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Theorem psgnunilem4 27420
Description: Lemma for psgnuni 27422. 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 11420 . . . . 5  |-  ( W  e. Word  T  ->  W  e.  Fin )
4 hashcl 11350 . . . . 5  |-  ( W  e.  Fin  ->  ( # `
 W )  e. 
NN0 )
51, 3, 43syl 18 . . . 4  |-  ( ph  ->  ( # `  W
)  e.  NN0 )
6 nn0uz 10262 . . . 4  |-  NN0  =  ( ZZ>= `  0 )
75, 6syl6eleq 2373 . . 3  |-  ( ph  ->  ( # `  W
)  e.  ( ZZ>= ` 
0 ) )
8 fveq2 5525 . . . . . . . . . 10  |-  ( w  =  (/)  ->  ( # `  w )  =  (
# `  (/) ) )
9 hash0 11355 . . . . . . . . . 10  |-  ( # `  (/) )  =  0
108, 9syl6eq 2331 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( # `  w )  =  0 )
1110oveq2d 5874 . . . . . . . 8  |-  ( w  =  (/)  ->  ( -u
1 ^ ( # `  w ) )  =  ( -u 1 ^ 0 ) )
12 neg1cn 9813 . . . . . . . . 9  |-  -u 1  e.  CC
13 exp0 11108 . . . . . . . . 9  |-  ( -u
1  e.  CC  ->  (
-u 1 ^ 0 )  =  1 )
1412, 13ax-mp 8 . . . . . . . 8  |-  ( -u
1 ^ 0 )  =  1
1511, 14syl6eq 2331 . . . . . . 7  |-  ( w  =  (/)  ->  ( -u
1 ^ ( # `  w ) )  =  1 )
1615a1d 22 . . . . . 6  |-  ( w  =  (/)  ->  ( ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
)  ->  ( -u 1 ^ ( # `  w
) )  =  1 ) )
1716a1d 22 . . . . 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 958 . . . . . . . . . . . . . 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 15 . . . . . . . . . . . . 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 1010 . . . . . . . . . . . . 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 2284 . . . . . . . . . . . . 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 11420 . . . . . . . . . . . . . . 15  |-  ( w  e. Word  T  ->  w  e.  Fin )
2623, 25syl 15 . . . . . . . . . . . . . 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 959 . . . . . . . . . . . . . 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 11354 . . . . . . . . . . . . . . 15  |-  ( w  e.  Fin  ->  (
( # `  w )  e.  NN  <->  w  =/=  (/) ) )
2928biimpar 471 . . . . . . . . . . . . . 14  |-  ( ( w  e.  Fin  /\  w  =/=  (/) )  ->  ( # `
 w )  e.  NN )
3026, 27, 29syl2anc 642 . . . . . . . . . . . . 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 1011 . . . . . . . . . . . . 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 5525 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  y  ->  ( # `
 x )  =  ( # `  y
) )
3332eqeq1d 2291 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  y  ->  (
( # `  x )  =  ( ( # `  w )  -  2 )  <->  ( # `  y
)  =  ( (
# `  w )  -  2 ) ) )
34 oveq2 5866 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  y  ->  ( G  gsumg  x )  =  ( G  gsumg  y ) )
3534eqeq1d 2291 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  y  ->  (
( G  gsumg  x )  =  (  _I  |`  D )  <->  ( G  gsumg  y )  =  (  _I  |`  D )
) )
3633, 35anbi12d 691 . . . . . . . . . . . . . . . . 17  |-  ( x  =  y  ->  (
( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
)  <->  ( ( # `  y )  =  ( ( # `  w
)  -  2 )  /\  ( G  gsumg  y )  =  (  _I  |`  D ) ) ) )
3736cbvrexv 2765 . . . . . . . . . . . . . . . 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 287 . . . . . . . . . . . . . . 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 186 . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . 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 27419 . . . . . . . . . . . 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 413 . . . . . . . . . . . 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 200 . . . . . . . . . . 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 2549 . . . . . . . . . . 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 188 . . . . . . . . . 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 732 . . . . . . . . . . . . . . . . 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 741 . . . . . . . . . . . . . . . . 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 518 . . . . . . . . . . . . . . . 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 11420 . . . . . . . . . . . . . . . . . 18  |-  ( x  e. Word  T  ->  x  e.  Fin )
50 hashcl 11350 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  Fin  ->  ( # `
 x )  e. 
NN0 )
5146, 49, 503syl 18 . . . . . . . . . . . . . . . . 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 983 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  w  =/=  (/) 
/\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D ) ) )  ->  w  e. Word  T )
5352, 25syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  w  =/=  (/) 
/\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D ) ) )  ->  w  e.  Fin )
54 simp2 956 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  w  =/=  (/) 
/\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D ) ) )  ->  w  =/=  (/) )
5553, 54, 29syl2anc 642 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  w  =/=  (/) 
/\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D ) ) )  ->  ( # `
 w )  e.  NN )
5655adantr 451 . . . . . . . . . . . . . . . . 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 740 . . . . . . . . . . . . . . . . . 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 9761 . . . . . . . . . . . . . . . . . . 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 10359 . . . . . . . . . . . . . . . . . . 19  |-  2  e.  RR+
60 ltsubrp 10385 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( # `  w
)  e.  RR  /\  2  e.  RR+ )  -> 
( ( # `  w
)  -  2 )  <  ( # `  w
) )
6158, 59, 60sylancl 643 . . . . . . . . . . . . . . . . . 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 4043 . . . . . . . . . . . . . . . . 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 10904 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  x )  e.  ( 0..^ ( # `  w ) )  <->  ( ( # `
 x )  e. 
NN0  /\  ( # `  w
)  e.  NN  /\  ( # `  x )  <  ( # `  w
) ) )
6451, 56, 62, 63syl3anbrc 1136 . . . . . . . . . . . . . . . 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 19 . . . . . . . . . . . . . . . . 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 74 . . . . . . . . . . . . . . . 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 56 . . . . . . . . . . . . . . 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 5874 . . . . . . . . . . . . . . . . 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 10 . . . . . . . . . . . . . . . . . 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 ax-1cn 8795 . . . . . . . . . . . . . . . . . . . 20  |-  1  e.  CC
71 ax-1ne0 8806 . . . . . . . . . . . . . . . . . . . 20  |-  1  =/=  0
7270, 71negne0i 9121 . . . . . . . . . . . . . . . . . . 19  |-  -u 1  =/=  0
7372a1i 10 . . . . . . . . . . . . . . . . . 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 )
74 2z 10054 . . . . . . . . . . . . . . . . . . 19  |-  2  e.  ZZ
7574a1i 10 . . . . . . . . . . . . . . . . . 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 )
7656nnzd 10116 . . . . . . . . . . . . . . . . . 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 )
7769, 73, 75, 76expsubd 11256 . . . . . . . . . . . . . . . . 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 ) ) )
78 sqneg 11164 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 1  e.  CC  ->  ( -u 1 ^ 2 )  =  ( 1 ^ 2 ) )
7970, 78ax-mp 8 . . . . . . . . . . . . . . . . . . . 20  |-  ( -u
1 ^ 2 )  =  ( 1 ^ 2 )
80 sq1 11198 . . . . . . . . . . . . . . . . . . . 20  |-  ( 1 ^ 2 )  =  1
8179, 80eqtri 2303 . . . . . . . . . . . . . . . . . . 19  |-  ( -u
1 ^ 2 )  =  1
8281oveq2i 5869 . . . . . . . . . . . . . . . . . 18  |-  ( (
-u 1 ^ ( # `
 w ) )  /  ( -u 1 ^ 2 ) )  =  ( ( -u
1 ^ ( # `  w ) )  / 
1 )
83 m1expcl 11126 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  w )  e.  ZZ  ->  ( -u 1 ^ ( # `  w
) )  e.  ZZ )
8483zcnd 10118 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  w )  e.  ZZ  ->  ( -u 1 ^ ( # `  w
) )  e.  CC )
8576, 84syl 15 . . . . . . . . . . . . . . . . . . 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 )
8685div1d 9528 . . . . . . . . . . . . . . . . . 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 ) ) )
8782, 86syl5eq 2327 . . . . . . . . . . . . . . . . 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 ) ) )
8868, 77, 873eqtrd 2319 . . . . . . . . . . . . . . . 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
) ) )
8988eqeq1d 2291 . . . . . . . . . . . . . . 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 ) )
9067, 89sylibd 205 . . . . . . . . . . . . . 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 ) )
9190ex 423 . . . . . . . . . . . . 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 ) ) )
9291com23 72 . . . . . . . . . . . 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 ) ) )
9392alimdv 1607 . . . . . . . . . . 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 ) ) )
94 19.23v 1832 . . . . . . . . . . 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 ) )
9593, 94syl6ib 217 . . . . . . . . . 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 ) ) )
9645, 95mpid 37 . . . . . . . . 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 ) )
97963exp 1150 . . . . . . . 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 ) ) ) )
9897com34 77 . . . . . . 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 ) ) ) )
9998com12 27 . . . . . 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 ) ) ) )
10099imp3a 420 . . . . 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 ) ) )
10117, 100pm2.61ine 2522 . . . 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 ) )
1021013adant2 974 . . 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 ) )
103 eleq1 2343 . . . . 5  |-  ( w  =  x  ->  (
w  e. Word  T  <->  x  e. Word  T ) )
104 oveq2 5866 . . . . . 6  |-  ( w  =  x  ->  ( G  gsumg  w )  =  ( G  gsumg  x ) )
105104eqeq1d 2291 . . . . 5  |-  ( w  =  x  ->  (
( G  gsumg  w )  =  (  _I  |`  D )  <->  ( G  gsumg  x )  =  (  _I  |`  D )
) )
106103, 105anbi12d 691 . . . 4  |-  ( w  =  x  ->  (
( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
)  <->  ( x  e. Word  T  /\  ( G  gsumg  x )  =  (  _I  |`  D ) ) ) )
107 fveq2 5525 . . . . . 6  |-  ( w  =  x  ->  ( # `
 w )  =  ( # `  x
) )
108107oveq2d 5874 . . . . 5  |-  ( w  =  x  ->  ( -u 1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  x
) ) )
109108eqeq1d 2291 . . . 4  |-  ( w  =  x  ->  (
( -u 1 ^ ( # `
 w ) )  =  1  <->  ( -u 1 ^ ( # `  x
) )  =  1 ) )
110106, 109imbi12d 311 . . 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 ) ) )
111 eleq1 2343 . . . . 5  |-  ( w  =  W  ->  (
w  e. Word  T  <->  W  e. Word  T ) )
112 oveq2 5866 . . . . . 6  |-  ( w  =  W  ->  ( G  gsumg  w )  =  ( G  gsumg  W ) )
113112eqeq1d 2291 . . . . 5  |-  ( w  =  W  ->  (
( G  gsumg  w )  =  (  _I  |`  D )  <->  ( G  gsumg  W )  =  (  _I  |`  D )
) )
114111, 113anbi12d 691 . . . 4  |-  ( w  =  W  ->  (
( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
)  <->  ( W  e. Word  T  /\  ( G  gsumg  W )  =  (  _I  |`  D ) ) ) )
115 fveq2 5525 . . . . . 6  |-  ( w  =  W  ->  ( # `
 w )  =  ( # `  W
) )
116115oveq2d 5874 . . . . 5  |-  ( w  =  W  ->  ( -u 1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  W
) ) )
117116eqeq1d 2291 . . . 4  |-  ( w  =  W  ->  (
( -u 1 ^ ( # `
 w ) )  =  1  <->  ( -u 1 ^ ( # `  W
) )  =  1 ) )
118114, 117imbi12d 311 . . 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 ) ) )
1191, 7, 102, 110, 118, 107, 115uzindi 11043 . 2  |-  ( ph  ->  ( ( W  e. Word  T  /\  ( G  gsumg  W )  =  (  _I  |`  D ) )  ->  ( -u 1 ^ ( # `  W
) )  =  1 ) )
1201, 2, 119mp2and 660 1  |-  ( ph  ->  ( -u 1 ^ ( # `  W
) )  =  1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   (/)c0 3455   class class class wbr 4023    _I cid 4304   ran crn 4690    |` cres 4691   ` cfv 5255  (class class class)co 5858   Fincfn 6863   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    < clt 8867    - cmin 9037   -ucneg 9038    / cdiv 9423   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354   ...cfz 10782  ..^cfzo 10870   ^cexp 11104   #chash 11337  Word cword 11403    gsumg cgsu 13401   SymGrpcsymg 14769  pmTrspcpmtr 27384
This theorem is referenced by:  psgnuni  27422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-xor 1296  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-ot 3650  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-word 11409  df-concat 11410  df-s1 11411  df-substr 11412  df-splice 11413  df-s2 11498  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-tset 13227  df-0g 13404  df-gsum 13405  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-subg 14618  df-symg 14770  df-pmtr 27385
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