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Theorem psgnunilem4 16002
Description: Lemma for psgnuni 16004. 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 12247 . . . . 5  |-  ( W  e. Word  T  ->  W  e.  Fin )
4 hashcl 12125 . . . . 5  |-  ( W  e.  Fin  ->  ( # `
 W )  e. 
NN0 )
51, 3, 43syl 20 . . . 4  |-  ( ph  ->  ( # `  W
)  e.  NN0 )
6 nn0uz 10894 . . . 4  |-  NN0  =  ( ZZ>= `  0 )
75, 6syl6eleq 2532 . . 3  |-  ( ph  ->  ( # `  W
)  e.  ( ZZ>= ` 
0 ) )
8 fveq2 5690 . . . . . . . . . 10  |-  ( w  =  (/)  ->  ( # `  w )  =  (
# `  (/) ) )
9 hash0 12134 . . . . . . . . . 10  |-  ( # `  (/) )  =  0
108, 9syl6eq 2490 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( # `  w )  =  0 )
1110oveq2d 6106 . . . . . . . 8  |-  ( w  =  (/)  ->  ( -u
1 ^ ( # `  w ) )  =  ( -u 1 ^ 0 ) )
12 neg1cn 10424 . . . . . . . . 9  |-  -u 1  e.  CC
13 exp0 11868 . . . . . . . . 9  |-  ( -u
1  e.  CC  ->  (
-u 1 ^ 0 )  =  1 )
1412, 13ax-mp 5 . . . . . . . 8  |-  ( -u
1 ^ 0 )  =  1
1511, 14syl6eq 2490 . . . . . . 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 991 . . . . . . . . . . . . . 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 1043 . . . . . . . . . . . . 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 2443 . . . . . . . . . . . . 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 12247 . . . . . . . . . . . . . . 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 992 . . . . . . . . . . . . . 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 12133 . . . . . . . . . . . . . . 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 1044 . . . . . . . . . . . . 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 5690 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  y  ->  ( # `
 x )  =  ( # `  y
) )
3332eqeq1d 2450 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  y  ->  (
( # `  x )  =  ( ( # `  w )  -  2 )  <->  ( # `  y
)  =  ( (
# `  w )  -  2 ) ) )
34 oveq2 6098 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  y  ->  ( G  gsumg  x )  =  ( G  gsumg  y ) )
3534eqeq1d 2450 . . . . . . . . . . . . . . . . . 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 2947 . . . . . . . . . . . . . . . 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 16001 . . . . . . . . . . . 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 2720 . . . . . . . . . . 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 12247 . . . . . . . . . . . . . . . . . 18  |-  ( x  e. Word  T  ->  x  e.  Fin )
50 hashcl 12125 . . . . . . . . . . . . . . . . . 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 1016 . . . . . . . . . . . . . . . . . . . 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 989 . . . . . . . . . . . . . . . . . . 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 10336 . . . . . . . . . . . . . . . . . . 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 10995 . . . . . . . . . . . . . . . . . . 19  |-  2  e.  RR+
60 ltsubrp 11021 . . . . . . . . . . . . . . . . . . 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 4311 . . . . . . . . . . . . . . . . 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 11586 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  x )  e.  ( 0..^ ( # `  w ) )  <->  ( ( # `
 x )  e. 
NN0  /\  ( # `  w
)  e.  NN  /\  ( # `  x )  <  ( # `  w
) ) )
6451, 56, 62, 63syl3anbrc 1172 . . . . . . . . . . . . . . . 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 6106 . . . . . . . . . . . . . . . . 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 10426 . . . . . . . . . . . . . . . . . . 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 10677 . . . . . . . . . . . . . . . . . . 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 10745 . . . . . . . . . . . . . . . . . 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 12018 . . . . . . . . . . . . . . . . 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 11960 . . . . . . . . . . . . . . . . . . 19  |-  ( -u
1 ^ 2 )  =  1
7776oveq2i 6101 . . . . . . . . . . . . . . . . . 18  |-  ( (
-u 1 ^ ( # `
 w ) )  /  ( -u 1 ^ 2 ) )  =  ( ( -u
1 ^ ( # `  w ) )  / 
1 )
78 m1expcl 11887 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  w )  e.  ZZ  ->  ( -u 1 ^ ( # `  w
) )  e.  ZZ )
7978zcnd 10747 . . . . . . . . . . . . . . . . . . . 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 10098 . . . . . . . . . . . . . . . . . 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 2486 . . . . . . . . . . . . . . . . 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 2478 . . . . . . . . . . . . . . . 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 2450 . . . . . . . . . . . . . . 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 1675 . . . . . . . . . . 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 1910 . . . . . . . . . . 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 1186 . . . . . . . 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 2686 . . . 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 1007 . . 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 2502 . . . . 5  |-  ( w  =  x  ->  (
w  e. Word  T  <->  x  e. Word  T ) )
99 oveq2 6098 . . . . . 6  |-  ( w  =  x  ->  ( G  gsumg  w )  =  ( G  gsumg  x ) )
10099eqeq1d 2450 . . . . 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 5690 . . . . . 6  |-  ( w  =  x  ->  ( # `
 w )  =  ( # `  x
) )
103102oveq2d 6106 . . . . 5  |-  ( w  =  x  ->  ( -u 1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  x
) ) )
104103eqeq1d 2450 . . . 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 2502 . . . . 5  |-  ( w  =  W  ->  (
w  e. Word  T  <->  W  e. Word  T ) )
107 oveq2 6098 . . . . . 6  |-  ( w  =  W  ->  ( G  gsumg  w )  =  ( G  gsumg  W ) )
108107eqeq1d 2450 . . . . 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 5690 . . . . . 6  |-  ( w  =  W  ->  ( # `
 w )  =  ( # `  W
) )
111110oveq2d 6106 . . . . 5  |-  ( w  =  W  ->  ( -u 1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  W
) ) )
112111eqeq1d 2450 . . . 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 11802 . 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 965   A.wal 1367    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2605   E.wrex 2715   (/)c0 3636   class class class wbr 4291    _I cid 4630   ran crn 4840    |` cres 4841   ` cfv 5417  (class class class)co 6090   Fincfn 7309   CCcc 9279   RRcr 9280   0cc0 9281   1c1 9282    < clt 9417    - cmin 9594   -ucneg 9595    / cdiv 9992   NNcn 10321   2c2 10370   NN0cn0 10578   ZZcz 10645   ZZ>=cuz 10860   RR+crp 10990   ...cfz 11436  ..^cfzo 11547   ^cexp 11864   #chash 12102  Word cword 12220    gsumg cgsu 14378   SymGrpcsymg 15881  pmTrspcpmtr 15946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-xor 1351  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-ot 3885  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-2o 6920  df-oadd 6923  df-er 7100  df-map 7215  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-card 8108  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-n0 10579  df-z 10646  df-uz 10861  df-rp 10991  df-fz 11437  df-fzo 11548  df-seq 11806  df-exp 11865  df-hash 12103  df-word 12228  df-concat 12230  df-s1 12231  df-substr 12232  df-splice 12233  df-s2 12474  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-tset 14256  df-0g 14379  df-gsum 14380  df-mnd 15414  df-submnd 15464  df-grp 15544  df-minusg 15545  df-subg 15677  df-symg 15882  df-pmtr 15947
This theorem is referenced by:  psgnuni  16004
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