  ( -u 1 ^ ( # `  w ) )  =  1 ) ) )
8887alimdv 1680 . . . . . . . . . . 11  |-  ( (
ph  /\  w  =/=  (/) 
/\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D ) ) )  ->  ( A. x ( ( # `  x )  e.  ( 0..^ ( # `  w
) )  ->  (
( x  e. Word  T  /\  ( G  gsumg  x )  =  (  _I  |`  D )
)  ->  ( -u 1 ^ ( # `  x
) )  =  1 ) )  ->  A. x
( ( x  e. Word  T  /\  ( ( # `  x )  =  ( ( # `  w
)  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D ) ) )  ->  ( -u 1 ^ ( # `  w ) )  =  1 ) ) )
89 19.23v 1927 . . . . . . . . . . 11  |-  ( A. x ( ( x  e. Word  T  /\  (
( # `  x )  =  ( ( # `  w )  -  2 )  /\  ( G 
gsumg  x )  =  (  _I  |`  D )
) )  ->  ( -u 1 ^ ( # `  w ) )  =  1 )  <->  ( E. x ( x  e. Word  T  /\  ( ( # `  x )  =  ( ( # `  w
)  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D ) ) )  ->  ( -u 1 ^ ( # `  w ) )  =  1 ) )
9088, 89syl6ib 226 . . . . . . . . . 10  |-  ( (
ph  /\  w  =/=  (/) 
/\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D ) ) )  ->  ( A. x ( ( # `  x )  e.  ( 0..^ ( # `  w
) )  ->  (
( x  e. Word  T  /\  ( G  gsumg  x )  =  (  _I  |`  D )
)  ->  ( -u 1 ^ ( # `  x
) )  =  1 ) )  ->  ( E. x ( x  e. Word  T  /\  ( ( # `  x )  =  ( ( # `  w
)  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D ) ) )  ->  ( -u 1 ^ ( # `  w ) )  =  1 ) ) )
9145, 90mpid 41 . . . . . . . . 9  |-  ( (
ph  /\  w  =/=  (/) 
/\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D ) ) )  ->  ( A. x ( ( # `  x )  e.  ( 0..^ ( # `  w
) )  ->  (
( x  e. Word  T  /\  ( G  gsumg  x )  =  (  _I  |`  D )
)  ->  ( -u 1 ^ ( # `  x
) )  =  1 ) )  ->  ( -u 1 ^ ( # `  w ) )  =  1 ) )
92913exp 1190 . . . . . . . 8  |-  ( ph  ->  ( w  =/=  (/)  ->  (
( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
)  ->  ( A. x ( ( # `  x )  e.  ( 0..^ ( # `  w
) )  ->  (
( x  e. Word  T  /\  ( G  gsumg  x )  =  (  _I  |`  D )
)  ->  ( -u 1 ^ ( # `  x
) )  =  1 ) )  ->  ( -u 1 ^ ( # `  w ) )  =  1 ) ) ) )
9392com34 83 . . . . . . 7  |-  ( ph  ->  ( w  =/=  (/)  ->  ( A. x ( ( # `  x )  e.  ( 0..^ ( # `  w
) )  ->  (
( x  e. Word  T  /\  ( G  gsumg  x )  =  (  _I  |`  D )
)  ->  ( -u 1 ^ ( # `  x
) )  =  1 ) )  ->  (
( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
)  ->  ( -u 1 ^ ( # `  w
) )  =  1 ) ) ) )
9493com12 31 . . . . . 6  |-  ( w  =/=  (/)  ->  ( ph  ->  ( A. x ( ( # `  x
)  e.  ( 0..^ ( # `  w
) )  ->  (
( x  e. Word  T  /\  ( G  gsumg  x )  =  (  _I  |`  D )
)  ->  ( -u 1 ^ ( # `  x
) )  =  1 ) )  ->  (
( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
)  ->  ( -u 1 ^ ( # `  w
) )  =  1 ) ) ) )
9594impd 431 . . . . 5  |-  ( w  =/=  (/)  ->  ( ( ph  /\  A. x ( ( # `  x
)  e.  ( 0..^ ( # `  w
) )  ->  (
( x  e. Word  T  /\  ( G  gsumg  x )  =  (  _I  |`  D )
)  ->  ( -u 1 ^ ( # `  x
) )  =  1 ) ) )  -> 
( ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D ) )  ->  ( -u 1 ^ ( # `  w
) )  =  1 ) ) )
9617, 95pm2.61ine 2775 . . . 4  |-  ( (
ph  /\  A. x
( ( # `  x
)  e.  ( 0..^ ( # `  w
) )  ->  (
( x  e. Word  T  /\  ( G  gsumg  x )  =  (  _I  |`  D )
)  ->  ( -u 1 ^ ( # `  x
) )  =  1 ) ) )  -> 
( ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D ) )  ->  ( -u 1 ^ ( # `  w
) )  =  1 ) )
97963adant2 1010 . . 3  |-  ( (
ph  /\  ( # `  w
)  e.  ( 0 ... ( # `  W
) )  /\  A. x ( ( # `  x )  e.  ( 0..^ ( # `  w
) )  ->  (
( x  e. Word  T  /\  ( G  gsumg  x )  =  (  _I  |`  D )
)  ->  ( -u 1 ^ ( # `  x
) )  =  1 ) ) )  -> 
( ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D ) )  ->  ( -u 1 ^ ( # `  w
) )  =  1 ) )
98 eleq1 2534 . . . . 5  |-  ( w  =  x  ->  (
w  e. Word  T  <->  x  e. Word  T ) )
99 oveq2 6285 . . . . . 6  |-  ( w  =  x  ->  ( G  gsumg  w )  =  ( G  gsumg  x ) )
10099eqeq1d 2464 . . . . 5  |-  ( w  =  x  ->  (
( G  gsumg  w )  =  (  _I  |`  D )  <->  ( G  gsumg  x )  =  (  _I  |`  D )
) )
10198, 100anbi12d 710 . . . 4  |-  ( w  =  x  ->  (
( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
)  <->  ( x  e. Word  T  /\  ( G  gsumg  x )  =  (  _I  |`  D ) ) ) )
102 fveq2 5859 . . . . . 6  |-  ( w  =  x  ->  ( # `
 w )  =  ( # `  x
) )
103102oveq2d 6293 . . . . 5  |-  ( w  =  x  ->  ( -u 1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  x
) ) )
104103eqeq1d 2464 . . . 4  |-  ( w  =  x  ->  (
( -u 1 ^ ( # `
 w ) )  =  1  <->  ( -u 1 ^ ( # `  x
) )  =  1 ) )
105101, 104imbi12d 320 . . 3  |-  ( w  =  x  ->  (
( ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D ) )  ->  ( -u 1 ^ ( # `  w
) )  =  1 )  <->  ( ( x  e. Word  T  /\  ( G  gsumg  x )  =  (  _I  |`  D )
)  ->  ( -u 1 ^ ( # `  x
) )  =  1 ) ) )
106 eleq1 2534 . . . . 5  |-  ( w  =  W  ->  (
w  e. Word  T  <->  W  e. Word  T ) )
107 oveq2 6285 . . . . . 6  |-  ( w  =  W  ->  ( G  gsumg  w )  =  ( G  gsumg  W ) )
108107eqeq1d 2464 . . . . 5  |-  ( w  =  W  ->  (
( G  gsumg  w )  =  (  _I  |`  D )  <->  ( G  gsumg  W )  =  (  _I  |`  D )
) )
109106, 108anbi12d 710 . . . 4  |-  ( w  =  W  ->  (
( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
)  <->  ( W  e. Word  T  /\  ( G  gsumg  W )  =  (  _I  |`  D ) ) ) )
110 fveq2 5859 . . . . . 6  |-  ( w  =  W  ->  ( # `
 w )  =  ( # `  W
) )
111110oveq2d 6293 . . . . 5  |-  ( w  =  W  ->  ( -u 1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  W
) ) )
112111eqeq1d 2464 . . . 4  |-  ( w  =  W  ->  (
( -u 1 ^ ( # `
 w ) )  =  1  <->  ( -u 1 ^ ( # `  W
) )  =  1 ) )
113109, 112imbi12d 320 . . 3  |-  ( w  =  W  ->  (
( ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D ) )  ->  ( -u 1 ^ ( # `  w
) )  =  1 )  <->  ( ( W  e. Word  T  /\  ( G  gsumg  W )  =  (  _I  |`  D )
)  ->  ( -u 1 ^ ( # `  W
) )  =  1 ) ) )
1141, 7, 97, 105, 113, 102, 110uzindi 12049 . 2  |-  ( ph  ->  ( ( W  e. Word  T  /\  ( G  gsumg  W )  =  (  _I  |`  D ) )  ->  ( -u 1 ^ ( # `  W
) )  =  1 ) )
1151, 2, 114mp2and 679 1  |-  ( ph  ->  ( -u 1 ^ ( # `  W
) )  =  1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 968   A.wal 1372    = wceq 1374   E.wex 1591    e. wcel 1762    =/= wne 2657   E.wrex 2810   (/)c0 3780   class class class wbr 4442    _I cid 4785   ran crn 4995    |` cres 4996   ` cfv 5581  (class class class)co 6277   Fincfn 7508   CCcc 9481   RRcr 9482   0cc0 9483   1c1 9484    < clt 9619    - cmin 9796   -ucneg 9797    / cdiv 10197   NNcn 10527   2c2 10576   NN0cn0 10786   ZZcz 10855   ZZ>=cuz 11073   RR+crp 11211   ...cfz 11663  ..^cfzo 11783   ^cexp 12124   #chash 12362  Word cword 12489    gsumg cgsu 14687   SymGrpcsymg 16192  pmTrspcpmtr 16257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-xor 1356  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-ot 4031  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-card 8311  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-n0 10787  df-z 10856  df-uz 11074  df-rp 11212  df-fz 11664  df-fzo 11784  df-seq 12066  df-exp 12125  df-hash 12363  df-word 12497  df-concat 12499  df-s1 12500  df-substr 12501  df-splice 12502  df-s2 12765  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-tset 14565  df-0g 14688  df-gsum 14689  df-mnd 15723  df-submnd 15773  df-grp 15853  df-minusg 15854  df-subg 15988  df-symg 16193  df-pmtr 16258
This theorem is referenced by:  psgnuni  16315
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