Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  psgnunilem4 Unicode version

Theorem psgnunilem4 27335
Description: Lemma for psgnuni 27337. 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 11722 . . . . 5  |-  ( W  e. Word  T  ->  W  e.  Fin )
4 hashcl 11627 . . . . 5  |-  ( W  e.  Fin  ->  ( # `
 W )  e. 
NN0 )
51, 3, 43syl 19 . . . 4  |-  ( ph  ->  ( # `  W
)  e.  NN0 )
6 nn0uz 10509 . . . 4  |-  NN0  =  ( ZZ>= `  0 )
75, 6syl6eleq 2525 . . 3  |-  ( ph  ->  ( # `  W
)  e.  ( ZZ>= ` 
0 ) )
8 fveq2 5719 . . . . . . . . . 10  |-  ( w  =  (/)  ->  ( # `  w )  =  (
# `  (/) ) )
9 hash0 11634 . . . . . . . . . 10  |-  ( # `  (/) )  =  0
108, 9syl6eq 2483 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( # `  w )  =  0 )
1110oveq2d 6088 . . . . . . . 8  |-  ( w  =  (/)  ->  ( -u
1 ^ ( # `  w ) )  =  ( -u 1 ^ 0 ) )
12 neg1cn 10056 . . . . . . . . 9  |-  -u 1  e.  CC
13 exp0 11374 . . . . . . . . 9  |-  ( -u
1  e.  CC  ->  (
-u 1 ^ 0 )  =  1 )
1412, 13ax-mp 8 . . . . . . . 8  |-  ( -u
1 ^ 0 )  =  1
1511, 14syl6eq 2483 . . . . . . 7  |-  ( w  =  (/)  ->  ( -u
1 ^ ( # `  w ) )  =  1 )
1615a1d 23 . . . . . 6  |-  ( w  =  (/)  ->  ( ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
)  ->  ( -u 1 ^ ( # `  w
) )  =  1 ) )
1716a1d 23 . . . . 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 960 . . . . . . . . . . . . . 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 1012 . . . . . . . . . . . . 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 2436 . . . . . . . . . . . . 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 11722 . . . . . . . . . . . . . . 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 961 . . . . . . . . . . . . . 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 11633 . . . . . . . . . . . . . . 15  |-  ( w  e.  Fin  ->  (
( # `  w )  e.  NN  <->  w  =/=  (/) ) )
2928biimpar 472 . . . . . . . . . . . . . 14  |-  ( ( w  e.  Fin  /\  w  =/=  (/) )  ->  ( # `
 w )  e.  NN )
3026, 27, 29syl2anc 643 . . . . . . . . . . . . 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 1013 . . . . . . . . . . . . 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 5719 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  y  ->  ( # `
 x )  =  ( # `  y
) )
3332eqeq1d 2443 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  y  ->  (
( # `  x )  =  ( ( # `  w )  -  2 )  <->  ( # `  y
)  =  ( (
# `  w )  -  2 ) ) )
34 oveq2 6080 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  y  ->  ( G  gsumg  x )  =  ( G  gsumg  y ) )
3534eqeq1d 2443 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  y  ->  (
( G  gsumg  x )  =  (  _I  |`  D )  <->  ( G  gsumg  y )  =  (  _I  |`  D )
) )
3633, 35anbi12d 692 . . . . . . . . . . . . . . . . 17  |-  ( x  =  y  ->  (
( ( # `  x
)  =  ( (
# `  w )  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
)  <->  ( ( # `  y )  =  ( ( # `  w
)  -  2 )  /\  ( G  gsumg  y )  =  (  _I  |`  D ) ) ) )
3736cbvrexv 2925 . . . . . . . . . . . . . . . 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 288 . . . . . . . . . . . . . . 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 187 . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . 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 27334 . . . . . . . . . . . 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 414 . . . . . . . . . . . 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 201 . . . . . . . . . . 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 2703 . . . . . . . . . . 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 189 . . . . . . . . . 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 733 . . . . . . . . . . . . . . . . 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 742 . . . . . . . . . . . . . . . . 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 519 . . . . . . . . . . . . . . . 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 11722 . . . . . . . . . . . . . . . . . 18  |-  ( x  e. Word  T  ->  x  e.  Fin )
50 hashcl 11627 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  Fin  ->  ( # `
 x )  e. 
NN0 )
5146, 49, 503syl 19 . . . . . . . . . . . . . . . . 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 985 . . . . . . . . . . . . . . . . . . . 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 958 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  w  =/=  (/) 
/\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D ) ) )  ->  w  =/=  (/) )
5553, 54, 29syl2anc 643 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  w  =/=  (/) 
/\  ( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D ) ) )  ->  ( # `
 w )  e.  NN )
5655adantr 452 . . . . . . . . . . . . . . . . 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 741 . . . . . . . . . . . . . . . . . 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 10004 . . . . . . . . . . . . . . . . . . 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 10606 . . . . . . . . . . . . . . . . . . 19  |-  2  e.  RR+
60 ltsubrp 10632 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( # `  w
)  e.  RR  /\  2  e.  RR+ )  -> 
( ( # `  w
)  -  2 )  <  ( # `  w
) )
6158, 59, 60sylancl 644 . . . . . . . . . . . . . . . . . 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 4224 . . . . . . . . . . . . . . . . 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 11159 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  x )  e.  ( 0..^ ( # `  w ) )  <->  ( ( # `
 x )  e. 
NN0  /\  ( # `  w
)  e.  NN  /\  ( # `  x )  <  ( # `  w
) ) )
6451, 56, 62, 63syl3anbrc 1138 . . . . . . . . . . . . . . . 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 20 . . . . . . . . . . . . . . . . 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 76 . . . . . . . . . . . . . . . 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 58 . . . . . . . . . . . . . . 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 6088 . . . . . . . . . . . . . . . . 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 ax-1cn 9037 . . . . . . . . . . . . . . . . . . . 20  |-  1  e.  CC
71 ax-1ne0 9048 . . . . . . . . . . . . . . . . . . . 20  |-  1  =/=  0
7270, 71negne0i 9364 . . . . . . . . . . . . . . . . . . 19  |-  -u 1  =/=  0
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 )
) ) )  ->  -u 1  =/=  0 )
74 2z 10301 . . . . . . . . . . . . . . . . . . 19  |-  2  e.  ZZ
7574a1i 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 )
7656nnzd 10363 . . . . . . . . . . . . . . . . . 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 11522 . . . . . . . . . . . . . . . . 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 11430 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 1  e.  CC  ->  ( -u 1 ^ 2 )  =  ( 1 ^ 2 ) )
7970, 78ax-mp 8 . . . . . . . . . . . . . . . . . . . 20  |-  ( -u
1 ^ 2 )  =  ( 1 ^ 2 )
80 sq1 11464 . . . . . . . . . . . . . . . . . . . 20  |-  ( 1 ^ 2 )  =  1
8179, 80eqtri 2455 . . . . . . . . . . . . . . . . . . 19  |-  ( -u
1 ^ 2 )  =  1
8281oveq2i 6083 . . . . . . . . . . . . . . . . . 18  |-  ( (
-u 1 ^ ( # `
 w ) )  /  ( -u 1 ^ 2 ) )  =  ( ( -u
1 ^ ( # `  w ) )  / 
1 )
83 m1expcl 11392 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  w )  e.  ZZ  ->  ( -u 1 ^ ( # `  w
) )  e.  ZZ )
8483zcnd 10365 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  w )  e.  ZZ  ->  ( -u 1 ^ ( # `  w
) )  e.  CC )
8576, 84syl 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 )
8685div1d 9771 . . . . . . . . . . . . . . . . . 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 2479 . . . . . . . . . . . . . . . . 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 2471 . . . . . . . . . . . . . . . 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 2443 . . . . . . . . . . . . . . 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 206 . . . . . . . . . . . . . 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 424 . . . . . . . . . . . . 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 74 . . . . . . . . . . . 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 1631 . . . . . . . . . . 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 1914 . . . . . . . . . . 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 218 . . . . . . . . . 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 39 . . . . . . . . 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 1152 . . . . . . . 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 79 . . . . . . 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 29 . . . . . 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 421 . . . . 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 2674 . . . 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 976 . . 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 2495 . . . . 5  |-  ( w  =  x  ->  (
w  e. Word  T  <->  x  e. Word  T ) )
104 oveq2 6080 . . . . . 6  |-  ( w  =  x  ->  ( G  gsumg  w )  =  ( G  gsumg  x ) )
105104eqeq1d 2443 . . . . 5  |-  ( w  =  x  ->  (
( G  gsumg  w )  =  (  _I  |`  D )  <->  ( G  gsumg  x )  =  (  _I  |`  D )
) )
106103, 105anbi12d 692 . . . 4  |-  ( w  =  x  ->  (
( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
)  <->  ( x  e. Word  T  /\  ( G  gsumg  x )  =  (  _I  |`  D ) ) ) )
107 fveq2 5719 . . . . . 6  |-  ( w  =  x  ->  ( # `
 w )  =  ( # `  x
) )
108107oveq2d 6088 . . . . 5  |-  ( w  =  x  ->  ( -u 1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  x
) ) )
109108eqeq1d 2443 . . . 4  |-  ( w  =  x  ->  (
( -u 1 ^ ( # `
 w ) )  =  1  <->  ( -u 1 ^ ( # `  x
) )  =  1 ) )
110106, 109imbi12d 312 . . 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 2495 . . . . 5  |-  ( w  =  W  ->  (
w  e. Word  T  <->  W  e. Word  T ) )
112 oveq2 6080 . . . . . 6  |-  ( w  =  W  ->  ( G  gsumg  w )  =  ( G  gsumg  W ) )
113112eqeq1d 2443 . . . . 5  |-  ( w  =  W  ->  (
( G  gsumg  w )  =  (  _I  |`  D )  <->  ( G  gsumg  W )  =  (  _I  |`  D )
) )
114111, 113anbi12d 692 . . . 4  |-  ( w  =  W  ->  (
( w  e. Word  T  /\  ( G  gsumg  w )  =  (  _I  |`  D )
)  <->  ( W  e. Word  T  /\  ( G  gsumg  W )  =  (  _I  |`  D ) ) ) )
115 fveq2 5719 . . . . . 6  |-  ( w  =  W  ->  ( # `
 w )  =  ( # `  W
) )
116115oveq2d 6088 . . . . 5  |-  ( w  =  W  ->  ( -u 1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  W
) ) )
117116eqeq1d 2443 . . . 4  |-  ( w  =  W  ->  (
( -u 1 ^ ( # `
 w ) )  =  1  <->  ( -u 1 ^ ( # `  W
) )  =  1 ) )
118114, 117imbi12d 312 . . 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 11308 . 2  |-  ( ph  ->  ( ( W  e. Word  T  /\  ( G  gsumg  W )  =  (  _I  |`  D ) )  ->  ( -u 1 ^ ( # `  W
) )  =  1 ) )
1201, 2, 119mp2and 661 1  |-  ( ph  ->  ( -u 1 ^ ( # `  W
) )  =  1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   (/)c0 3620   class class class wbr 4204    _I cid 4485   ran crn 4870    |` cres 4871   ` cfv 5445  (class class class)co 6072   Fincfn 7100   CCcc 8977   RRcr 8978   0cc0 8979   1c1 8980    < clt 9109    - cmin 9280   -ucneg 9281    / cdiv 9666   NNcn 9989   2c2 10038   NN0cn0 10210   ZZcz 10271   ZZ>=cuz 10477   RR+crp 10601   ...cfz 11032  ..^cfzo 11123   ^cexp 11370   #chash 11606  Word cword 11705    gsumg cgsu 13712   SymGrpcsymg 15080  pmTrspcpmtr 27299
This theorem is referenced by:  psgnuni  27337
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-xor 1314  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-ot 3816  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-2o 6716  df-oadd 6719  df-er 6896  df-map 7011  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-card 7815  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-5 10050  df-6 10051  df-7 10052  df-8 10053  df-9 10054  df-n0 10211  df-z 10272  df-uz 10478  df-rp 10602  df-fz 11033  df-fzo 11124  df-seq 11312  df-exp 11371  df-hash 11607  df-word 11711  df-concat 11712  df-s1 11713  df-substr 11714  df-splice 11715  df-s2 11800  df-struct 13459  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-tset 13536  df-0g 13715  df-gsum 13716  df-mnd 14678  df-submnd 14727  df-grp 14800  df-minusg 14801  df-subg 14929  df-symg 15081  df-pmtr 27300
  Copyright terms: Public domain W3C validator