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Theorem pgpfaclem2 16581
Description: Lemma for pgpfac 16583. (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 15702 . . . . . . 7  |-  ( U  e.  (SubGrp `  G
)  ->  ( W  e.  (SubGrp `  H )  <->  ( W  e.  (SubGrp `  G )  /\  W  C_  U ) ) )
52, 4syl 16 . . . . . 6  |-  ( ph  ->  ( W  e.  (SubGrp `  H )  <->  ( W  e.  (SubGrp `  G )  /\  W  C_  U ) ) )
61, 5mpbid 210 . . . . 5  |-  ( ph  ->  ( W  e.  (SubGrp `  G )  /\  W  C_  U ) )
76simpld 459 . . . 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 463 . . . . 5  |-  ( ph  ->  W  C_  U )
10 pgpfac.f . . . . . . . . . . 11  |-  ( ph  ->  B  e.  Fin )
11 pgpfac.b . . . . . . . . . . . . 13  |-  B  =  ( Base `  G
)
1211subgss 15680 . . . . . . . . . . . 12  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  B
)
132, 12syl 16 . . . . . . . . . . 11  |-  ( ph  ->  U  C_  B )
14 ssfi 7531 . . . . . . . . . . 11  |-  ( ( B  e.  Fin  /\  U  C_  B )  ->  U  e.  Fin )
1510, 13, 14syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  U  e.  Fin )
16 ssfi 7531 . . . . . . . . . 10  |-  ( ( U  e.  Fin  /\  W  C_  U )  ->  W  e.  Fin )
1715, 9, 16syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  W  e.  Fin )
18 hashcl 12124 . . . . . . . . 9  |-  ( W  e.  Fin  ->  ( # `
 W )  e. 
NN0 )
1917, 18syl 16 . . . . . . . 8  |-  ( ph  ->  ( # `  W
)  e.  NN0 )
2019nn0red 10635 . . . . . . 7  |-  ( ph  ->  ( # `  W
)  e.  RR )
21 pgpfac.0 . . . . . . . . . . . 12  |-  .0.  =  ( 0g `  H )
22 fvex 5699 . . . . . . . . . . . 12  |-  ( 0g
`  H )  e. 
_V
2321, 22eqeltri 2511 . . . . . . . . . . 11  |-  .0.  e.  _V
24 hashsng 12134 . . . . . . . . . . 11  |-  (  .0. 
e.  _V  ->  ( # `  {  .0.  } )  =  1 )
2523, 24ax-mp 5 . . . . . . . . . 10  |-  ( # `  {  .0.  } )  =  1
26 subgrcl 15684 . . . . . . . . . . . . . . . 16  |-  ( W  e.  (SubGrp `  H
)  ->  H  e.  Grp )
27 eqid 2441 . . . . . . . . . . . . . . . . 17  |-  ( Base `  H )  =  (
Base `  H )
2827subgacs 15714 . . . . . . . . . . . . . . . 16  |-  ( H  e.  Grp  ->  (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) ) )
29 acsmre 14588 . . . . . . . . . . . . . . . 16  |-  ( (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) )  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H
) ) )
301, 26, 28, 294syl 21 . . . . . . . . . . . . . . 15  |-  ( ph  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H ) ) )
31 pgpfac.k . . . . . . . . . . . . . . 15  |-  K  =  (mrCls `  (SubGrp `  H
) )
3230, 31mrcssvd 14559 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( K `  { X } )  C_  ( Base `  H ) )
333subgbas 15683 . . . . . . . . . . . . . . 15  |-  ( U  e.  (SubGrp `  G
)  ->  U  =  ( Base `  H )
)
342, 33syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  U  =  ( Base `  H ) )
3532, 34sseqtr4d 3391 . . . . . . . . . . . . 13  |-  ( ph  ->  ( K `  { X } )  C_  U
)
36 ssfi 7531 . . . . . . . . . . . . 13  |-  ( ( U  e.  Fin  /\  ( K `  { X } )  C_  U
)  ->  ( K `  { X } )  e.  Fin )
3715, 35, 36syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  ( K `  { X } )  e.  Fin )
38 pgpfac.x . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  X  e.  U )
3938, 34eleqtrd 2517 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  X  e.  ( Base `  H ) )
4031mrcsncl 14548 . . . . . . . . . . . . . . . 16  |-  ( ( (SubGrp `  H )  e.  (Moore `  ( Base `  H ) )  /\  X  e.  ( Base `  H ) )  -> 
( K `  { X } )  e.  (SubGrp `  H ) )
4130, 39, 40syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( K `  { X } )  e.  (SubGrp `  H ) )
4221subg0cl 15687 . . . . . . . . . . . . . . 15  |-  ( ( K `  { X } )  e.  (SubGrp `  H )  ->  .0.  e.  ( K `  { X } ) )
4341, 42syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  .0.  e.  ( K `
 { X }
) )
4443snssd 4016 . . . . . . . . . . . . 13  |-  ( ph  ->  {  .0.  }  C_  ( K `  { X } ) )
4539snssd 4016 . . . . . . . . . . . . . . 15  |-  ( ph  ->  { X }  C_  ( Base `  H )
)
4630, 31, 45mrcssidd 14561 . . . . . . . . . . . . . 14  |-  ( ph  ->  { X }  C_  ( K `  { X } ) )
47 snssg 4005 . . . . . . . . . . . . . . 15  |-  ( X  e.  U  ->  ( X  e.  ( K `  { X } )  <->  { X }  C_  ( K `  { X } ) ) )
4838, 47syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( X  e.  ( K `  { X } )  <->  { X }  C_  ( K `  { X } ) ) )
4946, 48mpbird 232 . . . . . . . . . . . . 13  |-  ( ph  ->  X  e.  ( K `
 { X }
) )
50 pgpfac.oe . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( O `  X
)  =  E )
51 pgpfac.1 . . . . . . . . . . . . . . 15  |-  ( ph  ->  E  =/=  1 )
5250, 51eqnetrd 2624 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( O `  X
)  =/=  1 )
53 pgpfac.o . . . . . . . . . . . . . . . . . 18  |-  O  =  ( od `  H
)
5453, 21od1 16058 . . . . . . . . . . . . . . . . 17  |-  ( H  e.  Grp  ->  ( O `  .0.  )  =  1 )
551, 26, 543syl 20 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( O `  .0.  )  =  1 )
56 elsni 3900 . . . . . . . . . . . . . . . . . 18  |-  ( X  e.  {  .0.  }  ->  X  =  .0.  )
5756fveq2d 5693 . . . . . . . . . . . . . . . . 17  |-  ( X  e.  {  .0.  }  ->  ( O `  X
)  =  ( O `
 .0.  ) )
5857eqeq1d 2449 . . . . . . . . . . . . . . . 16  |-  ( X  e.  {  .0.  }  ->  ( ( O `  X )  =  1  <-> 
( O `  .0.  )  =  1 ) )
5955, 58syl5ibrcom 222 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( X  e.  {  .0.  }  ->  ( O `  X )  =  1 ) )
6059necon3ad 2642 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( O `  X )  =/=  1  ->  -.  X  e.  {  .0.  } ) )
6152, 60mpd 15 . . . . . . . . . . . . 13  |-  ( ph  ->  -.  X  e.  {  .0.  } )
6244, 49, 61ssnelpssd 3741 . . . . . . . . . . . 12  |-  ( ph  ->  {  .0.  }  C.  ( K `  { X } ) )
63 php3 7495 . . . . . . . . . . . 12  |-  ( ( ( K `  { X } )  e.  Fin  /\ 
{  .0.  }  C.  ( K `  { X } ) )  ->  {  .0.  }  ~<  ( K `  { X } ) )
6437, 62, 63syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  {  .0.  }  ~<  ( K `  { X } ) )
65 snfi 7388 . . . . . . . . . . . 12  |-  {  .0.  }  e.  Fin
66 hashsdom 12142 . . . . . . . . . . . 12  |-  ( ( {  .0.  }  e.  Fin  /\  ( K `  { X } )  e. 
Fin )  ->  (
( # `  {  .0.  } )  <  ( # `  ( K `  { X } ) )  <->  {  .0.  } 
~<  ( K `  { X } ) ) )
6765, 37, 66sylancr 663 . . . . . . . . . . 11  |-  ( ph  ->  ( ( # `  {  .0.  } )  <  ( # `
 ( K `  { X } ) )  <->  {  .0.  }  ~<  ( K `  { X } ) ) )
6864, 67mpbird 232 . . . . . . . . . 10  |-  ( ph  ->  ( # `  {  .0.  } )  <  ( # `
 ( K `  { X } ) ) )
6925, 68syl5eqbrr 4324 . . . . . . . . 9  |-  ( ph  ->  1  <  ( # `  ( K `  { X } ) ) )
70 1red 9399 . . . . . . . . . 10  |-  ( ph  ->  1  e.  RR )
71 hashcl 12124 . . . . . . . . . . . 12  |-  ( ( K `  { X } )  e.  Fin  ->  ( # `  ( K `  { X } ) )  e. 
NN0 )
7237, 71syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  ( K `  { X } ) )  e. 
NN0 )
7372nn0red 10635 . . . . . . . . . 10  |-  ( ph  ->  ( # `  ( K `  { X } ) )  e.  RR )
7421subg0cl 15687 . . . . . . . . . . . . 13  |-  ( W  e.  (SubGrp `  H
)  ->  .0.  e.  W )
75 ne0i 3641 . . . . . . . . . . . . 13  |-  (  .0. 
e.  W  ->  W  =/=  (/) )
761, 74, 753syl 20 . . . . . . . . . . . 12  |-  ( ph  ->  W  =/=  (/) )
77 hashnncl 12132 . . . . . . . . . . . . 13  |-  ( W  e.  Fin  ->  (
( # `  W )  e.  NN  <->  W  =/=  (/) ) )
7817, 77syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( # `  W
)  e.  NN  <->  W  =/=  (/) ) )
7976, 78mpbird 232 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  W
)  e.  NN )
8079nngt0d 10363 . . . . . . . . . 10  |-  ( ph  ->  0  <  ( # `  W ) )
81 ltmul1 10177 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( # `  ( K `
 { X }
) )  e.  RR  /\  ( ( # `  W
)  e.  RR  /\  0  <  ( # `  W
) ) )  -> 
( 1  <  ( # `
 ( K `  { X } ) )  <-> 
( 1  x.  ( # `
 W ) )  <  ( ( # `  ( K `  { X } ) )  x.  ( # `  W
) ) ) )
8270, 73, 20, 80, 81syl112anc 1222 . . . . . . . . 9  |-  ( ph  ->  ( 1  <  ( # `
 ( K `  { X } ) )  <-> 
( 1  x.  ( # `
 W ) )  <  ( ( # `  ( K `  { X } ) )  x.  ( # `  W
) ) ) )
8369, 82mpbid 210 . . . . . . . 8  |-  ( ph  ->  ( 1  x.  ( # `
 W ) )  <  ( ( # `  ( K `  { X } ) )  x.  ( # `  W
) ) )
8420recnd 9410 . . . . . . . . 9  |-  ( ph  ->  ( # `  W
)  e.  CC )
8584mulid2d 9402 . . . . . . . 8  |-  ( ph  ->  ( 1  x.  ( # `
 W ) )  =  ( # `  W
) )
86 pgpfac.l . . . . . . . . . 10  |-  .(+)  =  (
LSSum `  H )
87 eqid 2441 . . . . . . . . . 10  |-  (Cntz `  H )  =  (Cntz `  H )
88 pgpfac.i . . . . . . . . . 10  |-  ( ph  ->  ( ( K `  { X } )  i^i 
W )  =  {  .0.  } )
89 pgpfac.g . . . . . . . . . . . 12  |-  ( ph  ->  G  e.  Abel )
903subgabl 16318 . . . . . . . . . . . 12  |-  ( ( G  e.  Abel  /\  U  e.  (SubGrp `  G )
)  ->  H  e.  Abel )
9189, 2, 90syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  H  e.  Abel )
9287, 91, 41, 1ablcntzd 16337 . . . . . . . . . 10  |-  ( ph  ->  ( K `  { X } )  C_  (
(Cntz `  H ) `  W ) )
9386, 21, 87, 41, 1, 88, 92, 37, 17lsmhash 16200 . . . . . . . . 9  |-  ( ph  ->  ( # `  (
( K `  { X } )  .(+)  W ) )  =  ( (
# `  ( K `  { X } ) )  x.  ( # `  W ) ) )
94 pgpfac.s . . . . . . . . . 10  |-  ( ph  ->  ( ( K `  { X } )  .(+)  W )  =  U )
9594fveq2d 5693 . . . . . . . . 9  |-  ( ph  ->  ( # `  (
( K `  { X } )  .(+)  W ) )  =  ( # `  U ) )
9693, 95eqtr3d 2475 . . . . . . . 8  |-  ( ph  ->  ( ( # `  ( K `  { X } ) )  x.  ( # `  W
) )  =  (
# `  U )
)
9783, 85, 963brtr3d 4319 . . . . . . 7  |-  ( ph  ->  ( # `  W
)  <  ( # `  U
) )
9820, 97ltned 9508 . . . . . 6  |-  ( ph  ->  ( # `  W
)  =/=  ( # `  U ) )
99 fveq2 5689 . . . . . . 7  |-  ( W  =  U  ->  ( # `
 W )  =  ( # `  U
) )
10099necon3i 2648 . . . . . 6  |-  ( (
# `  W )  =/=  ( # `  U
)  ->  W  =/=  U )
10198, 100syl 16 . . . . 5  |-  ( ph  ->  W  =/=  U )
102 df-pss 3342 . . . . 5  |-  ( W 
C.  U  <->  ( W  C_  U  /\  W  =/= 
U ) )
1039, 101, 102sylanbrc 664 . . . 4  |-  ( ph  ->  W  C.  U )
104 psseq1 3441 . . . . . 6  |-  ( t  =  W  ->  (
t  C.  U  <->  W  C.  U
) )
105 eqeq2 2450 . . . . . . . 8  |-  ( t  =  W  ->  (
( G DProd  s )  =  t  <->  ( G DProd  s
)  =  W ) )
106105anbi2d 703 . . . . . . 7  |-  ( t  =  W  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  t )  <->  ( G dom DProd  s  /\  ( G DProd 
s )  =  W ) ) )
107106rexbidv 2734 . . . . . 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 ) ) )
108104, 107imbi12d 320 . . . . 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 ) ) ) )
109108rspcv 3067 . . . 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 ) ) ) )
1107, 8, 103, 109syl3c 61 . . 3  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  W ) )
111 breq2 4294 . . . . 5  |-  ( s  =  a  ->  ( G dom DProd  s  <->  G dom DProd  a ) )
112 oveq2 6097 . . . . . 6  |-  ( s  =  a  ->  ( G DProd  s )  =  ( G DProd  a ) )
113112eqeq1d 2449 . . . . 5  |-  ( s  =  a  ->  (
( G DProd  s )  =  W  <->  ( G DProd  a
)  =  W ) )
114111, 113anbi12d 710 . . . 4  |-  ( s  =  a  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  W )  <->  ( G dom DProd  a  /\  ( G DProd 
a )  =  W ) ) )
115114cbvrexv 2946 . . 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 ) )
116110, 115sylib 196 . 2  |-  ( ph  ->  E. a  e. Word  C
( G dom DProd  a  /\  ( G DProd  a )  =  W ) )
117 pgpfac.c . . 3  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
11889adantr 465 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  G  e.  Abel )
119 pgpfac.p . . . 4  |-  ( ph  ->  P pGrp  G )
120119adantr 465 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  P pGrp  G )
12110adantr 465 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  B  e.  Fin )
1222adantr 465 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  U  e.  (SubGrp `  G
) )
1238adantr 465 . . 3  |-  ( (
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 ) ) )
124 pgpfac.e . . 3  |-  E  =  (gEx `  H )
12551adantr 465 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  E  =/=  1 )
12638adantr 465 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  X  e.  U )
12750adantr 465 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  -> 
( O `  X
)  =  E )
1281adantr 465 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  W  e.  (SubGrp `  H
) )
12988adantr 465 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  -> 
( ( K `  { X } )  i^i 
W )  =  {  .0.  } )
13094adantr 465 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  -> 
( ( K `  { X } )  .(+)  W )  =  U )
131 simprl 755 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  -> 
a  e. Word  C )
132 simprrl 763 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  G dom DProd  a )
133 simprrr 764 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  -> 
( G DProd  a )  =  W )
134 eqid 2441 . . 3  |-  ( a concat  <" ( K `  { X } ) "> )  =  ( a concat  <" ( K `
 { X }
) "> )
13511, 117, 118, 120, 121, 122, 123, 3, 31, 53, 124, 21, 86, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134pgpfaclem1 16580 . 2  |-  ( (
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 ) )
136116, 135rexlimddv 2843 1  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2604   A.wral 2713   E.wrex 2714   {crab 2717   _Vcvv 2970    i^i cin 3325    C_ wss 3326    C. wpss 3327   (/)c0 3635   {csn 3875   class class class wbr 4290   dom cdm 4838   ran crn 4839   ` cfv 5416  (class class class)co 6089    ~< csdm 7307   Fincfn 7308   RRcr 9279   0cc0 9280   1c1 9281    x. cmul 9285    < clt 9416   NNcn 10320   NN0cn0 10577   #chash 12101  Word cword 12219   concat cconcat 12221   <"cs1 12222   Basecbs 14172   ↾s cress 14173   0gc0g 14376  Moorecmre 14518  mrClscmrc 14519  ACScacs 14521   Grpcgrp 15408  SubGrpcsubg 15673  Cntzccntz 15831   odcod 16026  gExcgex 16027   pGrp cpgp 16028   LSSumclsm 16131   Abelcabel 16276  CycGrpccyg 16352   DProd cdprd 16473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-inf2 7845  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-iin 4172  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-se 4678  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-isom 5425  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-of 6318  df-om 6475  df-1st 6575  df-2nd 6576  df-supp 6689  df-tpos 6743  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-map 7214  df-ixp 7262  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-fsupp 7619  df-sup 7689  df-oi 7722  df-card 8107  df-cda 8335  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-n0 10578  df-z 10645  df-uz 10860  df-fz 11436  df-fzo 11547  df-seq 11805  df-hash 12102  df-word 12227  df-concat 12229  df-s1 12230  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-0g 14378  df-gsum 14379  df-mre 14522  df-mrc 14523  df-acs 14525  df-mnd 15413  df-mhm 15462  df-submnd 15463  df-grp 15543  df-minusg 15544  df-sbg 15545  df-mulg 15546  df-subg 15676  df-ghm 15743  df-gim 15785  df-cntz 15833  df-oppg 15859  df-od 16030  df-pgp 16032  df-lsm 16133  df-pj1 16134  df-cmn 16277  df-abl 16278  df-cyg 16353  df-dprd 16475
This theorem is referenced by:  pgpfaclem3  16582
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