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Theorem subgdmdprd 15269
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 15235 . . . 4  |-  Rel  dom DProd
21brrelex2i 4730 . . 3  |-  ( H dom DProd  S  ->  S  e. 
_V )
32a1i 10 . 2  |-  ( A  e.  (SubGrp `  G
)  ->  ( H dom DProd  S  ->  S  e.  _V ) )
41brrelex2i 4730 . . . 4  |-  ( G dom DProd  S  ->  S  e. 
_V )
54adantr 451 . . 3  |-  ( ( G dom DProd  S  /\  ran  S  C_  ~P A
)  ->  S  e.  _V )
65a1i 10 . 2  |-  ( A  e.  (SubGrp `  G
)  ->  ( ( G dom DProd  S  /\  ran  S  C_ 
~P A )  ->  S  e.  _V )
)
7 ffvelrn 5663 . . . . . . . . . . . . . . . 16  |-  ( ( S : dom  S --> (SubGrp `  H )  /\  x  e.  dom  S )  ->  ( S `  x )  e.  (SubGrp `  H ) )
87ad2ant2lr 728 . . . . . . . . . . . . . . 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 2283 . . . . . . . . . . . . . . . 16  |-  ( Base `  H )  =  (
Base `  H )
109subgss 14622 . . . . . . . . . . . . . . 15  |-  ( ( S `  x )  e.  (SubGrp `  H
)  ->  ( S `  x )  C_  ( Base `  H ) )
118, 10syl 15 . . . . . . . . . . . . . 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 14625 . . . . . . . . . . . . . . 15  |-  ( A  e.  (SubGrp `  G
)  ->  A  =  ( Base `  H )
)
1413ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  A  =  ( Base `  H )
)
1511, 14sseqtr4d 3215 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( S `  x )  C_  A
)
1615biantrud 493 . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . 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 731 . . . . . . . . . . . . . . . . . 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 3298 . . . . . . . . . . . . . . . . . . 19  |-  ( y  e.  ( dom  S  \  { x } )  ->  y  e.  dom  S )
2019ad2antll 709 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  y  e.  dom  S )
21 ffvelrn 5663 . . . . . . . . . . . . . . . . . 18  |-  ( ( S : dom  S --> (SubGrp `  H )  /\  y  e.  dom  S )  ->  ( S `  y )  e.  (SubGrp `  H ) )
2218, 20, 21syl2anc 642 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( S `  y )  e.  (SubGrp `  H ) )
239subgss 14622 . . . . . . . . . . . . . . . . 17  |-  ( ( S `  y )  e.  (SubGrp `  H
)  ->  ( S `  y )  C_  ( Base `  H ) )
2422, 23syl 15 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( S `  y )  C_  ( Base `  H ) )
2524, 14sseqtr4d 3215 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( S `  y )  C_  A
)
26 eqid 2283 . . . . . . . . . . . . . . . 16  |-  (Cntz `  G )  =  (Cntz `  G )
27 eqid 2283 . . . . . . . . . . . . . . . 16  |-  (Cntz `  H )  =  (Cntz `  H )
2812, 26, 27resscntz 14807 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  (SubGrp `  G )  /\  ( S `  y )  C_  A )  ->  (
(Cntz `  H ) `  ( S `  y
) )  =  ( ( (Cntz `  G
) `  ( S `  y ) )  i^i 
A ) )
2917, 25, 28syl2anc 642 . . . . . . . . . . . . . 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
) )
3029sseq2d 3206 . . . . . . . . . . . . 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 ) ) )
31 ssin 3391 . . . . . . . . . . . . 13  |-  ( ( ( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( S `  x )  C_  A
)  <->  ( S `  x )  C_  (
( (Cntz `  G
) `  ( S `  y ) )  i^i 
A ) )
3230, 31syl6bbr 254 . . . . . . . . . . . 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 ) ) )
3316, 32bitr4d 247 . . . . . . . . . . 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
) ) ) )
3433anassrs 629 . . . . . . . . . 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
) ) ) )
3534ralbidva 2559 . . . . . . . . 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 )
) ) )
36 subgrcl 14626 . . . . . . . . . . . . . . 15  |-  ( A  e.  (SubGrp `  G
)  ->  G  e.  Grp )
3736ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  G  e.  Grp )
38 eqid 2283 . . . . . . . . . . . . . . 15  |-  ( Base `  G )  =  (
Base `  G )
3938subgacs 14652 . . . . . . . . . . . . . 14  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
40 acsmre 13554 . . . . . . . . . . . . . 14  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
4137, 39, 403syl 18 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G ) ) )
4212subggrp 14624 . . . . . . . . . . . . . . . 16  |-  ( A  e.  (SubGrp `  G
)  ->  H  e.  Grp )
4342ad2antrr 706 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  H  e.  Grp )
449subgacs 14652 . . . . . . . . . . . . . . 15  |-  ( H  e.  Grp  ->  (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) ) )
45 acsmre 13554 . . . . . . . . . . . . . . 15  |-  ( (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) )  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H
) ) )
4643, 44, 453syl 18 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H ) ) )
47 imassrn 5025 . . . . . . . . . . . . . . . . 17  |-  ( S
" ( dom  S  \  { x } ) )  C_  ran  S
48 frn 5395 . . . . . . . . . . . . . . . . . 18  |-  ( S : dom  S --> (SubGrp `  H )  ->  ran  S 
C_  (SubGrp `  H )
)
4948ad2antlr 707 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ran  S  C_  (SubGrp `  H ) )
5047, 49syl5ss 3190 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( S "
( dom  S  \  {
x } ) ) 
C_  (SubGrp `  H )
)
51 mresspw 13494 . . . . . . . . . . . . . . . . 17  |-  ( (SubGrp `  H )  e.  (Moore `  ( Base `  H
) )  ->  (SubGrp `  H )  C_  ~P ( Base `  H )
)
5246, 51syl 15 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  (SubGrp `  H )  C_ 
~P ( Base `  H
) )
5350, 52sstrd 3189 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( S "
( dom  S  \  {
x } ) ) 
C_  ~P ( Base `  H
) )
54 sspwuni 3987 . . . . . . . . . . . . . . 15  |-  ( ( S " ( dom 
S  \  { x } ) )  C_  ~P ( Base `  H
)  <->  U. ( S "
( dom  S  \  {
x } ) ) 
C_  ( Base `  H
) )
5553, 54sylib 188 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  U. ( S "
( dom  S  \  {
x } ) ) 
C_  ( Base `  H
) )
56 eqid 2283 . . . . . . . . . . . . . . 15  |-  (mrCls `  (SubGrp `  H ) )  =  (mrCls `  (SubGrp `  H ) )
5756mrcssid 13519 . . . . . . . . . . . . . 14  |-  ( ( (SubGrp `  H )  e.  (Moore `  ( Base `  H ) )  /\  U. ( S " ( dom  S  \  { x } ) )  C_  ( Base `  H )
)  ->  U. ( S " ( dom  S  \  { x } ) )  C_  ( (mrCls `  (SubGrp `  H )
) `  U. ( S
" ( dom  S  \  { x } ) ) ) )
5846, 55, 57syl2anc 642 . . . . . . . . . . . . 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 } ) ) ) )
5956mrccl 13513 . . . . . . . . . . . . . . . 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 ) )
6046, 55, 59syl2anc 642 . . . . . . . . . . . . . . 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 ) )
6112subsubg 14640 . . . . . . . . . . . . . . . 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 ) ) )
6261ad2antrr 706 . . . . . . . . . . . . . . 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
) ) )
6360, 62mpbid 201 . . . . . . . . . . . . . 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 ) )
6463simpld 445 . . . . . . . . . . . . 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 ) )
65 eqid 2283 . . . . . . . . . . . . . 14  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
6665mrcsscl 13522 . . . . . . . . . . . . 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 } ) ) ) )
6741, 58, 64, 66syl3anc 1182 . . . . . . . . . . . 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 } ) ) ) )
6813ad2antrr 706 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  A  =  (
Base `  H )
)
6955, 68sseqtr4d 3215 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  U. ( S "
( dom  S  \  {
x } ) ) 
C_  A )
7038subgss 14622 . . . . . . . . . . . . . . . 16  |-  ( A  e.  (SubGrp `  G
)  ->  A  C_  ( Base `  G ) )
7170ad2antrr 706 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  A  C_  ( Base `  G ) )
7269, 71sstrd 3189 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  U. ( S "
( dom  S  \  {
x } ) ) 
C_  ( Base `  G
) )
7365mrcssid 13519 . . . . . . . . . . . . . 14  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U. ( S " ( dom  S  \  { x } ) )  C_  ( Base `  G )
)  ->  U. ( S " ( dom  S  \  { x } ) )  C_  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( dom  S  \  { x } ) ) ) )
7441, 72, 73syl2anc 642 . . . . . . . . . . . . 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 } ) ) ) )
7565mrccl 13513 . . . . . . . . . . . . . . 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 ) )
7641, 72, 75syl2anc 642 . . . . . . . . . . . . . 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 ) )
77 simpll 730 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  A  e.  (SubGrp `  G ) )
7865mrcsscl 13522 . . . . . . . . . . . . . . 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 )
7941, 69, 77, 78syl3anc 1182 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) )  C_  A
)
8012subsubg 14640 . . . . . . . . . . . . . . 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 ) ) )
8180ad2antrr 706 . . . . . . . . . . . . . 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
) ) )
8276, 79, 81mpbir2and 888 . . . . . . . . . . . . 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 ) )
8356mrcsscl 13522 . . . . . . . . . . . . 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 } ) ) ) )
8446, 74, 82, 83syl3anc 1182 . . . . . . . . . . . 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 } ) ) ) )
8567, 84eqssd 3196 . . . . . . . . . . 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 } ) ) ) )
8685ineq2d 3370 . . . . . . . . . 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 } ) ) ) ) )
87 eqid 2283 . . . . . . . . . . . . 13  |-  ( 0g
`  G )  =  ( 0g `  G
)
8812, 87subg0 14627 . . . . . . . . . . . 12  |-  ( A  e.  (SubGrp `  G
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
8988ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( 0g `  G )  =  ( 0g `  H ) )
9089sneqd 3653 . . . . . . . . . 10  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  { ( 0g
`  G ) }  =  { ( 0g
`  H ) } )
9186, 90eqeq12d 2297 . . . . . . . . 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 ) } ) )
9235, 91anbi12d 691 . . . . . . . 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 ) } ) ) )
9392ralbidva 2559 . . . . . . 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 ) } ) ) )
9493pm5.32da 622 . . . . . 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 ) } ) ) ) )
9512subsubg 14640 . . . . . . . . . . . . 13  |-  ( A  e.  (SubGrp `  G
)  ->  ( x  e.  (SubGrp `  H )  <->  ( x  e.  (SubGrp `  G )  /\  x  C_  A ) ) )
96 elin 3358 . . . . . . . . . . . . . 14  |-  ( x  e.  ( (SubGrp `  G )  i^i  ~P A )  <->  ( x  e.  (SubGrp `  G )  /\  x  e.  ~P A ) )
97 vex 2791 . . . . . . . . . . . . . . . 16  |-  x  e. 
_V
9897elpw 3631 . . . . . . . . . . . . . . 15  |-  ( x  e.  ~P A  <->  x  C_  A
)
9998anbi2i 675 . . . . . . . . . . . . . 14  |-  ( ( x  e.  (SubGrp `  G )  /\  x  e.  ~P A )  <->  ( x  e.  (SubGrp `  G )  /\  x  C_  A ) )
10096, 99bitri 240 . . . . . . . . . . . . 13  |-  ( x  e.  ( (SubGrp `  G )  i^i  ~P A )  <->  ( x  e.  (SubGrp `  G )  /\  x  C_  A ) )
10195, 100syl6bbr 254 . . . . . . . . . . . 12  |-  ( A  e.  (SubGrp `  G
)  ->  ( x  e.  (SubGrp `  H )  <->  x  e.  ( (SubGrp `  G )  i^i  ~P A ) ) )
102101eqrdv 2281 . . . . . . . . . . 11  |-  ( A  e.  (SubGrp `  G
)  ->  (SubGrp `  H
)  =  ( (SubGrp `  G )  i^i  ~P A ) )
103102sseq2d 3206 . . . . . . . . . 10  |-  ( A  e.  (SubGrp `  G
)  ->  ( ran  S 
C_  (SubGrp `  H )  <->  ran 
S  C_  ( (SubGrp `  G )  i^i  ~P A ) ) )
104 ssin 3391 . . . . . . . . . 10  |-  ( ( ran  S  C_  (SubGrp `  G )  /\  ran  S 
C_  ~P A )  <->  ran  S  C_  ( (SubGrp `  G )  i^i  ~P A ) )
105103, 104syl6bbr 254 . . . . . . . . 9  |-  ( A  e.  (SubGrp `  G
)  ->  ( ran  S 
C_  (SubGrp `  H )  <->  ( ran  S  C_  (SubGrp `  G )  /\  ran  S 
C_  ~P A ) ) )
106105anbi2d 684 . . . . . . . 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 ) ) ) )
107 df-f 5259 . . . . . . . 8  |-  ( S : dom  S --> (SubGrp `  H )  <->  ( S  Fn  dom  S  /\  ran  S 
C_  (SubGrp `  H )
) )
108 df-f 5259 . . . . . . . . . 10  |-  ( S : dom  S --> (SubGrp `  G )  <->  ( S  Fn  dom  S  /\  ran  S 
C_  (SubGrp `  G )
) )
109108anbi1i 676 . . . . . . . . 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 ) )
110 anass 630 . . . . . . . . 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 ) ) )
111109, 110bitri 240 . . . . . . . 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 ) ) )
112106, 107, 1113bitr4g 279 . . . . . . 7  |-  ( A  e.  (SubGrp `  G
)  ->  ( S : dom  S --> (SubGrp `  H )  <->  ( S : dom  S --> (SubGrp `  G )  /\  ran  S 
C_  ~P A ) ) )
113112anbi1d 685 . . . . . 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 ) } ) ) ) )
11494, 113bitr3d 246 . . . . 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 ) } ) ) ) )
115114adantr 451 . . . 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 ) } ) ) ) )
116 dmexg 4939 . . . . . 6  |-  ( S  e.  _V  ->  dom  S  e.  _V )
117116adantl 452 . . . . 5  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  dom  S  e.  _V )
118 eqidd 2284 . . . . 5  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  dom  S  =  dom  S )
11942adantr 451 . . . . 5  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  H  e.  Grp )
120 eqid 2283 . . . . . . . 8  |-  ( 0g
`  H )  =  ( 0g `  H
)
12127, 120, 56dmdprd 15236 . . . . . . 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 ) } ) ) ) )
122 3anass 938 . . . . . . 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 ) } ) ) ) )
123121, 122syl6bb 252 . . . . . 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 ) } ) ) ) ) )
124123baibd 875 . . . . 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 ) } ) ) ) )
125117, 118, 119, 124syl21anc 1181 . . . 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 ) } ) ) ) )
12636adantr 451 . . . . . . 7  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  G  e.  Grp )
12726, 87, 65dmdprd 15236 . . . . . . . . 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 ) } ) ) ) )
128 3anass 938 . . . . . . . . 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 ) } ) ) ) )
129127, 128syl6bb 252 . . . . . . . 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 ) } ) ) ) ) )
130129baibd 875 . . . . . . 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 ) } ) ) ) )
131117, 118, 126, 130syl21anc 1181 . . . . . 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 ) } ) ) ) )
132131anbi1d 685 . . . . 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
) ) )
133 an32 773 . . . . 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 ) } ) ) )
134132, 133syl6bb 252 . . . 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 ) } ) ) ) )
135115, 125, 1343bitr4d 276 . . 3  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  ( H dom DProd  S  <->  ( G dom DProd  S  /\  ran  S  C_  ~P A ) ) )
136135ex 423 . 2  |-  ( A  e.  (SubGrp `  G
)  ->  ( S  e.  _V  ->  ( H dom DProd  S  <->  ( G dom DProd  S  /\  ran  S  C_  ~P A ) ) ) )
1373, 6, 136pm5.21ndd 343 1  |-  ( A  e.  (SubGrp `  G
)  ->  ( H dom DProd  S  <->  ( G dom DProd  S  /\  ran  S  C_  ~P A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   {csn 3640   U.cuni 3827   class class class wbr 4023   dom cdm 4689   ran crn 4690   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149   0gc0g 13400  Moorecmre 13484  mrClscmrc 13485  ACScacs 13487   Grpcgrp 14362  SubGrpcsubg 14615  Cntzccntz 14791   DProd cdprd 15231
This theorem is referenced by:  subgdprd  15270  ablfaclem3  15322
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-subg 14618  df-cntz 14793  df-dprd 15233
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