MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pgpfaclem2 Structured version   Unicode version

Theorem pgpfaclem2 15645
Description: Lemma for pgpfac 15647. (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 14968 . . . . . . 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 203 . . . . 5  |-  ( ph  ->  ( W  e.  (SubGrp `  G )  /\  W  C_  U ) )
76simpld 447 . . . 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 451 . . . . 5  |-  ( ph  ->  W  C_  U )
10 pgpfac.f . . . . . . . . . . 11  |-  ( ph  ->  B  e.  Fin )
11 pgpfac.b . . . . . . . . . . . . 13  |-  B  =  ( Base `  G
)
1211subgss 14950 . . . . . . . . . . . 12  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  B
)
132, 12syl 16 . . . . . . . . . . 11  |-  ( ph  ->  U  C_  B )
14 ssfi 7332 . . . . . . . . . . 11  |-  ( ( B  e.  Fin  /\  U  C_  B )  ->  U  e.  Fin )
1510, 13, 14syl2anc 644 . . . . . . . . . 10  |-  ( ph  ->  U  e.  Fin )
16 ssfi 7332 . . . . . . . . . 10  |-  ( ( U  e.  Fin  /\  W  C_  U )  ->  W  e.  Fin )
1715, 9, 16syl2anc 644 . . . . . . . . 9  |-  ( ph  ->  W  e.  Fin )
18 hashcl 11644 . . . . . . . . 9  |-  ( W  e.  Fin  ->  ( # `
 W )  e. 
NN0 )
1917, 18syl 16 . . . . . . . 8  |-  ( ph  ->  ( # `  W
)  e.  NN0 )
2019nn0red 10280 . . . . . . 7  |-  ( ph  ->  ( # `  W
)  e.  RR )
21 pgpfac.0 . . . . . . . . . . . 12  |-  .0.  =  ( 0g `  H )
22 fvex 5745 . . . . . . . . . . . 12  |-  ( 0g
`  H )  e. 
_V
2321, 22eqeltri 2508 . . . . . . . . . . 11  |-  .0.  e.  _V
24 hashsng 11652 . . . . . . . . . . 11  |-  (  .0. 
e.  _V  ->  ( # `  {  .0.  } )  =  1 )
2523, 24ax-mp 5 . . . . . . . . . 10  |-  ( # `  {  .0.  } )  =  1
26 subgrcl 14954 . . . . . . . . . . . . . . . 16  |-  ( W  e.  (SubGrp `  H
)  ->  H  e.  Grp )
27 eqid 2438 . . . . . . . . . . . . . . . . 17  |-  ( Base `  H )  =  (
Base `  H )
2827subgacs 14980 . . . . . . . . . . . . . . . 16  |-  ( H  e.  Grp  ->  (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) ) )
29 acsmre 13882 . . . . . . . . . . . . . . . 16  |-  ( (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) )  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H
) ) )
301, 26, 28, 294syl 20 . . . . . . . . . . . . . . 15  |-  ( ph  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H ) ) )
31 pgpfac.k . . . . . . . . . . . . . . 15  |-  K  =  (mrCls `  (SubGrp `  H
) )
3230, 31mrcssvd 13853 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( K `  { X } )  C_  ( Base `  H ) )
333subgbas 14953 . . . . . . . . . . . . . . 15  |-  ( U  e.  (SubGrp `  G
)  ->  U  =  ( Base `  H )
)
342, 33syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  U  =  ( Base `  H ) )
3532, 34sseqtr4d 3387 . . . . . . . . . . . . 13  |-  ( ph  ->  ( K `  { X } )  C_  U
)
36 ssfi 7332 . . . . . . . . . . . . 13  |-  ( ( U  e.  Fin  /\  ( K `  { X } )  C_  U
)  ->  ( K `  { X } )  e.  Fin )
3715, 35, 36syl2anc 644 . . . . . . . . . . . 12  |-  ( ph  ->  ( K `  { X } )  e.  Fin )
38 pgpfac.x . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  X  e.  U )
3938, 34eleqtrd 2514 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  X  e.  ( Base `  H ) )
4031mrcsncl 13842 . . . . . . . . . . . . . . . 16  |-  ( ( (SubGrp `  H )  e.  (Moore `  ( Base `  H ) )  /\  X  e.  ( Base `  H ) )  -> 
( K `  { X } )  e.  (SubGrp `  H ) )
4130, 39, 40syl2anc 644 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( K `  { X } )  e.  (SubGrp `  H ) )
4221subg0cl 14957 . . . . . . . . . . . . . . 15  |-  ( ( K `  { X } )  e.  (SubGrp `  H )  ->  .0.  e.  ( K `  { X } ) )
4341, 42syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  .0.  e.  ( K `
 { X }
) )
4443snssd 3945 . . . . . . . . . . . . 13  |-  ( ph  ->  {  .0.  }  C_  ( K `  { X } ) )
4539snssd 3945 . . . . . . . . . . . . . . 15  |-  ( ph  ->  { X }  C_  ( Base `  H )
)
4630, 31, 45mrcssidd 13855 . . . . . . . . . . . . . 14  |-  ( ph  ->  { X }  C_  ( K `  { X } ) )
47 snssg 3934 . . . . . . . . . . . . . . 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 225 . . . . . . . . . . . . 13  |-  ( ph  ->  X  e.  ( K `
 { X }
) )
50 pgpfac.oe . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( O `  X
)  =  E )
51 pgpfac.1 . . . . . . . . . . . . . . 15  |-  ( ph  ->  E  =/=  1 )
5250, 51eqnetrd 2621 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( O `  X
)  =/=  1 )
53 pgpfac.o . . . . . . . . . . . . . . . . . 18  |-  O  =  ( od `  H
)
5453, 21od1 15200 . . . . . . . . . . . . . . . . 17  |-  ( H  e.  Grp  ->  ( O `  .0.  )  =  1 )
551, 26, 543syl 19 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( O `  .0.  )  =  1 )
56 elsni 3840 . . . . . . . . . . . . . . . . . 18  |-  ( X  e.  {  .0.  }  ->  X  =  .0.  )
5756fveq2d 5735 . . . . . . . . . . . . . . . . 17  |-  ( X  e.  {  .0.  }  ->  ( O `  X
)  =  ( O `
 .0.  ) )
5857eqeq1d 2446 . . . . . . . . . . . . . . . 16  |-  ( X  e.  {  .0.  }  ->  ( ( O `  X )  =  1  <-> 
( O `  .0.  )  =  1 ) )
5955, 58syl5ibrcom 215 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( X  e.  {  .0.  }  ->  ( O `  X )  =  1 ) )
6059necon3ad 2639 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( O `  X )  =/=  1  ->  -.  X  e.  {  .0.  } ) )
6152, 60mpd 15 . . . . . . . . . . . . 13  |-  ( ph  ->  -.  X  e.  {  .0.  } )
6244, 49, 61ssnelpssd 3694 . . . . . . . . . . . 12  |-  ( ph  ->  {  .0.  }  C.  ( K `  { X } ) )
63 php3 7296 . . . . . . . . . . . 12  |-  ( ( ( K `  { X } )  e.  Fin  /\ 
{  .0.  }  C.  ( K `  { X } ) )  ->  {  .0.  }  ~<  ( K `  { X } ) )
6437, 62, 63syl2anc 644 . . . . . . . . . . 11  |-  ( ph  ->  {  .0.  }  ~<  ( K `  { X } ) )
65 snfi 7190 . . . . . . . . . . . 12  |-  {  .0.  }  e.  Fin
66 hashsdom 11660 . . . . . . . . . . . 12  |-  ( ( {  .0.  }  e.  Fin  /\  ( K `  { X } )  e. 
Fin )  ->  (
( # `  {  .0.  } )  <  ( # `  ( K `  { X } ) )  <->  {  .0.  } 
~<  ( K `  { X } ) ) )
6765, 37, 66sylancr 646 . . . . . . . . . . 11  |-  ( ph  ->  ( ( # `  {  .0.  } )  <  ( # `
 ( K `  { X } ) )  <->  {  .0.  }  ~<  ( K `  { X } ) ) )
6864, 67mpbird 225 . . . . . . . . . 10  |-  ( ph  ->  ( # `  {  .0.  } )  <  ( # `
 ( K `  { X } ) ) )
6925, 68syl5eqbrr 4249 . . . . . . . . 9  |-  ( ph  ->  1  <  ( # `  ( K `  { X } ) ) )
70 1re 9095 . . . . . . . . . . 11  |-  1  e.  RR
7170a1i 11 . . . . . . . . . 10  |-  ( ph  ->  1  e.  RR )
72 hashcl 11644 . . . . . . . . . . . 12  |-  ( ( K `  { X } )  e.  Fin  ->  ( # `  ( K `  { X } ) )  e. 
NN0 )
7337, 72syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  ( K `  { X } ) )  e. 
NN0 )
7473nn0red 10280 . . . . . . . . . 10  |-  ( ph  ->  ( # `  ( K `  { X } ) )  e.  RR )
7521subg0cl 14957 . . . . . . . . . . . . 13  |-  ( W  e.  (SubGrp `  H
)  ->  .0.  e.  W )
76 ne0i 3636 . . . . . . . . . . . . 13  |-  (  .0. 
e.  W  ->  W  =/=  (/) )
771, 75, 763syl 19 . . . . . . . . . . . 12  |-  ( ph  ->  W  =/=  (/) )
78 hashnncl 11650 . . . . . . . . . . . . 13  |-  ( W  e.  Fin  ->  (
( # `  W )  e.  NN  <->  W  =/=  (/) ) )
7917, 78syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( # `  W
)  e.  NN  <->  W  =/=  (/) ) )
8077, 79mpbird 225 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  W
)  e.  NN )
8180nngt0d 10048 . . . . . . . . . 10  |-  ( ph  ->  0  <  ( # `  W ) )
82 ltmul1 9865 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( # `  ( K `
 { X }
) )  e.  RR  /\  ( ( # `  W
)  e.  RR  /\  0  <  ( # `  W
) ) )  -> 
( 1  <  ( # `
 ( K `  { X } ) )  <-> 
( 1  x.  ( # `
 W ) )  <  ( ( # `  ( K `  { X } ) )  x.  ( # `  W
) ) ) )
8371, 74, 20, 81, 82syl112anc 1189 . . . . . . . . 9  |-  ( ph  ->  ( 1  <  ( # `
 ( K `  { X } ) )  <-> 
( 1  x.  ( # `
 W ) )  <  ( ( # `  ( K `  { X } ) )  x.  ( # `  W
) ) ) )
8469, 83mpbid 203 . . . . . . . 8  |-  ( ph  ->  ( 1  x.  ( # `
 W ) )  <  ( ( # `  ( K `  { X } ) )  x.  ( # `  W
) ) )
8520recnd 9119 . . . . . . . . 9  |-  ( ph  ->  ( # `  W
)  e.  CC )
8685mulid2d 9111 . . . . . . . 8  |-  ( ph  ->  ( 1  x.  ( # `
 W ) )  =  ( # `  W
) )
87 pgpfac.l . . . . . . . . . 10  |-  .(+)  =  (
LSSum `  H )
88 eqid 2438 . . . . . . . . . 10  |-  (Cntz `  H )  =  (Cntz `  H )
89 pgpfac.i . . . . . . . . . 10  |-  ( ph  ->  ( ( K `  { X } )  i^i 
W )  =  {  .0.  } )
90 pgpfac.g . . . . . . . . . . . 12  |-  ( ph  ->  G  e.  Abel )
913subgabl 15460 . . . . . . . . . . . 12  |-  ( ( G  e.  Abel  /\  U  e.  (SubGrp `  G )
)  ->  H  e.  Abel )
9290, 2, 91syl2anc 644 . . . . . . . . . . 11  |-  ( ph  ->  H  e.  Abel )
9388, 92, 41, 1ablcntzd 15477 . . . . . . . . . 10  |-  ( ph  ->  ( K `  { X } )  C_  (
(Cntz `  H ) `  W ) )
9487, 21, 88, 41, 1, 89, 93, 37, 17lsmhash 15342 . . . . . . . . 9  |-  ( ph  ->  ( # `  (
( K `  { X } )  .(+)  W ) )  =  ( (
# `  ( K `  { X } ) )  x.  ( # `  W ) ) )
95 pgpfac.s . . . . . . . . . 10  |-  ( ph  ->  ( ( K `  { X } )  .(+)  W )  =  U )
9695fveq2d 5735 . . . . . . . . 9  |-  ( ph  ->  ( # `  (
( K `  { X } )  .(+)  W ) )  =  ( # `  U ) )
9794, 96eqtr3d 2472 . . . . . . . 8  |-  ( ph  ->  ( ( # `  ( K `  { X } ) )  x.  ( # `  W
) )  =  (
# `  U )
)
9884, 86, 973brtr3d 4244 . . . . . . 7  |-  ( ph  ->  ( # `  W
)  <  ( # `  U
) )
9920, 98ltned 9214 . . . . . 6  |-  ( ph  ->  ( # `  W
)  =/=  ( # `  U ) )
100 fveq2 5731 . . . . . . 7  |-  ( W  =  U  ->  ( # `
 W )  =  ( # `  U
) )
101100necon3i 2645 . . . . . 6  |-  ( (
# `  W )  =/=  ( # `  U
)  ->  W  =/=  U )
10299, 101syl 16 . . . . 5  |-  ( ph  ->  W  =/=  U )
103 df-pss 3338 . . . . 5  |-  ( W 
C.  U  <->  ( W  C_  U  /\  W  =/= 
U ) )
1049, 102, 103sylanbrc 647 . . . 4  |-  ( ph  ->  W  C.  U )
105 psseq1 3436 . . . . . 6  |-  ( t  =  W  ->  (
t  C.  U  <->  W  C.  U ) )
106 eqeq2 2447 . . . . . . . 8  |-  ( t  =  W  ->  (
( G DProd  s )  =  t  <->  ( G DProd  s
)  =  W ) )
107106anbi2d 686 . . . . . . 7  |-  ( t  =  W  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  t )  <->  ( G dom DProd  s  /\  ( G DProd 
s )  =  W ) ) )
108107rexbidv 2728 . . . . . 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 ) ) )
109105, 108imbi12d 313 . . . . 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 ) ) ) )
110109rspcv 3050 . . . 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 ) ) ) )
1117, 8, 104, 110syl3c 60 . . 3  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  W ) )
112 breq2 4219 . . . . 5  |-  ( s  =  a  ->  ( G dom DProd  s  <->  G dom DProd  a ) )
113 oveq2 6092 . . . . . 6  |-  ( s  =  a  ->  ( G DProd  s )  =  ( G DProd  a ) )
114113eqeq1d 2446 . . . . 5  |-  ( s  =  a  ->  (
( G DProd  s )  =  W  <->  ( G DProd  a
)  =  W ) )
115112, 114anbi12d 693 . . . 4  |-  ( s  =  a  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  W )  <->  ( G dom DProd  a  /\  ( G DProd 
a )  =  W ) ) )
116115cbvrexv 2935 . . 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 ) )
117111, 116sylib 190 . 2  |-  ( ph  ->  E. a  e. Word  C
( G dom DProd  a  /\  ( G DProd  a )  =  W ) )
118 pgpfac.c . . 3  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
11990adantr 453 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  G  e.  Abel )
120 pgpfac.p . . . 4  |-  ( ph  ->  P pGrp  G )
121120adantr 453 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  P pGrp  G )
12210adantr 453 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  B  e.  Fin )
1232adantr 453 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  U  e.  (SubGrp `  G
) )
1248adantr 453 . . 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 ) ) )
125 pgpfac.e . . 3  |-  E  =  (gEx `  H )
12651adantr 453 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  E  =/=  1 )
12738adantr 453 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  X  e.  U )
12850adantr 453 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  -> 
( O `  X
)  =  E )
1291adantr 453 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  W  e.  (SubGrp `  H
) )
13089adantr 453 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  -> 
( ( K `  { X } )  i^i 
W )  =  {  .0.  } )
13195adantr 453 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  -> 
( ( K `  { X } )  .(+)  W )  =  U )
132 simprl 734 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  -> 
a  e. Word  C )
133 simprrl 742 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  G dom DProd  a )
134 simprrr 743 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  -> 
( G DProd  a )  =  W )
135 eqid 2438 . . 3  |-  ( a concat  <" ( K `  { X } ) "> )  =  ( a concat  <" ( K `
 { X }
) "> )
13611, 118, 119, 121, 122, 123, 124, 3, 31, 53, 125, 21, 87, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135pgpfaclem1 15644 . 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 ) )
137117, 136rexlimddv 2836 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 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   {crab 2711   _Vcvv 2958    i^i cin 3321    C_ wss 3322    C. wpss 3323   (/)c0 3630   {csn 3816   class class class wbr 4215   dom cdm 4881   ran crn 4882   ` cfv 5457  (class class class)co 6084    ~< csdm 7111   Fincfn 7112   RRcr 8994   0cc0 8995   1c1 8996    x. cmul 9000    < clt 9125   NNcn 10005   NN0cn0 10226   #chash 11623  Word cword 11722   concat cconcat 11723   <"cs1 11724   Basecbs 13474   ↾s cress 13475   0gc0g 13728  Moorecmre 13812  mrClscmrc 13813  ACScacs 13815   Grpcgrp 14690  SubGrpcsubg 14943  Cntzccntz 15119   odcod 15168  gExcgex 15169   pGrp cpgp 15170   LSSumclsm 15273   Abelcabel 15418  CycGrpccyg 15492   DProd cdprd 15559
This theorem is referenced by:  pgpfaclem3  15646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-tpos 6482  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-n0 10227  df-z 10288  df-uz 10494  df-fz 11049  df-fzo 11141  df-seq 11329  df-hash 11624  df-word 11728  df-concat 11729  df-s1 11730  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-0g 13732  df-gsum 13733  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-mhm 14743  df-submnd 14744  df-grp 14817  df-minusg 14818  df-sbg 14819  df-mulg 14820  df-subg 14946  df-ghm 15009  df-gim 15051  df-cntz 15121  df-oppg 15147  df-od 15172  df-pgp 15174  df-lsm 15275  df-pj1 15276  df-cmn 15419  df-abl 15420  df-cyg 15493  df-dprd 15561
  Copyright terms: Public domain W3C validator