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Theorem ablfac2 15324
Description: Choose generators for each cyclic group in ablfac 15323. (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 15323 . 2  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  B ) )
6 wrdf 11419 . . . . . . . . . 10  |-  ( s  e. Word  C  ->  s : ( 0..^ (
# `  s )
) --> C )
76ad2antlr 707 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  s : ( 0..^ (
# `  s )
) --> C )
8 fdm 5393 . . . . . . . . 9  |-  ( s : ( 0..^ (
# `  s )
) --> C  ->  dom  s  =  ( 0..^ ( # `  s
) ) )
97, 8syl 15 . . . . . . . 8  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  dom  s  =  ( 0..^ ( # `  s
) ) )
10 fzofi 11036 . . . . . . . 8  |-  ( 0..^ ( # `  s
) )  e.  Fin
119, 10syl6eqel 2371 . . . . . . 7  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  dom  s  e.  Fin )
129feq2d 5380 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  (
s : dom  s --> C 
<->  s : ( 0..^ ( # `  s
) ) --> C ) )
137, 12mpbird 223 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  s : dom  s --> C )
14 ffvelrn 5663 . . . . . . . . . . . 12  |-  ( ( s : dom  s --> C  /\  k  e.  dom  s )  ->  (
s `  k )  e.  C )
1513, 14sylan 457 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( s `  k )  e.  C
)
16 oveq2 5866 . . . . . . . . . . . . . 14  |-  ( r  =  ( s `  k )  ->  ( Gs  r )  =  ( Gs  ( s `  k
) ) )
1716eleq1d 2349 . . . . . . . . . . . . 13  |-  ( r  =  ( s `  k )  ->  (
( Gs  r )  e.  (CycGrp  i^i  ran pGrp  )  <->  ( Gs  (
s `  k )
)  e.  (CycGrp  i^i  ran pGrp  ) ) )
1817, 2elrab2 2925 . . . . . . . . . . . 12  |-  ( ( s `  k )  e.  C  <->  ( (
s `  k )  e.  (SubGrp `  G )  /\  ( Gs  ( s `  k ) )  e.  (CycGrp  i^i  ran pGrp  ) ) )
1918simplbi 446 . . . . . . . . . . 11  |-  ( ( s `  k )  e.  C  ->  (
s `  k )  e.  (SubGrp `  G )
)
2015, 19syl 15 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( s `  k )  e.  (SubGrp `  G ) )
211subgss 14622 . . . . . . . . . 10  |-  ( ( s `  k )  e.  (SubGrp `  G
)  ->  ( s `  k )  C_  B
)
2220, 21syl 15 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( s `  k )  C_  B
)
23 inss1 3389 . . . . . . . . . . . . 13  |-  (CycGrp  i^i  ran pGrp  )  C_ CycGrp
2418simprbi 450 . . . . . . . . . . . . . 14  |-  ( ( s `  k )  e.  C  ->  ( Gs  ( s `  k
) )  e.  (CycGrp 
i^i  ran pGrp  ) )
2515, 24syl 15 . . . . . . . . . . . . 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  ) )
2623, 25sseldi 3178 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( Gs  ( s `
 k ) )  e. CycGrp )
27 eqid 2283 . . . . . . . . . . . . . 14  |-  ( Base `  ( Gs  ( s `  k ) ) )  =  ( Base `  ( Gs  ( s `  k
) ) )
28 eqid 2283 . . . . . . . . . . . . . 14  |-  (.g `  ( Gs  ( s `  k
) ) )  =  (.g `  ( Gs  ( s `
 k ) ) )
2927, 28iscyg 15166 . . . . . . . . . . . . 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
) ) ) ) )
3029simprbi 450 . . . . . . . . . . . 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
) ) ) )
3126, 30syl 15 . . . . . . . . . . 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
) ) ) )
32 eqid 2283 . . . . . . . . . . . . . 14  |-  ( Gs  ( s `  k ) )  =  ( Gs  ( s `  k ) )
3332subgbas 14625 . . . . . . . . . . . . 13  |-  ( ( s `  k )  e.  (SubGrp `  G
)  ->  ( s `  k )  =  (
Base `  ( Gs  (
s `  k )
) ) )
3420, 33syl 15 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( s `  k )  =  (
Base `  ( Gs  (
s `  k )
) ) )
3534rexeqdv 2743 . . . . . . . . . . 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 )
) ) ) )
3631, 35mpbird 223 . . . . . . . . . 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
) ) ) )
3720ad2antrr 706 . . . . . . . . . . . . . . 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 ) )
38 simpr 447 . . . . . . . . . . . . . . 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 )
39 simplr 731 . . . . . . . . . . . . . . 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 ) )
40 ablfac2.m . . . . . . . . . . . . . . . 16  |-  .x.  =  (.g
`  G )
4140, 32, 28subgmulg 14635 . . . . . . . . . . . . . . 15  |-  ( ( ( s `  k
)  e.  (SubGrp `  G )  /\  n  e.  ZZ  /\  x  e.  ( s `  k
) )  ->  (
n  .x.  x )  =  ( n (.g `  ( Gs  ( s `  k ) ) ) x ) )
4237, 38, 39, 41syl3anc 1182 . . . . . . . . . . . . . 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 ) )
4342mpteq2dva 4106 . . . . . . . . . . . . 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 ) ) )
4443rneqd 4906 . . . . . . . . . . . 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 ) ) )
4534adantr 451 . . . . . . . . . . . 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 ) ) ) )
4644, 45eqeq12d 2297 . . . . . . . . . . 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
) ) ) ) )
4746rexbidva 2560 . . . . . . . . . 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
) ) ) ) )
4836, 47mpbird 223 . . . . . . . . 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
) )
49 ssrexv 3238 . . . . . . . . 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 ) ) )
5022, 48, 49sylc 56 . . . . . . . 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 ) )
5150ralrimiva 2626 . . . . . . 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 ) )
52 oveq2 5866 . . . . . . . . . . 11  |-  ( x  =  ( w `  k )  ->  (
n  .x.  x )  =  ( n  .x.  ( w `  k
) ) )
5352mpteq2dv 4107 . . . . . . . . . 10  |-  ( x  =  ( w `  k )  ->  (
n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( n  e.  ZZ  |->  ( n 
.x.  ( w `  k ) ) ) )
5453rneqd 4906 . . . . . . . . 9  |-  ( x  =  ( w `  k )  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) ) )
5554eqeq1d 2291 . . . . . . . 8  |-  ( x  =  ( w `  k )  ->  ( ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( s `
 k )  <->  ran  ( n  e.  ZZ  |->  ( n 
.x.  ( w `  k ) ) )  =  ( s `  k ) ) )
5655ac6sfi 7101 . . . . . . 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 ) ) )
5711, 51, 56syl2anc 642 . . . . . 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 ) ) )
58 simprl 732 . . . . . . . . . . 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 )
599adantr 451 . . . . . . . . . . . 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 ) ) )
6059feq2d 5380 . . . . . . . . . . 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 ) )
6158, 60mpbid 201 . . . . . . . . . 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 )
62 iswrdi 11417 . . . . . . . . . 10  |-  ( w : ( 0..^ (
# `  s )
) --> B  ->  w  e. Word  B )
6361, 62syl 15 . . . . . . . . 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 )
64 fdm 5393 . . . . . . . . . . . . . . . 16  |-  ( w : ( 0..^ (
# `  s )
) --> B  ->  dom  w  =  ( 0..^ ( # `  s
) ) )
6561, 64syl 15 . . . . . . . . . . . . . . 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 ) ) )
6665, 59eqtr4d 2318 . . . . . . . . . . . . . 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 )
6766eleq2d 2350 . . . . . . . . . . . . 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 ) )
6867biimpa 470 . . . . . . . . . . . 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
)
69 simprr 733 . . . . . . . . . . . . . 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 ) )
70 simpl 443 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( k  =  j  /\  n  e.  ZZ )  ->  k  =  j )
7170fveq2d 5529 . . . . . . . . . . . . . . . . . . 19  |-  ( ( k  =  j  /\  n  e.  ZZ )  ->  ( w `  k
)  =  ( w `
 j ) )
7271oveq2d 5874 . . . . . . . . . . . . . . . . . 18  |-  ( ( k  =  j  /\  n  e.  ZZ )  ->  ( n  .x.  (
w `  k )
)  =  ( n 
.x.  ( w `  j ) ) )
7372mpteq2dva 4106 . . . . . . . . . . . . . . . . 17  |-  ( k  =  j  ->  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( n  e.  ZZ  |->  ( n 
.x.  ( w `  j ) ) ) )
7473rneqd 4906 . . . . . . . . . . . . . . . 16  |-  ( k  =  j  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 j ) ) ) )
75 fveq2 5525 . . . . . . . . . . . . . . . 16  |-  ( k  =  j  ->  (
s `  k )  =  ( s `  j ) )
7674, 75eqeq12d 2297 . . . . . . . . . . . . . . 15  |-  ( k  =  j  ->  ( ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k )  <->  ran  ( n  e.  ZZ  |->  ( n 
.x.  ( w `  j ) ) )  =  ( s `  j ) ) )
7776rspccva 2883 . . . . . . . . . . . . . 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 ) )
7869, 77sylan 457 . . . . . . . . . . . . 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 ) )
7913adantr 451 . . . . . . . . . . . . . 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 )
80 ffvelrn 5663 . . . . . . . . . . . . . 14  |-  ( ( s : dom  s --> C  /\  j  e.  dom  s )  ->  (
s `  j )  e.  C )
8179, 80sylan 457 . . . . . . . . . . . . 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 )
8278, 81eqeltrd 2357 . . . . . . . . . . . 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 )
8368, 82syldan 456 . . . . . . . . . . 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 )
84 ablfac2.s . . . . . . . . . . . 12  |-  S  =  ( k  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `  k
) ) ) )
85 fveq2 5525 . . . . . . . . . . . . . . . 16  |-  ( k  =  j  ->  (
w `  k )  =  ( w `  j ) )
8685oveq2d 5874 . . . . . . . . . . . . . . 15  |-  ( k  =  j  ->  (
n  .x.  ( w `  k ) )  =  ( n  .x.  (
w `  j )
) )
8786mpteq2dv 4107 . . . . . . . . . . . . . 14  |-  ( k  =  j  ->  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( n  e.  ZZ  |->  ( n 
.x.  ( w `  j ) ) ) )
8887rneqd 4906 . . . . . . . . . . . . 13  |-  ( k  =  j  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 j ) ) ) )
8988cbvmptv 4111 . . . . . . . . . . . 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
) ) ) )
9084, 89eqtri 2303 . . . . . . . . . . 11  |-  S  =  ( j  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `  j
) ) ) )
9183, 90fmptd 5684 . . . . . . . . . 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 )
92 simprl 732 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  G dom DProd  s )
9392adantr 451 . . . . . . . . . . 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 )
9466raleqdv 2742 . . . . . . . . . . . . . . 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 ) ) )
9569, 94mpbird 223 . . . . . . . . . . . . . 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 ) )
96 mpteq12 4099 . . . . . . . . . . . . . 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
) ) )
9766, 95, 96syl2anc 642 . . . . . . . . . . . . 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
) ) )
9884, 97syl5eq 2327 . . . . . . . . . . . 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 ) ) )
99 dprdf 15241 . . . . . . . . . . . . . 14  |-  ( G dom DProd  s  ->  s : dom  s --> (SubGrp `  G ) )
10093, 99syl 15 . . . . . . . . . . . . 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 )
)
101100feqmptd 5575 . . . . . . . . . . . 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 ) ) )
10298, 101eqtr4d 2318 . . . . . . . . . . 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 )
10393, 102breqtrrd 4049 . . . . . . . . . 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 )
104102oveq2d 5874 . . . . . . . . . . 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 ) )
105 simplrr 737 . . . . . . . . . . 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 )
106104, 105eqtrd 2315 . . . . . . . . . 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 )
10791, 103, 1063jca 1132 . . . . . . . . 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 ) )
10863, 107jca 518 . . . . . . . 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 ) ) )
109108ex 423 . . . . . . 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 ) ) ) )
110109eximdv 1608 . . . . . 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 ) ) ) )
11157, 110mpd 14 . . . . 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 ) ) )
112 df-rex 2549 . . . . 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 ) ) )
113111, 112sylibr 203 . . . 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 ) )
114113ex 423 . . 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 ) ) )
115114rexlimdva 2667 . 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 ) ) )
1165, 115mpd 14 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 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547    i^i cin 3151    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   Fincfn 6863   0cc0 8737   ZZcz 10024  ..^cfzo 10870   #chash 11337  Word cword 11403   Basecbs 13148   ↾s cress 13149   Grpcgrp 14362  .gcmg 14366  SubGrpcsubg 14615   pGrp cpgp 14842   Abelcabel 15090  CycGrpccyg 15164   DProd cdprd 15231
This theorem is referenced by:  dchrpt  20506
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-inf2 7342  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  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  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-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-disj 3994  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-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-rpss 6277  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  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-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-word 11409  df-concat 11410  df-s1 11411  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-eqg 14620  df-ghm 14681  df-gim 14723  df-ga 14744  df-cntz 14793  df-oppg 14819  df-od 14844  df-gex 14845  df-pgp 14846  df-lsm 14947  df-pj1 14948  df-cmn 15091  df-abl 15092  df-cyg 15165  df-dprd 15233
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