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Theorem pgpfaclem1 15594
Description: Lemma for pgpfac 15597. (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 )
pgpfac.2  |-  ( ph  ->  S  e. Word  C )
pgpfac.4  |-  ( ph  ->  G dom DProd  S )
pgpfac.5  |-  ( ph  ->  ( G DProd  S )  =  W )
pgpfac.t  |-  T  =  ( S concat  <" ( K `  { X } ) "> )
Assertion
Ref Expression
pgpfaclem1  |-  ( 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    T, s
Allowed substitution hints:    ph( s, r)    B( r)    C( r)    P( t, s, r)    .(+) ( t, s, r)    S( t, s, r)    T( t, 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 pgpfaclem1
StepHypRef Expression
1 pgpfac.t . . 3  |-  T  =  ( S concat  <" ( K `  { X } ) "> )
2 pgpfac.2 . . 3  |-  ( ph  ->  S  e. Word  C )
3 pgpfac.u . . . . . . . . 9  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
4 pgpfac.h . . . . . . . . . 10  |-  H  =  ( Gs  U )
54subggrp 14902 . . . . . . . . 9  |-  ( U  e.  (SubGrp `  G
)  ->  H  e.  Grp )
63, 5syl 16 . . . . . . . 8  |-  ( ph  ->  H  e.  Grp )
7 eqid 2404 . . . . . . . . 9  |-  ( Base `  H )  =  (
Base `  H )
87subgacs 14930 . . . . . . . 8  |-  ( H  e.  Grp  ->  (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) ) )
9 acsmre 13832 . . . . . . . 8  |-  ( (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) )  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H
) ) )
106, 8, 93syl 19 . . . . . . 7  |-  ( ph  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H ) ) )
11 pgpfac.x . . . . . . . 8  |-  ( ph  ->  X  e.  U )
124subgbas 14903 . . . . . . . . 9  |-  ( U  e.  (SubGrp `  G
)  ->  U  =  ( Base `  H )
)
133, 12syl 16 . . . . . . . 8  |-  ( ph  ->  U  =  ( Base `  H ) )
1411, 13eleqtrd 2480 . . . . . . 7  |-  ( ph  ->  X  e.  ( Base `  H ) )
15 pgpfac.k . . . . . . . 8  |-  K  =  (mrCls `  (SubGrp `  H
) )
1615mrcsncl 13792 . . . . . . 7  |-  ( ( (SubGrp `  H )  e.  (Moore `  ( Base `  H ) )  /\  X  e.  ( Base `  H ) )  -> 
( K `  { X } )  e.  (SubGrp `  H ) )
1710, 14, 16syl2anc 643 . . . . . 6  |-  ( ph  ->  ( K `  { X } )  e.  (SubGrp `  H ) )
184subsubg 14918 . . . . . . 7  |-  ( U  e.  (SubGrp `  G
)  ->  ( ( K `  { X } )  e.  (SubGrp `  H )  <->  ( ( K `  { X } )  e.  (SubGrp `  G )  /\  ( K `  { X } )  C_  U
) ) )
193, 18syl 16 . . . . . 6  |-  ( ph  ->  ( ( K `  { X } )  e.  (SubGrp `  H )  <->  ( ( K `  { X } )  e.  (SubGrp `  G )  /\  ( K `  { X } )  C_  U
) ) )
2017, 19mpbid 202 . . . . 5  |-  ( ph  ->  ( ( K `  { X } )  e.  (SubGrp `  G )  /\  ( K `  { X } )  C_  U
) )
2120simpld 446 . . . 4  |-  ( ph  ->  ( K `  { X } )  e.  (SubGrp `  G ) )
224oveq1i 6050 . . . . . . 7  |-  ( Hs  ( K `  { X } ) )  =  ( ( Gs  U )s  ( K `  { X } ) )
2320simprd 450 . . . . . . . 8  |-  ( ph  ->  ( K `  { X } )  C_  U
)
24 ressabs 13482 . . . . . . . 8  |-  ( ( U  e.  (SubGrp `  G )  /\  ( K `  { X } )  C_  U
)  ->  ( ( Gs  U )s  ( K `  { X } ) )  =  ( Gs  ( K `
 { X }
) ) )
253, 23, 24syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( ( Gs  U )s  ( K `  { X } ) )  =  ( Gs  ( K `  { X } ) ) )
2622, 25syl5eq 2448 . . . . . 6  |-  ( ph  ->  ( Hs  ( K `  { X } ) )  =  ( Gs  ( K `
 { X }
) ) )
277, 15cycsubgcyg2 15466 . . . . . . 7  |-  ( ( H  e.  Grp  /\  X  e.  ( Base `  H ) )  -> 
( Hs  ( K `  { X } ) )  e. CycGrp )
286, 14, 27syl2anc 643 . . . . . 6  |-  ( ph  ->  ( Hs  ( K `  { X } ) )  e. CycGrp )
2926, 28eqeltrrd 2479 . . . . 5  |-  ( ph  ->  ( Gs  ( K `  { X } ) )  e. CycGrp )
30 pgpfac.p . . . . . . 7  |-  ( ph  ->  P pGrp  G )
31 pgpprm 15182 . . . . . . 7  |-  ( P pGrp 
G  ->  P  e.  Prime )
3230, 31syl 16 . . . . . 6  |-  ( ph  ->  P  e.  Prime )
33 subgpgp 15186 . . . . . . 7  |-  ( ( P pGrp  G  /\  ( K `  { X } )  e.  (SubGrp `  G ) )  ->  P pGrp  ( Gs  ( K `  { X } ) ) )
3430, 21, 33syl2anc 643 . . . . . 6  |-  ( ph  ->  P pGrp  ( Gs  ( K `
 { X }
) ) )
35 brelrng 5058 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( Gs  ( K `  { X } ) )  e. CycGrp  /\  P pGrp  ( Gs  ( K `  { X } ) ) )  ->  ( Gs  ( K `
 { X }
) )  e.  ran pGrp  )
3632, 29, 34, 35syl3anc 1184 . . . . 5  |-  ( ph  ->  ( Gs  ( K `  { X } ) )  e.  ran pGrp  )
37 elin 3490 . . . . 5  |-  ( ( Gs  ( K `  { X } ) )  e.  (CycGrp  i^i  ran pGrp  )  <->  ( ( Gs  ( K `  { X } ) )  e. CycGrp  /\  ( Gs  ( K `  { X } ) )  e.  ran pGrp  ) )
3829, 36, 37sylanbrc 646 . . . 4  |-  ( ph  ->  ( Gs  ( K `  { X } ) )  e.  (CycGrp  i^i  ran pGrp  ) )
39 oveq2 6048 . . . . . 6  |-  ( r  =  ( K `  { X } )  -> 
( Gs  r )  =  ( Gs  ( K `  { X } ) ) )
4039eleq1d 2470 . . . . 5  |-  ( r  =  ( K `  { X } )  -> 
( ( Gs  r )  e.  (CycGrp  i^i  ran pGrp  )  <-> 
( Gs  ( K `  { X } ) )  e.  (CycGrp  i^i  ran pGrp  ) ) )
41 pgpfac.c . . . . 5  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
4240, 41elrab2 3054 . . . 4  |-  ( ( K `  { X } )  e.  C  <->  ( ( K `  { X } )  e.  (SubGrp `  G )  /\  ( Gs  ( K `  { X } ) )  e.  (CycGrp  i^i  ran pGrp  ) ) )
4321, 38, 42sylanbrc 646 . . 3  |-  ( ph  ->  ( K `  { X } )  e.  C
)
441, 2, 43cats1cld 11774 . 2  |-  ( ph  ->  T  e. Word  C )
45 wrdf 11688 . . . . 5  |-  ( T  e. Word  C  ->  T : ( 0..^ (
# `  T )
) --> C )
4644, 45syl 16 . . . 4  |-  ( ph  ->  T : ( 0..^ ( # `  T
) ) --> C )
47 ssrab2 3388 . . . . 5  |-  { r  e.  (SubGrp `  G
)  |  ( Gs  r )  e.  (CycGrp  i^i  ran pGrp  ) }  C_  (SubGrp `  G )
4841, 47eqsstri 3338 . . . 4  |-  C  C_  (SubGrp `  G )
49 fss 5558 . . . 4  |-  ( ( T : ( 0..^ ( # `  T
) ) --> C  /\  C  C_  (SubGrp `  G
) )  ->  T : ( 0..^ (
# `  T )
) --> (SubGrp `  G )
)
5046, 48, 49sylancl 644 . . 3  |-  ( ph  ->  T : ( 0..^ ( # `  T
) ) --> (SubGrp `  G ) )
51 fzodisj 11122 . . . 4  |-  ( ( 0..^ ( # `  S
) )  i^i  (
( # `  S )..^ ( ( # `  S
)  +  1 ) ) )  =  (/)
52 lencl 11690 . . . . . . . 8  |-  ( S  e. Word  C  ->  ( # `
 S )  e. 
NN0 )
532, 52syl 16 . . . . . . 7  |-  ( ph  ->  ( # `  S
)  e.  NN0 )
5453nn0zd 10329 . . . . . 6  |-  ( ph  ->  ( # `  S
)  e.  ZZ )
55 fzosn 11136 . . . . . 6  |-  ( (
# `  S )  e.  ZZ  ->  ( ( # `
 S )..^ ( ( # `  S
)  +  1 ) )  =  { (
# `  S ) } )
5654, 55syl 16 . . . . 5  |-  ( ph  ->  ( ( # `  S
)..^ ( ( # `  S )  +  1 ) )  =  {
( # `  S ) } )
5756ineq2d 3502 . . . 4  |-  ( ph  ->  ( ( 0..^ (
# `  S )
)  i^i  ( ( # `
 S )..^ ( ( # `  S
)  +  1 ) ) )  =  ( ( 0..^ ( # `  S ) )  i^i 
{ ( # `  S
) } ) )
5851, 57syl5reqr 2451 . . 3  |-  ( ph  ->  ( ( 0..^ (
# `  S )
)  i^i  { ( # `
 S ) } )  =  (/) )
591fveq2i 5690 . . . . . . 7  |-  ( # `  T )  =  (
# `  ( S concat  <" ( K `  { X } ) "> ) )
6043s1cld 11711 . . . . . . . 8  |-  ( ph  ->  <" ( K `
 { X }
) ">  e. Word  C )
61 ccatlen 11699 . . . . . . . 8  |-  ( ( S  e. Word  C  /\  <" ( K `  { X } ) ">  e. Word  C )  ->  ( # `  ( S concat  <" ( K `
 { X }
) "> )
)  =  ( (
# `  S )  +  ( # `  <" ( K `  { X } ) "> ) ) )
622, 60, 61syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( # `  ( S concat  <" ( K `
 { X }
) "> )
)  =  ( (
# `  S )  +  ( # `  <" ( K `  { X } ) "> ) ) )
6359, 62syl5eq 2448 . . . . . 6  |-  ( ph  ->  ( # `  T
)  =  ( (
# `  S )  +  ( # `  <" ( K `  { X } ) "> ) ) )
64 s1len 11713 . . . . . . 7  |-  ( # `  <" ( K `
 { X }
) "> )  =  1
6564oveq2i 6051 . . . . . 6  |-  ( (
# `  S )  +  ( # `  <" ( K `  { X } ) "> ) )  =  ( ( # `  S
)  +  1 )
6663, 65syl6eq 2452 . . . . 5  |-  ( ph  ->  ( # `  T
)  =  ( (
# `  S )  +  1 ) )
6766oveq2d 6056 . . . 4  |-  ( ph  ->  ( 0..^ ( # `  T ) )  =  ( 0..^ ( (
# `  S )  +  1 ) ) )
68 nn0uz 10476 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
6953, 68syl6eleq 2494 . . . . 5  |-  ( ph  ->  ( # `  S
)  e.  ( ZZ>= ` 
0 ) )
70 fzosplitsn 11150 . . . . 5  |-  ( (
# `  S )  e.  ( ZZ>= `  0 )  ->  ( 0..^ ( (
# `  S )  +  1 ) )  =  ( ( 0..^ ( # `  S
) )  u.  {
( # `  S ) } ) )
7169, 70syl 16 . . . 4  |-  ( ph  ->  ( 0..^ ( (
# `  S )  +  1 ) )  =  ( ( 0..^ ( # `  S
) )  u.  {
( # `  S ) } ) )
7267, 71eqtrd 2436 . . 3  |-  ( ph  ->  ( 0..^ ( # `  T ) )  =  ( ( 0..^ (
# `  S )
)  u.  { (
# `  S ) } ) )
73 eqid 2404 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
74 eqid 2404 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
75 pgpfac.4 . . . 4  |-  ( ph  ->  G dom DProd  S )
76 cats1un 11745 . . . . . . . 8  |-  ( ( S  e. Word  C  /\  ( K `  { X } )  e.  C
)  ->  ( S concat  <" ( K `  { X } ) "> )  =  ( S  u.  { <. (
# `  S ) ,  ( K `  { X } ) >. } ) )
772, 43, 76syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( S concat  <" ( K `  { X } ) "> )  =  ( S  u.  { <. ( # `  S
) ,  ( K `
 { X }
) >. } ) )
781, 77syl5eq 2448 . . . . . 6  |-  ( ph  ->  T  =  ( S  u.  { <. ( # `
 S ) ,  ( K `  { X } ) >. } ) )
7978reseq1d 5104 . . . . 5  |-  ( ph  ->  ( T  |`  (
0..^ ( # `  S
) ) )  =  ( ( S  u.  {
<. ( # `  S
) ,  ( K `
 { X }
) >. } )  |`  ( 0..^ ( # `  S
) ) ) )
80 wrdf 11688 . . . . . . 7  |-  ( S  e. Word  C  ->  S : ( 0..^ (
# `  S )
) --> C )
81 ffn 5550 . . . . . . 7  |-  ( S : ( 0..^ (
# `  S )
) --> C  ->  S  Fn  ( 0..^ ( # `  S ) ) )
822, 80, 813syl 19 . . . . . 6  |-  ( ph  ->  S  Fn  ( 0..^ ( # `  S
) ) )
83 fzonel 11107 . . . . . 6  |-  -.  ( # `
 S )  e.  ( 0..^ ( # `  S ) )
84 fsnunres 5893 . . . . . 6  |-  ( ( S  Fn  ( 0..^ ( # `  S
) )  /\  -.  ( # `  S )  e.  ( 0..^ (
# `  S )
) )  ->  (
( S  u.  { <. ( # `  S
) ,  ( K `
 { X }
) >. } )  |`  ( 0..^ ( # `  S
) ) )  =  S )
8582, 83, 84sylancl 644 . . . . 5  |-  ( ph  ->  ( ( S  u.  {
<. ( # `  S
) ,  ( K `
 { X }
) >. } )  |`  ( 0..^ ( # `  S
) ) )  =  S )
8679, 85eqtrd 2436 . . . 4  |-  ( ph  ->  ( T  |`  (
0..^ ( # `  S
) ) )  =  S )
8775, 86breqtrrd 4198 . . 3  |-  ( ph  ->  G dom DProd  ( T  |`  ( 0..^ ( # `  S ) ) ) )
88 fvex 5701 . . . . . 6  |-  ( # `  S )  e.  _V
89 dprdsn 15549 . . . . . 6  |-  ( ( ( # `  S
)  e.  _V  /\  ( K `  { X } )  e.  (SubGrp `  G ) )  -> 
( G dom DProd  { <. (
# `  S ) ,  ( K `  { X } ) >. }  /\  ( G DProd  { <. (
# `  S ) ,  ( K `  { X } ) >. } )  =  ( K `  { X } ) ) )
9088, 21, 89sylancr 645 . . . . 5  |-  ( ph  ->  ( G dom DProd  { <. (
# `  S ) ,  ( K `  { X } ) >. }  /\  ( G DProd  { <. (
# `  S ) ,  ( K `  { X } ) >. } )  =  ( K `  { X } ) ) )
9190simpld 446 . . . 4  |-  ( ph  ->  G dom DProd  { <. ( # `
 S ) ,  ( K `  { X } ) >. } )
92 ffn 5550 . . . . . . 7  |-  ( T : ( 0..^ (
# `  T )
) --> C  ->  T  Fn  ( 0..^ ( # `  T ) ) )
9344, 45, 923syl 19 . . . . . 6  |-  ( ph  ->  T  Fn  ( 0..^ ( # `  T
) ) )
94 ssun2 3471 . . . . . . . 8  |-  { (
# `  S ) }  C_  ( ( 0..^ ( # `  S
) )  u.  {
( # `  S ) } )
9588snss 3886 . . . . . . . 8  |-  ( (
# `  S )  e.  ( ( 0..^ (
# `  S )
)  u.  { (
# `  S ) } )  <->  { ( # `
 S ) } 
C_  ( ( 0..^ ( # `  S
) )  u.  {
( # `  S ) } ) )
9694, 95mpbir 201 . . . . . . 7  |-  ( # `  S )  e.  ( ( 0..^ ( # `  S ) )  u. 
{ ( # `  S
) } )
9796, 72syl5eleqr 2491 . . . . . 6  |-  ( ph  ->  ( # `  S
)  e.  ( 0..^ ( # `  T
) ) )
98 fnressn 5877 . . . . . 6  |-  ( ( T  Fn  ( 0..^ ( # `  T
) )  /\  ( # `
 S )  e.  ( 0..^ ( # `  T ) ) )  ->  ( T  |`  { ( # `  S
) } )  =  { <. ( # `  S
) ,  ( T `
 ( # `  S
) ) >. } )
9993, 97, 98syl2anc 643 . . . . 5  |-  ( ph  ->  ( T  |`  { (
# `  S ) } )  =  { <. ( # `  S
) ,  ( T `
 ( # `  S
) ) >. } )
1001fveq1i 5688 . . . . . . . . 9  |-  ( T `
 ( # `  S
) )  =  ( ( S concat  <" ( K `  { X } ) "> ) `  ( # `  S
) )
10153nn0cnd 10232 . . . . . . . . . . . 12  |-  ( ph  ->  ( # `  S
)  e.  CC )
102101addid2d 9223 . . . . . . . . . . 11  |-  ( ph  ->  ( 0  +  (
# `  S )
)  =  ( # `  S ) )
103102eqcomd 2409 . . . . . . . . . 10  |-  ( ph  ->  ( # `  S
)  =  ( 0  +  ( # `  S
) ) )
104103fveq2d 5691 . . . . . . . . 9  |-  ( ph  ->  ( ( S concat  <" ( K `  { X } ) "> ) `  ( # `  S
) )  =  ( ( S concat  <" ( K `  { X } ) "> ) `  ( 0  +  ( # `  S
) ) ) )
105100, 104syl5eq 2448 . . . . . . . 8  |-  ( ph  ->  ( T `  ( # `
 S ) )  =  ( ( S concat  <" ( K `  { X } ) "> ) `  (
0  +  ( # `  S ) ) ) )
106 1nn 9967 . . . . . . . . . . . 12  |-  1  e.  NN
107106a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  1  e.  NN )
10864, 107syl5eqel 2488 . . . . . . . . . 10  |-  ( ph  ->  ( # `  <" ( K `  { X } ) "> )  e.  NN )
109 lbfzo0 11125 . . . . . . . . . 10  |-  ( 0  e.  ( 0..^ (
# `  <" ( K `  { X } ) "> ) )  <->  ( # `  <" ( K `  { X } ) "> )  e.  NN )
110108, 109sylibr 204 . . . . . . . . 9  |-  ( ph  ->  0  e.  ( 0..^ ( # `  <" ( K `  { X } ) "> ) ) )
111 ccatval3 11702 . . . . . . . . 9  |-  ( ( S  e. Word  C  /\  <" ( K `  { X } ) ">  e. Word  C  /\  0  e.  ( 0..^ ( # `  <" ( K `  { X } ) "> ) ) )  -> 
( ( S concat  <" ( K `  { X } ) "> ) `  ( 0  +  ( # `  S
) ) )  =  ( <" ( K `  { X } ) "> `  0 ) )
1122, 60, 110, 111syl3anc 1184 . . . . . . . 8  |-  ( ph  ->  ( ( S concat  <" ( K `  { X } ) "> ) `  ( 0  +  ( # `  S
) ) )  =  ( <" ( K `  { X } ) "> `  0 ) )
113 fvex 5701 . . . . . . . . 9  |-  ( K `
 { X }
)  e.  _V
114 s1fv 11715 . . . . . . . . 9  |-  ( ( K `  { X } )  e.  _V  ->  ( <" ( K `  { X } ) "> `  0 )  =  ( K `  { X } ) )
115113, 114mp1i 12 . . . . . . . 8  |-  ( ph  ->  ( <" ( K `  { X } ) "> `  0 )  =  ( K `  { X } ) )
116105, 112, 1153eqtrd 2440 . . . . . . 7  |-  ( ph  ->  ( T `  ( # `
 S ) )  =  ( K `  { X } ) )
117116opeq2d 3951 . . . . . 6  |-  ( ph  -> 
<. ( # `  S
) ,  ( T `
 ( # `  S
) ) >.  =  <. (
# `  S ) ,  ( K `  { X } ) >.
)
118117sneqd 3787 . . . . 5  |-  ( ph  ->  { <. ( # `  S
) ,  ( T `
 ( # `  S
) ) >. }  =  { <. ( # `  S
) ,  ( K `
 { X }
) >. } )
11999, 118eqtrd 2436 . . . 4  |-  ( ph  ->  ( T  |`  { (
# `  S ) } )  =  { <. ( # `  S
) ,  ( K `
 { X }
) >. } )
12091, 119breqtrrd 4198 . . 3  |-  ( ph  ->  G dom DProd  ( T  |` 
{ ( # `  S
) } ) )
121 pgpfac.g . . . 4  |-  ( ph  ->  G  e.  Abel )
122 dprdsubg 15537 . . . . 5  |-  ( G dom DProd  ( T  |`  ( 0..^ ( # `  S
) ) )  -> 
( G DProd  ( T  |`  ( 0..^ ( # `  S ) ) ) )  e.  (SubGrp `  G ) )
12387, 122syl 16 . . . 4  |-  ( ph  ->  ( G DProd  ( T  |`  ( 0..^ ( # `  S ) ) ) )  e.  (SubGrp `  G ) )
124 dprdsubg 15537 . . . . 5  |-  ( G dom DProd  ( T  |`  { ( # `  S
) } )  -> 
( G DProd  ( T  |` 
{ ( # `  S
) } ) )  e.  (SubGrp `  G
) )
125120, 124syl 16 . . . 4  |-  ( ph  ->  ( G DProd  ( T  |`  { ( # `  S
) } ) )  e.  (SubGrp `  G
) )
12673, 121, 123, 125ablcntzd 15427 . . 3  |-  ( ph  ->  ( G DProd  ( T  |`  ( 0..^ ( # `  S ) ) ) )  C_  ( (Cntz `  G ) `  ( G DProd  ( T  |`  { (
# `  S ) } ) ) ) )
127 pgpfac.i . . . 4  |-  ( ph  ->  ( ( K `  { X } )  i^i 
W )  =  {  .0.  } )
12886oveq2d 6056 . . . . . . 7  |-  ( ph  ->  ( G DProd  ( T  |`  ( 0..^ ( # `  S ) ) ) )  =  ( G DProd 
S ) )
129 pgpfac.5 . . . . . . 7  |-  ( ph  ->  ( G DProd  S )  =  W )
130128, 129eqtrd 2436 . . . . . 6  |-  ( ph  ->  ( G DProd  ( T  |`  ( 0..^ ( # `  S ) ) ) )  =  W )
131119oveq2d 6056 . . . . . . 7  |-  ( ph  ->  ( G DProd  ( T  |`  { ( # `  S
) } ) )  =  ( G DProd  { <. (
# `  S ) ,  ( K `  { X } ) >. } ) )
13290simprd 450 . . . . . . 7  |-  ( ph  ->  ( G DProd  { <. (
# `  S ) ,  ( K `  { X } ) >. } )  =  ( K `  { X } ) )
133131, 132eqtrd 2436 . . . . . 6  |-  ( ph  ->  ( G DProd  ( T  |`  { ( # `  S
) } ) )  =  ( K `  { X } ) )
134130, 133ineq12d 3503 . . . . 5  |-  ( ph  ->  ( ( G DProd  ( T  |`  ( 0..^ (
# `  S )
) ) )  i^i  ( G DProd  ( T  |`  { ( # `  S
) } ) ) )  =  ( W  i^i  ( K `  { X } ) ) )
135 incom 3493 . . . . 5  |-  ( W  i^i  ( K `  { X } ) )  =  ( ( K `
 { X }
)  i^i  W )
136134, 135syl6eq 2452 . . . 4  |-  ( ph  ->  ( ( G DProd  ( T  |`  ( 0..^ (
# `  S )
) ) )  i^i  ( G DProd  ( T  |`  { ( # `  S
) } ) ) )  =  ( ( K `  { X } )  i^i  W
) )
1374, 74subg0 14905 . . . . . . 7  |-  ( U  e.  (SubGrp `  G
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
1383, 137syl 16 . . . . . 6  |-  ( ph  ->  ( 0g `  G
)  =  ( 0g
`  H ) )
139 pgpfac.0 . . . . . 6  |-  .0.  =  ( 0g `  H )
140138, 139syl6eqr 2454 . . . . 5  |-  ( ph  ->  ( 0g `  G
)  =  .0.  )
141140sneqd 3787 . . . 4  |-  ( ph  ->  { ( 0g `  G ) }  =  {  .0.  } )
142127, 136, 1413eqtr4d 2446 . . 3  |-  ( ph  ->  ( ( G DProd  ( T  |`  ( 0..^ (
# `  S )
) ) )  i^i  ( G DProd  ( T  |`  { ( # `  S
) } ) ) )  =  { ( 0g `  G ) } )
14350, 58, 72, 73, 74, 87, 120, 126, 142dmdprdsplit2 15559 . 2  |-  ( ph  ->  G dom DProd  T )
144 eqid 2404 . . . . 5  |-  ( LSSum `  G )  =  (
LSSum `  G )
14550, 58, 72, 144, 143dprdsplit 15561 . . . 4  |-  ( ph  ->  ( G DProd  T )  =  ( ( G DProd 
( T  |`  (
0..^ ( # `  S
) ) ) ) ( LSSum `  G )
( G DProd  ( T  |` 
{ ( # `  S
) } ) ) ) )
146130, 133oveq12d 6058 . . . 4  |-  ( ph  ->  ( ( G DProd  ( T  |`  ( 0..^ (
# `  S )
) ) ) (
LSSum `  G ) ( G DProd  ( T  |`  { ( # `  S
) } ) ) )  =  ( W ( LSSum `  G )
( K `  { X } ) ) )
147130, 123eqeltrrd 2479 . . . . 5  |-  ( ph  ->  W  e.  (SubGrp `  G ) )
148144lsmcom 15428 . . . . 5  |-  ( ( G  e.  Abel  /\  W  e.  (SubGrp `  G )  /\  ( K `  { X } )  e.  (SubGrp `  G ) )  -> 
( W ( LSSum `  G ) ( K `
 { X }
) )  =  ( ( K `  { X } ) ( LSSum `  G ) W ) )
149121, 147, 21, 148syl3anc 1184 . . . 4  |-  ( ph  ->  ( W ( LSSum `  G ) ( K `
 { X }
) )  =  ( ( K `  { X } ) ( LSSum `  G ) W ) )
150145, 146, 1493eqtrd 2440 . . 3  |-  ( ph  ->  ( G DProd  T )  =  ( ( K `
 { X }
) ( LSSum `  G
) W ) )
151 pgpfac.w . . . . . 6  |-  ( ph  ->  W  e.  (SubGrp `  H ) )
1527subgss 14900 . . . . . 6  |-  ( W  e.  (SubGrp `  H
)  ->  W  C_  ( Base `  H ) )
153151, 152syl 16 . . . . 5  |-  ( ph  ->  W  C_  ( Base `  H ) )
154153, 13sseqtr4d 3345 . . . 4  |-  ( ph  ->  W  C_  U )
155 pgpfac.l . . . . 5  |-  .(+)  =  (
LSSum `  H )
1564, 144, 155subglsm 15260 . . . 4  |-  ( ( U  e.  (SubGrp `  G )  /\  ( K `  { X } )  C_  U  /\  W  C_  U )  ->  ( ( K `
 { X }
) ( LSSum `  G
) W )  =  ( ( K `  { X } )  .(+)  W ) )
1573, 23, 154, 156syl3anc 1184 . . 3  |-  ( ph  ->  ( ( K `  { X } ) (
LSSum `  G ) W )  =  ( ( K `  { X } )  .(+)  W ) )
158 pgpfac.s . . 3  |-  ( ph  ->  ( ( K `  { X } )  .(+)  W )  =  U )
159150, 157, 1583eqtrd 2440 . 2  |-  ( ph  ->  ( G DProd  T )  =  U )
160 breq2 4176 . . . 4  |-  ( s  =  T  ->  ( G dom DProd  s  <->  G dom DProd  T ) )
161 oveq2 6048 . . . . 5  |-  ( s  =  T  ->  ( G DProd  s )  =  ( G DProd  T ) )
162161eqeq1d 2412 . . . 4  |-  ( s  =  T  ->  (
( G DProd  s )  =  U  <->  ( G DProd  T
)  =  U ) )
163160, 162anbi12d 692 . . 3  |-  ( s  =  T  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  U )  <->  ( G dom DProd  T  /\  ( G DProd 
T )  =  U ) ) )
164163rspcev 3012 . 2  |-  ( ( T  e. Word  C  /\  ( G dom DProd  T  /\  ( G DProd  T )  =  U ) )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
16544, 143, 159, 164syl12anc 1182 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 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   {crab 2670   _Vcvv 2916    u. cun 3278    i^i cin 3279    C_ wss 3280    C. wpss 3281   (/)c0 3588   {csn 3774   <.cop 3777   class class class wbr 4172   dom cdm 4837   ran crn 4838    |` cres 4839    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   Fincfn 7068   0cc0 8946   1c1 8947    + caddc 8949   NNcn 9956   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444  ..^cfzo 11090   #chash 11573  Word cword 11672   concat cconcat 11673   <"cs1 11674   Primecprime 13034   Basecbs 13424   ↾s cress 13425   0gc0g 13678  Moorecmre 13762  mrClscmrc 13763  ACScacs 13765   Grpcgrp 14640  SubGrpcsubg 14893  Cntzccntz 15069   odcod 15118  gExcgex 15119   pGrp cpgp 15120   LSSumclsm 15223   Abelcabel 15368  CycGrpccyg 15442   DProd cdprd 15509
This theorem is referenced by:  pgpfaclem2  15595
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-word 11678  df-concat 11679  df-s1 11680  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-0g 13682  df-gsum 13683  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-ghm 14959  df-gim 15001  df-cntz 15071  df-oppg 15097  df-od 15122  df-pgp 15124  df-lsm 15225  df-cmn 15369  df-abl 15370  df-cyg 15443  df-dprd 15511
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