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Theorem pgpfaclem2 15416
Description: Lemma for pgpfac 15418. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
Hypotheses
Ref Expression
pgpfac.b  |-  B  =  ( Base `  G
)
pgpfac.c  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
pgpfac.g  |-  ( ph  ->  G  e.  Abel )
pgpfac.p  |-  ( ph  ->  P pGrp  G )
pgpfac.f  |-  ( ph  ->  B  e.  Fin )
pgpfac.u  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
pgpfac.a  |-  ( ph  ->  A. t  e.  (SubGrp `  G ) ( t 
C.  U  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  t ) ) )
pgpfac.h  |-  H  =  ( Gs  U )
pgpfac.k  |-  K  =  (mrCls `  (SubGrp `  H
) )
pgpfac.o  |-  O  =  ( od `  H
)
pgpfac.e  |-  E  =  (gEx `  H )
pgpfac.0  |-  .0.  =  ( 0g `  H )
pgpfac.l  |-  .(+)  =  (
LSSum `  H )
pgpfac.1  |-  ( ph  ->  E  =/=  1 )
pgpfac.x  |-  ( ph  ->  X  e.  U )
pgpfac.oe  |-  ( ph  ->  ( O `  X
)  =  E )
pgpfac.w  |-  ( ph  ->  W  e.  (SubGrp `  H ) )
pgpfac.i  |-  ( ph  ->  ( ( K `  { X } )  i^i 
W )  =  {  .0.  } )
pgpfac.s  |-  ( ph  ->  ( ( K `  { X } )  .(+)  W )  =  U )
Assertion
Ref Expression
pgpfaclem2  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
Distinct variable groups:    t, s, C    s, r, t, G    K, r, s    ph, t    B, s, t    U, r, s, t    W, s, t    X, r, s
Allowed substitution hints:    ph( s, r)    B( r)    C( r)    P( t, s, r)    .(+) ( t, s, r)    E( t, s, r)    H( t, s, r)    K( t)    O( t, s, r)    W( r)    X( t)    .0. ( t, s, r)

Proof of Theorem pgpfaclem2
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 pgpfac.w . . . . . 6  |-  ( ph  ->  W  e.  (SubGrp `  H ) )
2 pgpfac.u . . . . . . 7  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
3 pgpfac.h . . . . . . . 8  |-  H  =  ( Gs  U )
43subsubg 14739 . . . . . . 7  |-  ( U  e.  (SubGrp `  G
)  ->  ( W  e.  (SubGrp `  H )  <->  ( W  e.  (SubGrp `  G )  /\  W  C_  U ) ) )
52, 4syl 15 . . . . . 6  |-  ( ph  ->  ( W  e.  (SubGrp `  H )  <->  ( W  e.  (SubGrp `  G )  /\  W  C_  U ) ) )
61, 5mpbid 201 . . . . 5  |-  ( ph  ->  ( W  e.  (SubGrp `  G )  /\  W  C_  U ) )
76simpld 445 . . . 4  |-  ( ph  ->  W  e.  (SubGrp `  G ) )
8 pgpfac.a . . . 4  |-  ( ph  ->  A. t  e.  (SubGrp `  G ) ( t 
C.  U  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  t ) ) )
96simprd 449 . . . . 5  |-  ( ph  ->  W  C_  U )
10 pgpfac.f . . . . . . . . . . 11  |-  ( ph  ->  B  e.  Fin )
11 pgpfac.b . . . . . . . . . . . . 13  |-  B  =  ( Base `  G
)
1211subgss 14721 . . . . . . . . . . . 12  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  B
)
132, 12syl 15 . . . . . . . . . . 11  |-  ( ph  ->  U  C_  B )
14 ssfi 7171 . . . . . . . . . . 11  |-  ( ( B  e.  Fin  /\  U  C_  B )  ->  U  e.  Fin )
1510, 13, 14syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  U  e.  Fin )
16 ssfi 7171 . . . . . . . . . 10  |-  ( ( U  e.  Fin  /\  W  C_  U )  ->  W  e.  Fin )
1715, 9, 16syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  W  e.  Fin )
18 hashcl 11443 . . . . . . . . 9  |-  ( W  e.  Fin  ->  ( # `
 W )  e. 
NN0 )
1917, 18syl 15 . . . . . . . 8  |-  ( ph  ->  ( # `  W
)  e.  NN0 )
2019nn0red 10111 . . . . . . 7  |-  ( ph  ->  ( # `  W
)  e.  RR )
21 pgpfac.0 . . . . . . . . . . . 12  |-  .0.  =  ( 0g `  H )
22 fvex 5622 . . . . . . . . . . . 12  |-  ( 0g
`  H )  e. 
_V
2321, 22eqeltri 2428 . . . . . . . . . . 11  |-  .0.  e.  _V
24 hashsng 11449 . . . . . . . . . . 11  |-  (  .0. 
e.  _V  ->  ( # `  {  .0.  } )  =  1 )
2523, 24ax-mp 8 . . . . . . . . . 10  |-  ( # `  {  .0.  } )  =  1
26 subgrcl 14725 . . . . . . . . . . . . . . . . . 18  |-  ( W  e.  (SubGrp `  H
)  ->  H  e.  Grp )
271, 26syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  H  e.  Grp )
28 eqid 2358 . . . . . . . . . . . . . . . . . 18  |-  ( Base `  H )  =  (
Base `  H )
2928subgacs 14751 . . . . . . . . . . . . . . . . 17  |-  ( H  e.  Grp  ->  (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) ) )
30 acsmre 13653 . . . . . . . . . . . . . . . . 17  |-  ( (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) )  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H
) ) )
3127, 29, 303syl 18 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H ) ) )
32 pgpfac.x . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  X  e.  U )
333subgbas 14724 . . . . . . . . . . . . . . . . . 18  |-  ( U  e.  (SubGrp `  G
)  ->  U  =  ( Base `  H )
)
342, 33syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  U  =  ( Base `  H ) )
3532, 34eleqtrd 2434 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  X  e.  ( Base `  H ) )
36 pgpfac.k . . . . . . . . . . . . . . . . 17  |-  K  =  (mrCls `  (SubGrp `  H
) )
3736mrcsncl 13613 . . . . . . . . . . . . . . . 16  |-  ( ( (SubGrp `  H )  e.  (Moore `  ( Base `  H ) )  /\  X  e.  ( Base `  H ) )  -> 
( K `  { X } )  e.  (SubGrp `  H ) )
3831, 35, 37syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( K `  { X } )  e.  (SubGrp `  H ) )
3928subgss 14721 . . . . . . . . . . . . . . 15  |-  ( ( K `  { X } )  e.  (SubGrp `  H )  ->  ( K `  { X } )  C_  ( Base `  H ) )
4038, 39syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( K `  { X } )  C_  ( Base `  H ) )
4140, 34sseqtr4d 3291 . . . . . . . . . . . . 13  |-  ( ph  ->  ( K `  { X } )  C_  U
)
42 ssfi 7171 . . . . . . . . . . . . 13  |-  ( ( U  e.  Fin  /\  ( K `  { X } )  C_  U
)  ->  ( K `  { X } )  e.  Fin )
4315, 41, 42syl2anc 642 . . . . . . . . . . . 12  |-  ( ph  ->  ( K `  { X } )  e.  Fin )
4435snssd 3839 . . . . . . . . . . . . . . 15  |-  ( ph  ->  { X }  C_  ( Base `  H )
)
4536mrcssid 13618 . . . . . . . . . . . . . . 15  |-  ( ( (SubGrp `  H )  e.  (Moore `  ( Base `  H ) )  /\  { X }  C_  ( Base `  H ) )  ->  { X }  C_  ( K `  { X } ) )
4631, 44, 45syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ph  ->  { X }  C_  ( K `  { X } ) )
47 snssg 3830 . . . . . . . . . . . . . . 15  |-  ( X  e.  U  ->  ( X  e.  ( K `  { X } )  <->  { X }  C_  ( K `  { X } ) ) )
4832, 47syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( X  e.  ( K `  { X } )  <->  { X }  C_  ( K `  { X } ) ) )
4946, 48mpbird 223 . . . . . . . . . . . . 13  |-  ( ph  ->  X  e.  ( K `
 { X }
) )
50 pgpfac.oe . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( O `  X
)  =  E )
51 pgpfac.1 . . . . . . . . . . . . . . 15  |-  ( ph  ->  E  =/=  1 )
5250, 51eqnetrd 2539 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( O `  X
)  =/=  1 )
53 pgpfac.o . . . . . . . . . . . . . . . . . 18  |-  O  =  ( od `  H
)
5453, 21od1 14971 . . . . . . . . . . . . . . . . 17  |-  ( H  e.  Grp  ->  ( O `  .0.  )  =  1 )
551, 26, 543syl 18 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( O `  .0.  )  =  1 )
56 elsni 3740 . . . . . . . . . . . . . . . . . 18  |-  ( X  e.  {  .0.  }  ->  X  =  .0.  )
5756fveq2d 5612 . . . . . . . . . . . . . . . . 17  |-  ( X  e.  {  .0.  }  ->  ( O `  X
)  =  ( O `
 .0.  ) )
5857eqeq1d 2366 . . . . . . . . . . . . . . . 16  |-  ( X  e.  {  .0.  }  ->  ( ( O `  X )  =  1  <-> 
( O `  .0.  )  =  1 ) )
5955, 58syl5ibrcom 213 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( X  e.  {  .0.  }  ->  ( O `  X )  =  1 ) )
6059necon3ad 2557 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( O `  X )  =/=  1  ->  -.  X  e.  {  .0.  } ) )
6152, 60mpd 14 . . . . . . . . . . . . 13  |-  ( ph  ->  -.  X  e.  {  .0.  } )
6221subg0cl 14728 . . . . . . . . . . . . . . . 16  |-  ( ( K `  { X } )  e.  (SubGrp `  H )  ->  .0.  e.  ( K `  { X } ) )
6338, 62syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  .0.  e.  ( K `
 { X }
) )
6463snssd 3839 . . . . . . . . . . . . . 14  |-  ( ph  ->  {  .0.  }  C_  ( K `  { X } ) )
65 ssnelpss 3593 . . . . . . . . . . . . . 14  |-  ( {  .0.  }  C_  ( K `  { X } )  ->  (
( X  e.  ( K `  { X } )  /\  -.  X  e.  {  .0.  } )  ->  {  .0.  } 
C.  ( K `  { X } ) ) )
6664, 65syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( X  e.  ( K `  { X } )  /\  -.  X  e.  {  .0.  } )  ->  {  .0.  } 
C.  ( K `  { X } ) ) )
6749, 61, 66mp2and 660 . . . . . . . . . . . 12  |-  ( ph  ->  {  .0.  }  C.  ( K `  { X } ) )
68 php3 7135 . . . . . . . . . . . 12  |-  ( ( ( K `  { X } )  e.  Fin  /\ 
{  .0.  }  C.  ( K `  { X } ) )  ->  {  .0.  }  ~<  ( K `  { X } ) )
6943, 67, 68syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  {  .0.  }  ~<  ( K `  { X } ) )
70 snfi 7029 . . . . . . . . . . . 12  |-  {  .0.  }  e.  Fin
71 hashsdom 11456 . . . . . . . . . . . 12  |-  ( ( {  .0.  }  e.  Fin  /\  ( K `  { X } )  e. 
Fin )  ->  (
( # `  {  .0.  } )  <  ( # `  ( K `  { X } ) )  <->  {  .0.  } 
~<  ( K `  { X } ) ) )
7270, 43, 71sylancr 644 . . . . . . . . . . 11  |-  ( ph  ->  ( ( # `  {  .0.  } )  <  ( # `
 ( K `  { X } ) )  <->  {  .0.  }  ~<  ( K `  { X } ) ) )
7369, 72mpbird 223 . . . . . . . . . 10  |-  ( ph  ->  ( # `  {  .0.  } )  <  ( # `
 ( K `  { X } ) ) )
7425, 73syl5eqbrr 4138 . . . . . . . . 9  |-  ( ph  ->  1  <  ( # `  ( K `  { X } ) ) )
75 1re 8927 . . . . . . . . . . 11  |-  1  e.  RR
7675a1i 10 . . . . . . . . . 10  |-  ( ph  ->  1  e.  RR )
77 hashcl 11443 . . . . . . . . . . . 12  |-  ( ( K `  { X } )  e.  Fin  ->  ( # `  ( K `  { X } ) )  e. 
NN0 )
7843, 77syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  ( K `  { X } ) )  e. 
NN0 )
7978nn0red 10111 . . . . . . . . . 10  |-  ( ph  ->  ( # `  ( K `  { X } ) )  e.  RR )
8021subg0cl 14728 . . . . . . . . . . . . 13  |-  ( W  e.  (SubGrp `  H
)  ->  .0.  e.  W )
81 ne0i 3537 . . . . . . . . . . . . 13  |-  (  .0. 
e.  W  ->  W  =/=  (/) )
821, 80, 813syl 18 . . . . . . . . . . . 12  |-  ( ph  ->  W  =/=  (/) )
83 hashnncl 11447 . . . . . . . . . . . . 13  |-  ( W  e.  Fin  ->  (
( # `  W )  e.  NN  <->  W  =/=  (/) ) )
8417, 83syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( # `  W
)  e.  NN  <->  W  =/=  (/) ) )
8582, 84mpbird 223 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  W
)  e.  NN )
8685nngt0d 9879 . . . . . . . . . 10  |-  ( ph  ->  0  <  ( # `  W ) )
87 ltmul1 9696 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( # `  ( K `
 { X }
) )  e.  RR  /\  ( ( # `  W
)  e.  RR  /\  0  <  ( # `  W
) ) )  -> 
( 1  <  ( # `
 ( K `  { X } ) )  <-> 
( 1  x.  ( # `
 W ) )  <  ( ( # `  ( K `  { X } ) )  x.  ( # `  W
) ) ) )
8876, 79, 20, 86, 87syl112anc 1186 . . . . . . . . 9  |-  ( ph  ->  ( 1  <  ( # `
 ( K `  { X } ) )  <-> 
( 1  x.  ( # `
 W ) )  <  ( ( # `  ( K `  { X } ) )  x.  ( # `  W
) ) ) )
8974, 88mpbid 201 . . . . . . . 8  |-  ( ph  ->  ( 1  x.  ( # `
 W ) )  <  ( ( # `  ( K `  { X } ) )  x.  ( # `  W
) ) )
9020recnd 8951 . . . . . . . . 9  |-  ( ph  ->  ( # `  W
)  e.  CC )
9190mulid2d 8943 . . . . . . . 8  |-  ( ph  ->  ( 1  x.  ( # `
 W ) )  =  ( # `  W
) )
92 pgpfac.l . . . . . . . . . 10  |-  .(+)  =  (
LSSum `  H )
93 eqid 2358 . . . . . . . . . 10  |-  (Cntz `  H )  =  (Cntz `  H )
94 pgpfac.i . . . . . . . . . 10  |-  ( ph  ->  ( ( K `  { X } )  i^i 
W )  =  {  .0.  } )
95 pgpfac.g . . . . . . . . . . . 12  |-  ( ph  ->  G  e.  Abel )
963subgabl 15231 . . . . . . . . . . . 12  |-  ( ( G  e.  Abel  /\  U  e.  (SubGrp `  G )
)  ->  H  e.  Abel )
9795, 2, 96syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  H  e.  Abel )
9893, 97, 38, 1ablcntzd 15248 . . . . . . . . . 10  |-  ( ph  ->  ( K `  { X } )  C_  (
(Cntz `  H ) `  W ) )
9992, 21, 93, 38, 1, 94, 98, 43, 17lsmhash 15113 . . . . . . . . 9  |-  ( ph  ->  ( # `  (
( K `  { X } )  .(+)  W ) )  =  ( (
# `  ( K `  { X } ) )  x.  ( # `  W ) ) )
100 pgpfac.s . . . . . . . . . 10  |-  ( ph  ->  ( ( K `  { X } )  .(+)  W )  =  U )
101100fveq2d 5612 . . . . . . . . 9  |-  ( ph  ->  ( # `  (
( K `  { X } )  .(+)  W ) )  =  ( # `  U ) )
10299, 101eqtr3d 2392 . . . . . . . 8  |-  ( ph  ->  ( ( # `  ( K `  { X } ) )  x.  ( # `  W
) )  =  (
# `  U )
)
10389, 91, 1023brtr3d 4133 . . . . . . 7  |-  ( ph  ->  ( # `  W
)  <  ( # `  U
) )
10420, 103ltned 9045 . . . . . 6  |-  ( ph  ->  ( # `  W
)  =/=  ( # `  U ) )
105 fveq2 5608 . . . . . . 7  |-  ( W  =  U  ->  ( # `
 W )  =  ( # `  U
) )
106105necon3i 2560 . . . . . 6  |-  ( (
# `  W )  =/=  ( # `  U
)  ->  W  =/=  U )
107104, 106syl 15 . . . . 5  |-  ( ph  ->  W  =/=  U )
108 df-pss 3244 . . . . 5  |-  ( W 
C.  U  <->  ( W  C_  U  /\  W  =/= 
U ) )
1099, 107, 108sylanbrc 645 . . . 4  |-  ( ph  ->  W  C.  U )
110 psseq1 3339 . . . . . 6  |-  ( t  =  W  ->  (
t  C.  U  <->  W  C.  U ) )
111 eqeq2 2367 . . . . . . . 8  |-  ( t  =  W  ->  (
( G DProd  s )  =  t  <->  ( G DProd  s
)  =  W ) )
112111anbi2d 684 . . . . . . 7  |-  ( t  =  W  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  t )  <->  ( G dom DProd  s  /\  ( G DProd 
s )  =  W ) ) )
113112rexbidv 2640 . . . . . 6  |-  ( t  =  W  ->  ( E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s )  =  t )  <->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  W ) ) )
114110, 113imbi12d 311 . . . . 5  |-  ( t  =  W  ->  (
( t  C.  U  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  t ) )  <-> 
( W  C.  U  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  W ) ) ) )
115114rspcv 2956 . . . 4  |-  ( W  e.  (SubGrp `  G
)  ->  ( A. t  e.  (SubGrp `  G
) ( t  C.  U  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  t ) )  ->  ( W  C.  U  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  W ) ) ) )
1167, 8, 109, 115syl3c 57 . . 3  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  W ) )
117 breq2 4108 . . . . 5  |-  ( s  =  a  ->  ( G dom DProd  s  <->  G dom DProd  a ) )
118 oveq2 5953 . . . . . 6  |-  ( s  =  a  ->  ( G DProd  s )  =  ( G DProd  a ) )
119118eqeq1d 2366 . . . . 5  |-  ( s  =  a  ->  (
( G DProd  s )  =  W  <->  ( G DProd  a
)  =  W ) )
120117, 119anbi12d 691 . . . 4  |-  ( s  =  a  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  W )  <->  ( G dom DProd  a  /\  ( G DProd 
a )  =  W ) ) )
121120cbvrexv 2841 . . 3  |-  ( E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s )  =  W )  <->  E. a  e. Word  C ( G dom DProd  a  /\  ( G DProd  a
)  =  W ) )
122116, 121sylib 188 . 2  |-  ( ph  ->  E. a  e. Word  C
( G dom DProd  a  /\  ( G DProd  a )  =  W ) )
123 pgpfac.c . . . . 5  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
12495adantr 451 . . . . 5  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  G  e.  Abel )
125 pgpfac.p . . . . . 6  |-  ( ph  ->  P pGrp  G )
126125adantr 451 . . . . 5  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  P pGrp  G )
12710adantr 451 . . . . 5  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  B  e.  Fin )
1282adantr 451 . . . . 5  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  U  e.  (SubGrp `  G
) )
1298adantr 451 . . . . 5  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  A. t  e.  (SubGrp `  G ) ( t 
C.  U  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  t ) ) )
130 pgpfac.e . . . . 5  |-  E  =  (gEx `  H )
13151adantr 451 . . . . 5  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  E  =/=  1 )
13232adantr 451 . . . . 5  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  X  e.  U )
13350adantr 451 . . . . 5  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  -> 
( O `  X
)  =  E )
1341adantr 451 . . . . 5  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  W  e.  (SubGrp `  H
) )
13594adantr 451 . . . . 5  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  -> 
( ( K `  { X } )  i^i 
W )  =  {  .0.  } )
136100adantr 451 . . . . 5  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  -> 
( ( K `  { X } )  .(+)  W )  =  U )
137 simprl 732 . . . . 5  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  -> 
a  e. Word  C )
138 simprrl 740 . . . . 5  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  G dom DProd  a )
139 simprrr 741 . . . . 5  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  -> 
( G DProd  a )  =  W )
140 eqid 2358 . . . . 5  |-  ( a concat  <" ( K `  { X } ) "> )  =  ( a concat  <" ( K `
 { X }
) "> )
14111, 123, 124, 126, 127, 128, 129, 3, 36, 53, 130, 21, 92, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140pgpfaclem1 15415 . . . 4  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
142141expr 598 . . 3  |-  ( (
ph  /\  a  e. Word  C )  ->  ( ( G dom DProd  a  /\  ( G DProd  a )  =  W )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  U ) ) )
143142rexlimdva 2743 . 2  |-  ( ph  ->  ( E. a  e. Word  C ( G dom DProd  a  /\  ( G DProd  a
)  =  W )  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  U ) ) )
144122, 143mpd 14 1  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   A.wral 2619   E.wrex 2620   {crab 2623   _Vcvv 2864    i^i cin 3227    C_ wss 3228    C. wpss 3229   (/)c0 3531   {csn 3716   class class class wbr 4104   dom cdm 4771   ran crn 4772   ` cfv 5337  (class class class)co 5945    ~< csdm 6950   Fincfn 6951   RRcr 8826   0cc0 8827   1c1 8828    x. cmul 8832    < clt 8957   NNcn 9836   NN0cn0 10057   #chash 11430  Word cword 11499   concat cconcat 11500   <"cs1 11501   Basecbs 13245   ↾s cress 13246   0gc0g 13499  Moorecmre 13583  mrClscmrc 13584  ACScacs 13586   Grpcgrp 14461  SubGrpcsubg 14714  Cntzccntz 14890   odcod 14939  gExcgex 14940   pGrp cpgp 14941   LSSumclsm 15044   Abelcabel 15189  CycGrpccyg 15263   DProd cdprd 15330
This theorem is referenced by:  pgpfaclem3  15417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-1st 6209  df-2nd 6210  df-tpos 6321  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-map 6862  df-ixp 6906  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-sup 7284  df-oi 7315  df-card 7662  df-cda 7884  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-2 9894  df-n0 10058  df-z 10117  df-uz 10323  df-fz 10875  df-fzo 10963  df-seq 11139  df-hash 11431  df-word 11505  df-concat 11506  df-s1 11507  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-0g 13503  df-gsum 13504  df-mre 13587  df-mrc 13588  df-acs 13590  df-mnd 14466  df-mhm 14514  df-submnd 14515  df-grp 14588  df-minusg 14589  df-sbg 14590  df-mulg 14591  df-subg 14717  df-ghm 14780  df-gim 14822  df-cntz 14892  df-oppg 14918  df-od 14943  df-pgp 14945  df-lsm 15046  df-pj1 15047  df-cmn 15190  df-abl 15191  df-cyg 15264  df-dprd 15332
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