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Theorem ablfac2 15637
Description: Choose generators for each cyclic group in ablfac 15636. (Contributed by Mario Carneiro, 28-Apr-2016.)
Hypotheses
Ref Expression
ablfac.b  |-  B  =  ( Base `  G
)
ablfac.c  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
ablfac.1  |-  ( ph  ->  G  e.  Abel )
ablfac.2  |-  ( ph  ->  B  e.  Fin )
ablfac2.m  |-  .x.  =  (.g
`  G )
ablfac2.s  |-  S  =  ( k  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `  k
) ) ) )
Assertion
Ref Expression
ablfac2  |-  ( ph  ->  E. w  e. Word  B
( S : dom  w
--> C  /\  G dom DProd  S  /\  ( G DProd  S
)  =  B ) )
Distinct variable groups:    S, r    k, n, r, w, B    .x. , k, w    C, k, n, w    ph, k, n, w    k, G, n, r, w
Allowed substitution hints:    ph( r)    C( r)    S( w, k, n)    .x. ( n, r)

Proof of Theorem ablfac2
Dummy variables  s  x  j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ablfac.b . . 3  |-  B  =  ( Base `  G
)
2 ablfac.c . . 3  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
3 ablfac.1 . . 3  |-  ( ph  ->  G  e.  Abel )
4 ablfac.2 . . 3  |-  ( ph  ->  B  e.  Fin )
51, 2, 3, 4ablfac 15636 . 2  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  B ) )
6 wrdf 11723 . . . . . . . . . 10  |-  ( s  e. Word  C  ->  s : ( 0..^ (
# `  s )
) --> C )
76ad2antlr 708 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  s : ( 0..^ (
# `  s )
) --> C )
8 fdm 5587 . . . . . . . . 9  |-  ( s : ( 0..^ (
# `  s )
) --> C  ->  dom  s  =  ( 0..^ ( # `  s
) ) )
97, 8syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  dom  s  =  ( 0..^ ( # `  s
) ) )
10 fzofi 11303 . . . . . . . 8  |-  ( 0..^ ( # `  s
) )  e.  Fin
119, 10syl6eqel 2523 . . . . . . 7  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  dom  s  e.  Fin )
129feq2d 5573 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  (
s : dom  s --> C 
<->  s : ( 0..^ ( # `  s
) ) --> C ) )
137, 12mpbird 224 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  s : dom  s --> C )
1413ffvelrnda 5862 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( s `  k )  e.  C
)
15 oveq2 6081 . . . . . . . . . . . . . 14  |-  ( r  =  ( s `  k )  ->  ( Gs  r )  =  ( Gs  ( s `  k
) ) )
1615eleq1d 2501 . . . . . . . . . . . . 13  |-  ( r  =  ( s `  k )  ->  (
( Gs  r )  e.  (CycGrp  i^i  ran pGrp  )  <->  ( Gs  (
s `  k )
)  e.  (CycGrp  i^i  ran pGrp  ) ) )
1716, 2elrab2 3086 . . . . . . . . . . . 12  |-  ( ( s `  k )  e.  C  <->  ( (
s `  k )  e.  (SubGrp `  G )  /\  ( Gs  ( s `  k ) )  e.  (CycGrp  i^i  ran pGrp  ) ) )
1817simplbi 447 . . . . . . . . . . 11  |-  ( ( s `  k )  e.  C  ->  (
s `  k )  e.  (SubGrp `  G )
)
1914, 18syl 16 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( s `  k )  e.  (SubGrp `  G ) )
201subgss 14935 . . . . . . . . . 10  |-  ( ( s `  k )  e.  (SubGrp `  G
)  ->  ( s `  k )  C_  B
)
2119, 20syl 16 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( s `  k )  C_  B
)
22 inss1 3553 . . . . . . . . . . . . 13  |-  (CycGrp  i^i  ran pGrp  )  C_ CycGrp
2317simprbi 451 . . . . . . . . . . . . . 14  |-  ( ( s `  k )  e.  C  ->  ( Gs  ( s `  k
) )  e.  (CycGrp 
i^i  ran pGrp  ) )
2414, 23syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( Gs  ( s `
 k ) )  e.  (CycGrp  i^i  ran pGrp  ) )
2522, 24sseldi 3338 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( Gs  ( s `
 k ) )  e. CycGrp )
26 eqid 2435 . . . . . . . . . . . . . 14  |-  ( Base `  ( Gs  ( s `  k ) ) )  =  ( Base `  ( Gs  ( s `  k
) ) )
27 eqid 2435 . . . . . . . . . . . . . 14  |-  (.g `  ( Gs  ( s `  k
) ) )  =  (.g `  ( Gs  ( s `
 k ) ) )
2826, 27iscyg 15479 . . . . . . . . . . . . 13  |-  ( ( Gs  ( s `  k
) )  e. CycGrp  <->  ( ( Gs  ( s `  k
) )  e.  Grp  /\ 
E. x  e.  (
Base `  ( Gs  (
s `  k )
) ) ran  (
n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `
 k ) ) ) x ) )  =  ( Base `  ( Gs  ( s `  k
) ) ) ) )
2928simprbi 451 . . . . . . . . . . . 12  |-  ( ( Gs  ( s `  k
) )  e. CycGrp  ->  E. x  e.  ( Base `  ( Gs  ( s `  k ) ) ) ran  ( n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `  k ) ) ) x ) )  =  ( Base `  ( Gs  ( s `  k
) ) ) )
3025, 29syl 16 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  E. x  e.  (
Base `  ( Gs  (
s `  k )
) ) ran  (
n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `
 k ) ) ) x ) )  =  ( Base `  ( Gs  ( s `  k
) ) ) )
31 eqid 2435 . . . . . . . . . . . . . 14  |-  ( Gs  ( s `  k ) )  =  ( Gs  ( s `  k ) )
3231subgbas 14938 . . . . . . . . . . . . 13  |-  ( ( s `  k )  e.  (SubGrp `  G
)  ->  ( s `  k )  =  (
Base `  ( Gs  (
s `  k )
) ) )
3319, 32syl 16 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( s `  k )  =  (
Base `  ( Gs  (
s `  k )
) ) )
3433rexeqdv 2903 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( E. x  e.  ( s `  k
) ran  ( n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `  k ) ) ) x ) )  =  ( Base `  ( Gs  ( s `  k
) ) )  <->  E. x  e.  ( Base `  ( Gs  ( s `  k
) ) ) ran  ( n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `  k
) ) ) x ) )  =  (
Base `  ( Gs  (
s `  k )
) ) ) )
3530, 34mpbird 224 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  E. x  e.  ( s `  k ) ran  ( n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `  k ) ) ) x ) )  =  ( Base `  ( Gs  ( s `  k
) ) ) )
3619ad2antrr 707 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  /\  x  e.  ( s `  k ) )  /\  n  e.  ZZ )  ->  ( s `  k
)  e.  (SubGrp `  G ) )
37 simpr 448 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  /\  x  e.  ( s `  k ) )  /\  n  e.  ZZ )  ->  n  e.  ZZ )
38 simplr 732 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  /\  x  e.  ( s `  k ) )  /\  n  e.  ZZ )  ->  x  e.  ( s `
 k ) )
39 ablfac2.m . . . . . . . . . . . . . . . 16  |-  .x.  =  (.g
`  G )
4039, 31, 27subgmulg 14948 . . . . . . . . . . . . . . 15  |-  ( ( ( s `  k
)  e.  (SubGrp `  G )  /\  n  e.  ZZ  /\  x  e.  ( s `  k
) )  ->  (
n  .x.  x )  =  ( n (.g `  ( Gs  ( s `  k ) ) ) x ) )
4136, 37, 38, 40syl3anc 1184 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  /\  x  e.  ( s `  k ) )  /\  n  e.  ZZ )  ->  ( n  .x.  x
)  =  ( n (.g `  ( Gs  ( s `
 k ) ) ) x ) )
4241mpteq2dva 4287 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  k  e. 
dom  s )  /\  x  e.  ( s `  k ) )  -> 
( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `
 k ) ) ) x ) ) )
4342rneqd 5089 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  k  e. 
dom  s )  /\  x  e.  ( s `  k ) )  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ran  (
n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `
 k ) ) ) x ) ) )
4433adantr 452 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  k  e. 
dom  s )  /\  x  e.  ( s `  k ) )  -> 
( s `  k
)  =  ( Base `  ( Gs  ( s `  k ) ) ) )
4543, 44eqeq12d 2449 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  k  e. 
dom  s )  /\  x  e.  ( s `  k ) )  -> 
( ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( s `  k
)  <->  ran  ( n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `  k ) ) ) x ) )  =  ( Base `  ( Gs  ( s `  k
) ) ) ) )
4645rexbidva 2714 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( E. x  e.  ( s `  k
) ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( s `  k
)  <->  E. x  e.  ( s `  k ) ran  ( n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `  k ) ) ) x ) )  =  ( Base `  ( Gs  ( s `  k
) ) ) ) )
4735, 46mpbird 224 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  E. x  e.  ( s `  k ) ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( s `  k
) )
48 ssrexv 3400 . . . . . . . . 9  |-  ( ( s `  k ) 
C_  B  ->  ( E. x  e.  (
s `  k ) ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( s `
 k )  ->  E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( s `
 k ) ) )
4921, 47, 48sylc 58 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( s `
 k ) )
5049ralrimiva 2781 . . . . . . 7  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  A. k  e.  dom  s E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n 
.x.  x ) )  =  ( s `  k ) )
51 oveq2 6081 . . . . . . . . . . 11  |-  ( x  =  ( w `  k )  ->  (
n  .x.  x )  =  ( n  .x.  ( w `  k
) ) )
5251mpteq2dv 4288 . . . . . . . . . 10  |-  ( x  =  ( w `  k )  ->  (
n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( n  e.  ZZ  |->  ( n 
.x.  ( w `  k ) ) ) )
5352rneqd 5089 . . . . . . . . 9  |-  ( x  =  ( w `  k )  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) ) )
5453eqeq1d 2443 . . . . . . . 8  |-  ( x  =  ( w `  k )  ->  ( ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( s `
 k )  <->  ran  ( n  e.  ZZ  |->  ( n 
.x.  ( w `  k ) ) )  =  ( s `  k ) ) )
5554ac6sfi 7343 . . . . . . 7  |-  ( ( dom  s  e.  Fin  /\ 
A. k  e.  dom  s E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( s `
 k ) )  ->  E. w ( w : dom  s --> B  /\  A. k  e. 
dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )
5611, 50, 55syl2anc 643 . . . . . 6  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  E. w
( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )
57 simprl 733 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  w : dom  s --> B )
589adantr 452 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  dom  s  =  ( 0..^ ( # `  s ) ) )
5958feq2d 5573 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( w : dom  s --> B  <->  w :
( 0..^ ( # `  s ) ) --> B ) )
6057, 59mpbid 202 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  w :
( 0..^ ( # `  s ) ) --> B )
61 iswrdi 11721 . . . . . . . . . 10  |-  ( w : ( 0..^ (
# `  s )
) --> B  ->  w  e. Word  B )
6260, 61syl 16 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  w  e. Word  B )
63 fdm 5587 . . . . . . . . . . . . . . . 16  |-  ( w : ( 0..^ (
# `  s )
) --> B  ->  dom  w  =  ( 0..^ ( # `  s
) ) )
6460, 63syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  dom  w  =  ( 0..^ ( # `  s ) ) )
6564, 58eqtr4d 2470 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  dom  w  =  dom  s )
6665eleq2d 2502 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( j  e.  dom  w  <->  j  e.  dom  s ) )
6766biimpa 471 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  ( w : dom  s --> B  /\  A. k  e. 
dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  /\  j  e. 
dom  w )  -> 
j  e.  dom  s
)
68 simprr 734 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) )
69 simpl 444 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( k  =  j  /\  n  e.  ZZ )  ->  k  =  j )
7069fveq2d 5724 . . . . . . . . . . . . . . . . . . 19  |-  ( ( k  =  j  /\  n  e.  ZZ )  ->  ( w `  k
)  =  ( w `
 j ) )
7170oveq2d 6089 . . . . . . . . . . . . . . . . . 18  |-  ( ( k  =  j  /\  n  e.  ZZ )  ->  ( n  .x.  (
w `  k )
)  =  ( n 
.x.  ( w `  j ) ) )
7271mpteq2dva 4287 . . . . . . . . . . . . . . . . 17  |-  ( k  =  j  ->  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( n  e.  ZZ  |->  ( n 
.x.  ( w `  j ) ) ) )
7372rneqd 5089 . . . . . . . . . . . . . . . 16  |-  ( k  =  j  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 j ) ) ) )
74 fveq2 5720 . . . . . . . . . . . . . . . 16  |-  ( k  =  j  ->  (
s `  k )  =  ( s `  j ) )
7573, 74eqeq12d 2449 . . . . . . . . . . . . . . 15  |-  ( k  =  j  ->  ( ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k )  <->  ran  ( n  e.  ZZ  |->  ( n 
.x.  ( w `  j ) ) )  =  ( s `  j ) ) )
7675rspccva 3043 . . . . . . . . . . . . . 14  |-  ( ( A. k  e.  dom  s ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `  k
) ) )  =  ( s `  k
)  /\  j  e.  dom  s )  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 j ) ) )  =  ( s `
 j ) )
7768, 76sylan 458 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  ( w : dom  s --> B  /\  A. k  e. 
dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  /\  j  e. 
dom  s )  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 j ) ) )  =  ( s `
 j ) )
7813adantr 452 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  s : dom  s --> C )
7978ffvelrnda 5862 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  ( w : dom  s --> B  /\  A. k  e. 
dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  /\  j  e. 
dom  s )  -> 
( s `  j
)  e.  C )
8077, 79eqeltrd 2509 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  ( w : dom  s --> B  /\  A. k  e. 
dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  /\  j  e. 
dom  s )  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 j ) ) )  e.  C )
8167, 80syldan 457 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  ( w : dom  s --> B  /\  A. k  e. 
dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  /\  j  e. 
dom  w )  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 j ) ) )  e.  C )
82 ablfac2.s . . . . . . . . . . . 12  |-  S  =  ( k  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `  k
) ) ) )
83 fveq2 5720 . . . . . . . . . . . . . . . 16  |-  ( k  =  j  ->  (
w `  k )  =  ( w `  j ) )
8483oveq2d 6089 . . . . . . . . . . . . . . 15  |-  ( k  =  j  ->  (
n  .x.  ( w `  k ) )  =  ( n  .x.  (
w `  j )
) )
8584mpteq2dv 4288 . . . . . . . . . . . . . 14  |-  ( k  =  j  ->  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( n  e.  ZZ  |->  ( n 
.x.  ( w `  j ) ) ) )
8685rneqd 5089 . . . . . . . . . . . . 13  |-  ( k  =  j  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 j ) ) ) )
8786cbvmptv 4292 . . . . . . . . . . . 12  |-  ( k  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) ) )  =  ( j  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `  j
) ) ) )
8882, 87eqtri 2455 . . . . . . . . . . 11  |-  S  =  ( j  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `  j
) ) ) )
8981, 88fmptd 5885 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  S : dom  w --> C )
90 simprl 733 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  G dom DProd  s )
9190adantr 452 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  G dom DProd  s )
9265raleqdv 2902 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( A. k  e.  dom  w ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k )  <->  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )
9368, 92mpbird 224 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  A. k  e.  dom  w ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) )
94 mpteq12 4280 . . . . . . . . . . . . . 14  |-  ( ( dom  w  =  dom  s  /\  A. k  e. 
dom  w ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) )  ->  ( k  e. 
dom  w  |->  ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) ) )  =  ( k  e.  dom  s  |->  ( s `  k
) ) )
9565, 93, 94syl2anc 643 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( k  e.  dom  w  |->  ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) ) )  =  ( k  e.  dom  s  |->  ( s `  k
) ) )
9682, 95syl5eq 2479 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  S  =  ( k  e.  dom  s  |->  ( s `  k ) ) )
97 dprdf 15554 . . . . . . . . . . . . . 14  |-  ( G dom DProd  s  ->  s : dom  s --> (SubGrp `  G ) )
9891, 97syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  s : dom  s --> (SubGrp `  G )
)
9998feqmptd 5771 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  s  =  ( k  e.  dom  s  |->  ( s `  k ) ) )
10096, 99eqtr4d 2470 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  S  =  s )
10191, 100breqtrrd 4230 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  G dom DProd  S )
102100oveq2d 6089 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( G DProd  S )  =  ( G DProd 
s ) )
103 simplrr 738 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( G DProd  s )  =  B )
104102, 103eqtrd 2467 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( G DProd  S )  =  B )
10589, 101, 1043jca 1134 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( S : dom  w --> C  /\  G dom DProd  S  /\  ( G DProd  S )  =  B ) )
10662, 105jca 519 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( w  e. Word  B  /\  ( S : dom  w --> C  /\  G dom DProd  S  /\  ( G DProd  S )  =  B ) ) )
107106ex 424 . . . . . . 7  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  (
( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) )  ->  ( w  e. Word  B  /\  ( S : dom  w --> C  /\  G dom DProd  S  /\  ( G DProd 
S )  =  B ) ) ) )
108107eximdv 1632 . . . . . 6  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  ( E. w ( w : dom  s --> B  /\  A. k  e.  dom  s ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) )  ->  E. w ( w  e. Word  B  /\  ( S : dom  w --> C  /\  G dom DProd  S  /\  ( G DProd  S )  =  B ) ) ) )
10956, 108mpd 15 . . . . 5  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  E. w
( w  e. Word  B  /\  ( S : dom  w
--> C  /\  G dom DProd  S  /\  ( G DProd  S
)  =  B ) ) )
110 df-rex 2703 . . . . 5  |-  ( E. w  e. Word  B ( S : dom  w --> C  /\  G dom DProd  S  /\  ( G DProd  S )  =  B )  <->  E. w
( w  e. Word  B  /\  ( S : dom  w
--> C  /\  G dom DProd  S  /\  ( G DProd  S
)  =  B ) ) )
111109, 110sylibr 204 . . . 4  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  E. w  e. Word  B ( S : dom  w --> C  /\  G dom DProd  S  /\  ( G DProd 
S )  =  B ) )
112111ex 424 . . 3  |-  ( (
ph  /\  s  e. Word  C )  ->  ( ( G dom DProd  s  /\  ( G DProd  s )  =  B )  ->  E. w  e. Word  B ( S : dom  w --> C  /\  G dom DProd  S  /\  ( G DProd 
S )  =  B ) ) )
113112rexlimdva 2822 . 2  |-  ( ph  ->  ( E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  B )  ->  E. w  e. Word  B
( S : dom  w
--> C  /\  G dom DProd  S  /\  ( G DProd  S
)  =  B ) ) )
1145, 113mpd 15 1  |-  ( ph  ->  E. w  e. Word  B
( S : dom  w
--> C  /\  G dom DProd  S  /\  ( G DProd  S
)  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   {crab 2701    i^i cin 3311    C_ wss 3312   class class class wbr 4204    e. cmpt 4258   dom cdm 4870   ran crn 4871   -->wf 5442   ` cfv 5446  (class class class)co 6073   Fincfn 7101   0cc0 8980   ZZcz 10272  ..^cfzo 11125   #chash 11608  Word cword 11707   Basecbs 13459   ↾s cress 13460   Grpcgrp 14675  .gcmg 14679  SubGrpcsubg 14928   pGrp cpgp 15155   Abelcabel 15403  CycGrpccyg 15477   DProd cdprd 15544
This theorem is referenced by:  dchrpt  21041
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 4693  ax-inf2 7586  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  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-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-disj 4175  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 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-tpos 6471  df-rpss 6514  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-omul 6721  df-er 6897  df-ec 6899  df-qs 6903  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7469  df-card 7816  df-acn 7819  df-cda 8038  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-n0 10212  df-z 10273  df-uz 10479  df-q 10565  df-rp 10603  df-fz 11034  df-fzo 11126  df-fl 11192  df-mod 11241  df-seq 11314  df-exp 11373  df-fac 11557  df-bc 11584  df-hash 11609  df-word 11713  df-concat 11714  df-s1 11715  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-clim 12272  df-sum 12470  df-dvds 12843  df-gcd 12997  df-prm 13070  df-pc 13201  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-ress 13466  df-plusg 13532  df-0g 13717  df-gsum 13718  df-mre 13801  df-mrc 13802  df-acs 13804  df-mnd 14680  df-mhm 14728  df-submnd 14729  df-grp 14802  df-minusg 14803  df-sbg 14804  df-mulg 14805  df-subg 14931  df-eqg 14933  df-ghm 14994  df-gim 15036  df-ga 15057  df-cntz 15106  df-oppg 15132  df-od 15157  df-gex 15158  df-pgp 15159  df-lsm 15260  df-pj1 15261  df-cmn 15404  df-abl 15405  df-cyg 15478  df-dprd 15546
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