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Theorem subgdmdprd 17194
Description: A direct product in a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
Hypothesis
Ref Expression
subgdprd.1  |-  H  =  ( Gs  A )
Assertion
Ref Expression
subgdmdprd  |-  ( A  e.  (SubGrp `  G
)  ->  ( H dom DProd  S  <->  ( G dom DProd  S  /\  ran  S  C_  ~P A ) ) )

Proof of Theorem subgdmdprd
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reldmdprd 17141 . . . 4  |-  Rel  dom DProd
21brrelex2i 4955 . . 3  |-  ( H dom DProd  S  ->  S  e. 
_V )
32a1i 11 . 2  |-  ( A  e.  (SubGrp `  G
)  ->  ( H dom DProd  S  ->  S  e.  _V ) )
41brrelex2i 4955 . . . 4  |-  ( G dom DProd  S  ->  S  e. 
_V )
54adantr 463 . . 3  |-  ( ( G dom DProd  S  /\  ran  S  C_  ~P A
)  ->  S  e.  _V )
65a1i 11 . 2  |-  ( A  e.  (SubGrp `  G
)  ->  ( ( G dom DProd  S  /\  ran  S  C_ 
~P A )  ->  S  e.  _V )
)
7 ffvelrn 5931 . . . . . . . . . . . . . . . 16  |-  ( ( S : dom  S --> (SubGrp `  H )  /\  x  e.  dom  S )  ->  ( S `  x )  e.  (SubGrp `  H ) )
87ad2ant2lr 745 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( S `  x )  e.  (SubGrp `  H ) )
9 eqid 2382 . . . . . . . . . . . . . . . 16  |-  ( Base `  H )  =  (
Base `  H )
109subgss 16319 . . . . . . . . . . . . . . 15  |-  ( ( S `  x )  e.  (SubGrp `  H
)  ->  ( S `  x )  C_  ( Base `  H ) )
118, 10syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( S `  x )  C_  ( Base `  H ) )
12 subgdprd.1 . . . . . . . . . . . . . . . 16  |-  H  =  ( Gs  A )
1312subgbas 16322 . . . . . . . . . . . . . . 15  |-  ( A  e.  (SubGrp `  G
)  ->  A  =  ( Base `  H )
)
1413ad2antrr 723 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  A  =  ( Base `  H )
)
1511, 14sseqtr4d 3454 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( S `  x )  C_  A
)
1615biantrud 505 . . . . . . . . . . . 12  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( ( S `  x )  C_  ( (Cntz `  G
) `  ( S `  y ) )  <->  ( ( S `  x )  C_  ( (Cntz `  G
) `  ( S `  y ) )  /\  ( S `  x ) 
C_  A ) ) )
17 simpll 751 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  A  e.  (SubGrp `  G ) )
18 simplr 753 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  S : dom  S --> (SubGrp `  H )
)
19 eldifi 3540 . . . . . . . . . . . . . . . . . . 19  |-  ( y  e.  ( dom  S  \  { x } )  ->  y  e.  dom  S )
2019ad2antll 726 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  y  e.  dom  S )
2118, 20ffvelrnd 5934 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( S `  y )  e.  (SubGrp `  H ) )
229subgss 16319 . . . . . . . . . . . . . . . . 17  |-  ( ( S `  y )  e.  (SubGrp `  H
)  ->  ( S `  y )  C_  ( Base `  H ) )
2321, 22syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( S `  y )  C_  ( Base `  H ) )
2423, 14sseqtr4d 3454 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( S `  y )  C_  A
)
25 eqid 2382 . . . . . . . . . . . . . . . 16  |-  (Cntz `  G )  =  (Cntz `  G )
26 eqid 2382 . . . . . . . . . . . . . . . 16  |-  (Cntz `  H )  =  (Cntz `  H )
2712, 25, 26resscntz 16486 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  (SubGrp `  G )  /\  ( S `  y )  C_  A )  ->  (
(Cntz `  H ) `  ( S `  y
) )  =  ( ( (Cntz `  G
) `  ( S `  y ) )  i^i 
A ) )
2817, 24, 27syl2anc 659 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( (Cntz `  H ) `  ( S `  y )
)  =  ( ( (Cntz `  G ) `  ( S `  y
) )  i^i  A
) )
2928sseq2d 3445 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( ( S `  x )  C_  ( (Cntz `  H
) `  ( S `  y ) )  <->  ( S `  x )  C_  (
( (Cntz `  G
) `  ( S `  y ) )  i^i 
A ) ) )
30 ssin 3634 . . . . . . . . . . . . 13  |-  ( ( ( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( S `  x )  C_  A
)  <->  ( S `  x )  C_  (
( (Cntz `  G
) `  ( S `  y ) )  i^i 
A ) )
3129, 30syl6bbr 263 . . . . . . . . . . . 12  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( ( S `  x )  C_  ( (Cntz `  H
) `  ( S `  y ) )  <->  ( ( S `  x )  C_  ( (Cntz `  G
) `  ( S `  y ) )  /\  ( S `  x ) 
C_  A ) ) )
3216, 31bitr4d 256 . . . . . . . . . . 11  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( ( S `  x )  C_  ( (Cntz `  G
) `  ( S `  y ) )  <->  ( S `  x )  C_  (
(Cntz `  H ) `  ( S `  y
) ) ) )
3332anassrs 646 . . . . . . . . . 10  |-  ( ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  /\  y  e.  ( dom  S  \  { x } ) )  ->  ( ( S `  x )  C_  ( (Cntz `  G
) `  ( S `  y ) )  <->  ( S `  x )  C_  (
(Cntz `  H ) `  ( S `  y
) ) ) )
3433ralbidva 2818 . . . . . . . . 9  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
)  <->  A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  H ) `  ( S `  y )
) ) )
35 subgrcl 16323 . . . . . . . . . . . . . . 15  |-  ( A  e.  (SubGrp `  G
)  ->  G  e.  Grp )
3635ad2antrr 723 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  G  e.  Grp )
37 eqid 2382 . . . . . . . . . . . . . . 15  |-  ( Base `  G )  =  (
Base `  G )
3837subgacs 16353 . . . . . . . . . . . . . 14  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
39 acsmre 15059 . . . . . . . . . . . . . 14  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
4036, 38, 393syl 20 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G ) ) )
4112subggrp 16321 . . . . . . . . . . . . . . . 16  |-  ( A  e.  (SubGrp `  G
)  ->  H  e.  Grp )
4241ad2antrr 723 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  H  e.  Grp )
439subgacs 16353 . . . . . . . . . . . . . . 15  |-  ( H  e.  Grp  ->  (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) ) )
44 acsmre 15059 . . . . . . . . . . . . . . 15  |-  ( (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) )  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H
) ) )
4542, 43, 443syl 20 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H ) ) )
46 eqid 2382 . . . . . . . . . . . . . 14  |-  (mrCls `  (SubGrp `  H ) )  =  (mrCls `  (SubGrp `  H ) )
47 imassrn 5260 . . . . . . . . . . . . . . . . 17  |-  ( S
" ( dom  S  \  { x } ) )  C_  ran  S
48 frn 5645 . . . . . . . . . . . . . . . . . 18  |-  ( S : dom  S --> (SubGrp `  H )  ->  ran  S 
C_  (SubGrp `  H )
)
4948ad2antlr 724 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ran  S  C_  (SubGrp `  H ) )
5047, 49syl5ss 3428 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( S "
( dom  S  \  {
x } ) ) 
C_  (SubGrp `  H )
)
51 mresspw 14999 . . . . . . . . . . . . . . . . 17  |-  ( (SubGrp `  H )  e.  (Moore `  ( Base `  H
) )  ->  (SubGrp `  H )  C_  ~P ( Base `  H )
)
5245, 51syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  (SubGrp `  H )  C_ 
~P ( Base `  H
) )
5350, 52sstrd 3427 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( S "
( dom  S  \  {
x } ) ) 
C_  ~P ( Base `  H
) )
54 sspwuni 4332 . . . . . . . . . . . . . . 15  |-  ( ( S " ( dom 
S  \  { x } ) )  C_  ~P ( Base `  H
)  <->  U. ( S "
( dom  S  \  {
x } ) ) 
C_  ( Base `  H
) )
5553, 54sylib 196 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  U. ( S "
( dom  S  \  {
x } ) ) 
C_  ( Base `  H
) )
5645, 46, 55mrcssidd 15032 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  U. ( S "
( dom  S  \  {
x } ) ) 
C_  ( (mrCls `  (SubGrp `  H ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) ) )
5746mrccl 15018 . . . . . . . . . . . . . . . 16  |-  ( ( (SubGrp `  H )  e.  (Moore `  ( Base `  H ) )  /\  U. ( S " ( dom  S  \  { x } ) )  C_  ( Base `  H )
)  ->  ( (mrCls `  (SubGrp `  H )
) `  U. ( S
" ( dom  S  \  { x } ) ) )  e.  (SubGrp `  H ) )
5845, 55, 57syl2anc 659 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( (mrCls `  (SubGrp `  H ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) )  e.  (SubGrp `  H ) )
5912subsubg 16341 . . . . . . . . . . . . . . . 16  |-  ( A  e.  (SubGrp `  G
)  ->  ( (
(mrCls `  (SubGrp `  H
) ) `  U. ( S " ( dom 
S  \  { x } ) ) )  e.  (SubGrp `  H
)  <->  ( ( (mrCls `  (SubGrp `  H )
) `  U. ( S
" ( dom  S  \  { x } ) ) )  e.  (SubGrp `  G )  /\  (
(mrCls `  (SubGrp `  H
) ) `  U. ( S " ( dom 
S  \  { x } ) ) ) 
C_  A ) ) )
6059ad2antrr 723 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( ( (mrCls `  (SubGrp `  H )
) `  U. ( S
" ( dom  S  \  { x } ) ) )  e.  (SubGrp `  H )  <->  ( (
(mrCls `  (SubGrp `  H
) ) `  U. ( S " ( dom 
S  \  { x } ) ) )  e.  (SubGrp `  G
)  /\  ( (mrCls `  (SubGrp `  H )
) `  U. ( S
" ( dom  S  \  { x } ) ) )  C_  A
) ) )
6158, 60mpbid 210 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( ( (mrCls `  (SubGrp `  H )
) `  U. ( S
" ( dom  S  \  { x } ) ) )  e.  (SubGrp `  G )  /\  (
(mrCls `  (SubGrp `  H
) ) `  U. ( S " ( dom 
S  \  { x } ) ) ) 
C_  A ) )
6261simpld 457 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( (mrCls `  (SubGrp `  H ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) )  e.  (SubGrp `  G ) )
63 eqid 2382 . . . . . . . . . . . . . 14  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
6463mrcsscl 15027 . . . . . . . . . . . . 13  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U. ( S " ( dom  S  \  { x } ) )  C_  ( (mrCls `  (SubGrp `  H
) ) `  U. ( S " ( dom 
S  \  { x } ) ) )  /\  ( (mrCls `  (SubGrp `  H ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) )  e.  (SubGrp `  G ) )  -> 
( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) 
C_  ( (mrCls `  (SubGrp `  H ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) ) )
6540, 56, 62, 64syl3anc 1226 . . . . . . . . . . . 12  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) )  C_  (
(mrCls `  (SubGrp `  H
) ) `  U. ( S " ( dom 
S  \  { x } ) ) ) )
6613ad2antrr 723 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  A  =  (
Base `  H )
)
6755, 66sseqtr4d 3454 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  U. ( S "
( dom  S  \  {
x } ) ) 
C_  A )
6837subgss 16319 . . . . . . . . . . . . . . . 16  |-  ( A  e.  (SubGrp `  G
)  ->  A  C_  ( Base `  G ) )
6968ad2antrr 723 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  A  C_  ( Base `  G ) )
7067, 69sstrd 3427 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  U. ( S "
( dom  S  \  {
x } ) ) 
C_  ( Base `  G
) )
7140, 63, 70mrcssidd 15032 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  U. ( S "
( dom  S  \  {
x } ) ) 
C_  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) ) )
7263mrccl 15018 . . . . . . . . . . . . . . 15  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U. ( S " ( dom  S  \  { x } ) )  C_  ( Base `  G )
)  ->  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( dom  S  \  { x } ) ) )  e.  (SubGrp `  G ) )
7340, 70, 72syl2anc 659 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) )  e.  (SubGrp `  G ) )
74 simpll 751 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  A  e.  (SubGrp `  G ) )
7563mrcsscl 15027 . . . . . . . . . . . . . . 15  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U. ( S " ( dom  S  \  { x } ) )  C_  A  /\  A  e.  (SubGrp `  G ) )  -> 
( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) 
C_  A )
7640, 67, 74, 75syl3anc 1226 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) )  C_  A
)
7712subsubg 16341 . . . . . . . . . . . . . . 15  |-  ( A  e.  (SubGrp `  G
)  ->  ( (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( dom 
S  \  { x } ) ) )  e.  (SubGrp `  H
)  <->  ( ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( dom  S  \  { x } ) ) )  e.  (SubGrp `  G )  /\  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( dom 
S  \  { x } ) ) ) 
C_  A ) ) )
7877ad2antrr 723 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( dom  S  \  { x } ) ) )  e.  (SubGrp `  H )  <->  ( (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( dom 
S  \  { x } ) ) )  e.  (SubGrp `  G
)  /\  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( dom  S  \  { x } ) ) )  C_  A
) ) )
7973, 76, 78mpbir2and 920 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) )  e.  (SubGrp `  H ) )
8046mrcsscl 15027 . . . . . . . . . . . . 13  |-  ( ( (SubGrp `  H )  e.  (Moore `  ( Base `  H ) )  /\  U. ( S " ( dom  S  \  { x } ) )  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ( S " ( dom 
S  \  { x } ) ) )  /\  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) )  e.  (SubGrp `  H ) )  -> 
( (mrCls `  (SubGrp `  H ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) 
C_  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) ) )
8145, 71, 79, 80syl3anc 1226 . . . . . . . . . . . 12  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( (mrCls `  (SubGrp `  H ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) )  C_  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( dom 
S  \  { x } ) ) ) )
8265, 81eqssd 3434 . . . . . . . . . . 11  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) )  =  ( (mrCls `  (SubGrp `  H
) ) `  U. ( S " ( dom 
S  \  { x } ) ) ) )
8382ineq2d 3614 . . . . . . . . . 10  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( ( S `
 x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  H ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) ) ) )
84 eqid 2382 . . . . . . . . . . . . 13  |-  ( 0g
`  G )  =  ( 0g `  G
)
8512, 84subg0 16324 . . . . . . . . . . . 12  |-  ( A  e.  (SubGrp `  G
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
8685ad2antrr 723 . . . . . . . . . . 11  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( 0g `  G )  =  ( 0g `  H ) )
8786sneqd 3956 . . . . . . . . . 10  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  { ( 0g
`  G ) }  =  { ( 0g
`  H ) } )
8883, 87eqeq12d 2404 . . . . . . . . 9  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) }  <->  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  H ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  H ) } ) )
8934, 88anbi12d 708 . . . . . . . 8  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( ( A. y  e.  ( dom  S 
\  { x }
) ( S `  x )  C_  (
(Cntz `  G ) `  ( S `  y
) )  /\  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } )  <-> 
( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  H ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  H ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  H ) } ) ) )
9089ralbidva 2818 . . . . . . 7  |-  ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  -> 
( A. x  e. 
dom  S ( A. y  e.  ( dom  S 
\  { x }
) ( S `  x )  C_  (
(Cntz `  G ) `  ( S `  y
) )  /\  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } )  <->  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  H ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  H ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  H ) } ) ) )
9190pm5.32da 639 . . . . . 6  |-  ( A  e.  (SubGrp `  G
)  ->  ( ( S : dom  S --> (SubGrp `  H )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) )  <->  ( S : dom  S --> (SubGrp `  H )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  H ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  H ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  H ) } ) ) ) )
9212subsubg 16341 . . . . . . . . . . . . 13  |-  ( A  e.  (SubGrp `  G
)  ->  ( x  e.  (SubGrp `  H )  <->  ( x  e.  (SubGrp `  G )  /\  x  C_  A ) ) )
93 elin 3601 . . . . . . . . . . . . . 14  |-  ( x  e.  ( (SubGrp `  G )  i^i  ~P A )  <->  ( x  e.  (SubGrp `  G )  /\  x  e.  ~P A ) )
94 selpw 3934 . . . . . . . . . . . . . . 15  |-  ( x  e.  ~P A  <->  x  C_  A
)
9594anbi2i 692 . . . . . . . . . . . . . 14  |-  ( ( x  e.  (SubGrp `  G )  /\  x  e.  ~P A )  <->  ( x  e.  (SubGrp `  G )  /\  x  C_  A ) )
9693, 95bitri 249 . . . . . . . . . . . . 13  |-  ( x  e.  ( (SubGrp `  G )  i^i  ~P A )  <->  ( x  e.  (SubGrp `  G )  /\  x  C_  A ) )
9792, 96syl6bbr 263 . . . . . . . . . . . 12  |-  ( A  e.  (SubGrp `  G
)  ->  ( x  e.  (SubGrp `  H )  <->  x  e.  ( (SubGrp `  G )  i^i  ~P A ) ) )
9897eqrdv 2379 . . . . . . . . . . 11  |-  ( A  e.  (SubGrp `  G
)  ->  (SubGrp `  H
)  =  ( (SubGrp `  G )  i^i  ~P A ) )
9998sseq2d 3445 . . . . . . . . . 10  |-  ( A  e.  (SubGrp `  G
)  ->  ( ran  S 
C_  (SubGrp `  H )  <->  ran 
S  C_  ( (SubGrp `  G )  i^i  ~P A ) ) )
100 ssin 3634 . . . . . . . . . 10  |-  ( ( ran  S  C_  (SubGrp `  G )  /\  ran  S 
C_  ~P A )  <->  ran  S  C_  ( (SubGrp `  G )  i^i  ~P A ) )
10199, 100syl6bbr 263 . . . . . . . . 9  |-  ( A  e.  (SubGrp `  G
)  ->  ( ran  S 
C_  (SubGrp `  H )  <->  ( ran  S  C_  (SubGrp `  G )  /\  ran  S 
C_  ~P A ) ) )
102101anbi2d 701 . . . . . . . 8  |-  ( A  e.  (SubGrp `  G
)  ->  ( ( S  Fn  dom  S  /\  ran  S  C_  (SubGrp `  H
) )  <->  ( S  Fn  dom  S  /\  ( ran  S  C_  (SubGrp `  G
)  /\  ran  S  C_  ~P A ) ) ) )
103 df-f 5500 . . . . . . . 8  |-  ( S : dom  S --> (SubGrp `  H )  <->  ( S  Fn  dom  S  /\  ran  S 
C_  (SubGrp `  H )
) )
104 df-f 5500 . . . . . . . . . 10  |-  ( S : dom  S --> (SubGrp `  G )  <->  ( S  Fn  dom  S  /\  ran  S 
C_  (SubGrp `  G )
) )
105104anbi1i 693 . . . . . . . . 9  |-  ( ( S : dom  S --> (SubGrp `  G )  /\  ran  S  C_  ~P A
)  <->  ( ( S  Fn  dom  S  /\  ran  S  C_  (SubGrp `  G
) )  /\  ran  S 
C_  ~P A ) )
106 anass 647 . . . . . . . . 9  |-  ( ( ( S  Fn  dom  S  /\  ran  S  C_  (SubGrp `  G ) )  /\  ran  S  C_  ~P A )  <->  ( S  Fn  dom  S  /\  ( ran  S  C_  (SubGrp `  G
)  /\  ran  S  C_  ~P A ) ) )
107105, 106bitri 249 . . . . . . . 8  |-  ( ( S : dom  S --> (SubGrp `  G )  /\  ran  S  C_  ~P A
)  <->  ( S  Fn  dom  S  /\  ( ran 
S  C_  (SubGrp `  G
)  /\  ran  S  C_  ~P A ) ) )
108102, 103, 1073bitr4g 288 . . . . . . 7  |-  ( A  e.  (SubGrp `  G
)  ->  ( S : dom  S --> (SubGrp `  H )  <->  ( S : dom  S --> (SubGrp `  G )  /\  ran  S 
C_  ~P A ) ) )
109108anbi1d 702 . . . . . 6  |-  ( A  e.  (SubGrp `  G
)  ->  ( ( S : dom  S --> (SubGrp `  H )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) )  <->  ( ( S : dom  S --> (SubGrp `  G )  /\  ran  S 
C_  ~P A )  /\  A. x  e.  dom  S
( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) )
11091, 109bitr3d 255 . . . . 5  |-  ( A  e.  (SubGrp `  G
)  ->  ( ( S : dom  S --> (SubGrp `  H )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  H ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  H ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  H ) } ) )  <->  ( ( S : dom  S --> (SubGrp `  G )  /\  ran  S 
C_  ~P A )  /\  A. x  e.  dom  S
( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) )
111110adantr 463 . . . 4  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  (
( S : dom  S --> (SubGrp `  H )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  H ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  H ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  H ) } ) )  <->  ( ( S : dom  S --> (SubGrp `  G )  /\  ran  S 
C_  ~P A )  /\  A. x  e.  dom  S
( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) )
112 dmexg 6630 . . . . . 6  |-  ( S  e.  _V  ->  dom  S  e.  _V )
113112adantl 464 . . . . 5  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  dom  S  e.  _V )
114 eqidd 2383 . . . . 5  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  dom  S  =  dom  S )
11541adantr 463 . . . . 5  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  H  e.  Grp )
116 eqid 2382 . . . . . . . 8  |-  ( 0g
`  H )  =  ( 0g `  H
)
11726, 116, 46dmdprd 17142 . . . . . . 7  |-  ( ( dom  S  e.  _V  /\ 
dom  S  =  dom  S )  ->  ( H dom DProd  S  <->  ( H  e. 
Grp  /\  S : dom  S --> (SubGrp `  H )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  H ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  H ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  H ) } ) ) ) )
118 3anass 975 . . . . . . 7  |-  ( ( H  e.  Grp  /\  S : dom  S --> (SubGrp `  H )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  H ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  H ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  H ) } ) )  <->  ( H  e.  Grp  /\  ( S : dom  S --> (SubGrp `  H )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  H ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  H ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  H ) } ) ) ) )
119117, 118syl6bb 261 . . . . . 6  |-  ( ( dom  S  e.  _V  /\ 
dom  S  =  dom  S )  ->  ( H dom DProd  S  <->  ( H  e. 
Grp  /\  ( S : dom  S --> (SubGrp `  H )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  H ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  H ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  H ) } ) ) ) ) )
120119baibd 907 . . . . 5  |-  ( ( ( dom  S  e. 
_V  /\  dom  S  =  dom  S )  /\  H  e.  Grp )  ->  ( H dom DProd  S  <->  ( S : dom  S --> (SubGrp `  H )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  H ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  H ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  H ) } ) ) ) )
121113, 114, 115, 120syl21anc 1225 . . . 4  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  ( H dom DProd  S  <->  ( S : dom  S --> (SubGrp `  H )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  H ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  H ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  H ) } ) ) ) )
12235adantr 463 . . . . . . 7  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  G  e.  Grp )
12325, 84, 63dmdprd 17142 . . . . . . . . 9  |-  ( ( dom  S  e.  _V  /\ 
dom  S  =  dom  S )  ->  ( G dom DProd  S  <->  ( G  e. 
Grp  /\  S : dom  S --> (SubGrp `  G )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) )
124 3anass 975 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  S : dom  S --> (SubGrp `  G )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) )  <->  ( G  e.  Grp  /\  ( S : dom  S --> (SubGrp `  G )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) )
125123, 124syl6bb 261 . . . . . . . 8  |-  ( ( dom  S  e.  _V  /\ 
dom  S  =  dom  S )  ->  ( G dom DProd  S  <->  ( G  e. 
Grp  /\  ( S : dom  S --> (SubGrp `  G )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) ) )
126125baibd 907 . . . . . . 7  |-  ( ( ( dom  S  e. 
_V  /\  dom  S  =  dom  S )  /\  G  e.  Grp )  ->  ( G dom DProd  S  <->  ( S : dom  S --> (SubGrp `  G )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) )
127113, 114, 122, 126syl21anc 1225 . . . . . 6  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  ( G dom DProd  S  <->  ( S : dom  S --> (SubGrp `  G )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) )
128127anbi1d 702 . . . . 5  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  (
( G dom DProd  S  /\  ran  S  C_  ~P A
)  <->  ( ( S : dom  S --> (SubGrp `  G )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) )  /\  ran  S  C_  ~P A
) ) )
129 an32 796 . . . . 5  |-  ( ( ( S : dom  S --> (SubGrp `  G )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) )  /\  ran  S  C_  ~P A
)  <->  ( ( S : dom  S --> (SubGrp `  G )  /\  ran  S 
C_  ~P A )  /\  A. x  e.  dom  S
( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) )
130128, 129syl6bb 261 . . . 4  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  (
( G dom DProd  S  /\  ran  S  C_  ~P A
)  <->  ( ( S : dom  S --> (SubGrp `  G )  /\  ran  S 
C_  ~P A )  /\  A. x  e.  dom  S
( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) )
131111, 121, 1303bitr4d 285 . . 3  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  ( H dom DProd  S  <->  ( G dom DProd  S  /\  ran  S  C_  ~P A ) ) )
132131ex 432 . 2  |-  ( A  e.  (SubGrp `  G
)  ->  ( S  e.  _V  ->  ( H dom DProd  S  <->  ( G dom DProd  S  /\  ran  S  C_  ~P A ) ) ) )
1333, 6, 132pm5.21ndd 352 1  |-  ( A  e.  (SubGrp `  G
)  ->  ( H dom DProd  S  <->  ( G dom DProd  S  /\  ran  S  C_  ~P A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826   A.wral 2732   _Vcvv 3034    \ cdif 3386    i^i cin 3388    C_ wss 3389   ~Pcpw 3927   {csn 3944   U.cuni 4163   class class class wbr 4367   dom cdm 4913   ran crn 4914   "cima 4916    Fn wfn 5491   -->wf 5492   ` cfv 5496  (class class class)co 6196   Basecbs 14634   ↾s cress 14635   0gc0g 14847  Moorecmre 14989  mrClscmrc 14990  ACScacs 14992   Grpcgrp 16170  SubGrpcsubg 16312  Cntzccntz 16470   DProd cdprd 17137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-ixp 7389  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-0g 14849  df-mre 14993  df-mrc 14994  df-acs 14996  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-submnd 16084  df-grp 16174  df-minusg 16175  df-subg 16315  df-cntz 16472  df-dprd 17139
This theorem is referenced by:  subgdprd  17195  ablfaclem3  17251
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