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Theorem psgnunilem4 27523
Description: Lemma for psgnuni 27525. 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 11436 . . . . 5  |-  ( W  e. Word  T  ->  W  e.  Fin )
4 hashcl 11366 . . . . 5  |-  ( W  e.  Fin  ->  ( # `
 W )  e. 
NN0 )
51, 3, 43syl 18 . . . 4  |-  ( ph  ->  ( # `  W
)  e.  NN0 )
6 nn0uz 10278 . . . 4  |-  NN0  =  ( ZZ>= `  0 )
75, 6syl6eleq 2386 . . 3  |-  ( ph  ->  ( # `  W
)  e.  ( ZZ>= ` 
0 ) )
8 fveq2 5541 . . . . . . . . . 10  |-  ( w  =  (/)  ->  ( # `  w )  =  (
# `  (/) ) )
9 hash0 11371 . . . . . . . . . 10  |-  ( # `  (/) )  =  0
108, 9syl6eq 2344 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( # `  w )  =  0 )
1110oveq2d 5890 . . . . . . . 8  |-  ( w  =  (/)  ->  ( -u
1 ^ ( # `  w ) )  =  ( -u 1 ^ 0 ) )
12 neg1cn 9829 . . . . . . . . 9  |-  -u 1  e.  CC
13 exp0 11124 . . . . . . . . 9  |-  ( -u
1  e.  CC  ->  (
-u 1 ^ 0 )  =  1 )
1412, 13ax-mp 8 . . . . . . . 8  |-  ( -u
1 ^ 0 )  =  1
1511, 14syl6eq 2344 . . . . . . 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 2297 . . . . . . . . . . . . 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 11436 . . . . . . . . . . . . . . 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 11370 . . . . . . . . . . . . . . 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 5541 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  y  ->  ( # `
 x )  =  ( # `  y
) )
3332eqeq1d 2304 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  y  ->  (
( # `  x )  =  ( ( # `  w )  -  2 )  <->  ( # `  y
)  =  ( (
# `  w )  -  2 ) ) )
34 oveq2 5882 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  y  ->  ( G  gsumg  x )  =  ( G  gsumg  y ) )
3534eqeq1d 2304 . . . . . . . . . . . . . . . . . 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 2778 . . . . . . . . . . . . . . . 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 27522 . . . . . . . . . . . 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 2562 . . . . . . . . . . 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 11436 . . . . . . . . . . . . . . . . . 18  |-  ( x  e. Word  T  ->  x  e.  Fin )
50 hashcl 11366 . . . . . . . . . . . . . . . . . 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 9777 . . . . . . . . . . . . . . . . . . 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 10375 . . . . . . . . . . . . . . . . . . 19  |-  2  e.  RR+
60 ltsubrp 10401 . . . . . . . . . . . . . . . . . . 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 4059 . . . . . . . . . . . . . . . . 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 10920 . . . . . . . . . . . . . . . . 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 5890 . . . . . . . . . . . . . . . . 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 8811 . . . . . . . . . . . . . . . . . . . 20  |-  1  e.  CC
71 ax-1ne0 8822 . . . . . . . . . . . . . . . . . . . 20  |-  1  =/=  0
7270, 71negne0i 9137 . . . . . . . . . . . . . . . . . . 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 10070 . . . . . . . . . . . . . . . . . . 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 10132 . . . . . . . . . . . . . . . . . 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 11272 . . . . . . . . . . . . . . . . 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 11180 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 1  e.  CC  ->  ( -u 1 ^ 2 )  =  ( 1 ^ 2 ) )
7970, 78ax-mp 8 . . . . . . . . . . . . . . . . . . . 20  |-  ( -u
1 ^ 2 )  =  ( 1 ^ 2 )
80 sq1 11214 . . . . . . . . . . . . . . . . . . . 20  |-  ( 1 ^ 2 )  =  1
8179, 80eqtri 2316 . . . . . . . . . . . . . . . . . . 19  |-  ( -u
1 ^ 2 )  =  1
8281oveq2i 5885 . . . . . . . . . . . . . . . . . 18  |-  ( (
-u 1 ^ ( # `
 w ) )  /  ( -u 1 ^ 2 ) )  =  ( ( -u
1 ^ ( # `  w ) )  / 
1 )
83 m1expcl 11142 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  w )  e.  ZZ  ->  ( -u 1 ^ ( # `  w
) )  e.  ZZ )
8483zcnd 10134 . . . . . . . . . . . . . . . . . . . 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 9544 . . . . . . . . . . . . . . . . . 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 2340 . . . . . . . . . . . . . . . . 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 2332 . . . . . . . . . . . . . . . 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 2304 . . . . . . . . . . . . . . 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 1611 . . . . . . . . . . 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 1844 . . . . . . . . . . 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 2535 . . . 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 2356 . . . . 5  |-  ( w  =  x  ->  (
w  e. Word  T  <->  x  e. Word  T ) )
104 oveq2 5882 . . . . . 6  |-  ( w  =  x  ->  ( G  gsumg  w )  =  ( G  gsumg  x ) )
105104eqeq1d 2304 . . . . 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 5541 . . . . . 6  |-  ( w  =  x  ->  ( # `
 w )  =  ( # `  x
) )
108107oveq2d 5890 . . . . 5  |-  ( w  =  x  ->  ( -u 1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  x
) ) )
109108eqeq1d 2304 . . . 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 2356 . . . . 5  |-  ( w  =  W  ->  (
w  e. Word  T  <->  W  e. Word  T ) )
112 oveq2 5882 . . . . . 6  |-  ( w  =  W  ->  ( G  gsumg  w )  =  ( G  gsumg  W ) )
113112eqeq1d 2304 . . . . 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 5541 . . . . . 6  |-  ( w  =  W  ->  ( # `
 w )  =  ( # `  W
) )
116115oveq2d 5890 . . . . 5  |-  ( w  =  W  ->  ( -u 1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  W
) ) )
117116eqeq1d 2304 . . . 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 11059 . 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 1530   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   (/)c0 3468   class class class wbr 4039    _I cid 4320   ran crn 4706    |` cres 4707   ` cfv 5271  (class class class)co 5874   Fincfn 6879   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    < clt 8883    - cmin 9053   -ucneg 9054    / cdiv 9439   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   RR+crp 10370   ...cfz 10798  ..^cfzo 10886   ^cexp 11120   #chash 11353  Word cword 11419    gsumg cgsu 13417   SymGrpcsymg 14785  pmTrspcpmtr 27487
This theorem is referenced by:  psgnuni  27525
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-ot 3663  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-word 11425  df-concat 11426  df-s1 11427  df-substr 11428  df-splice 11429  df-s2 11514  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-tset 13243  df-0g 13420  df-gsum 13421  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-subg 14634  df-symg 14786  df-pmtr 27488
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