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Theorem dprdres 17273
Description: Restriction of a direct product (dropping factors). (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
dprdres.1  |-  ( ph  ->  G dom DProd  S )
dprdres.2  |-  ( ph  ->  dom  S  =  I )
dprdres.3  |-  ( ph  ->  A  C_  I )
Assertion
Ref Expression
dprdres  |-  ( ph  ->  ( G dom DProd  ( S  |`  A )  /\  ( G DProd  ( S  |`  A ) )  C_  ( G DProd  S ) ) )

Proof of Theorem dprdres
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dprdres.1 . . . 4  |-  ( ph  ->  G dom DProd  S )
2 dprdgrp 17236 . . . 4  |-  ( G dom DProd  S  ->  G  e. 
Grp )
31, 2syl 16 . . 3  |-  ( ph  ->  G  e.  Grp )
4 dprdres.2 . . . . 5  |-  ( ph  ->  dom  S  =  I )
51, 4dprdf2 17238 . . . 4  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
6 dprdres.3 . . . 4  |-  ( ph  ->  A  C_  I )
75, 6fssresd 5734 . . 3  |-  ( ph  ->  ( S  |`  A ) : A --> (SubGrp `  G ) )
81ad2antrr 723 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  G dom DProd  S )
94ad2antrr 723 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  dom  S  =  I )
106ad2antrr 723 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  A  C_  I
)
11 simplr 753 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  x  e.  A
)
1210, 11sseldd 3490 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  x  e.  I
)
13 eldifi 3612 . . . . . . . . . 10  |-  ( y  e.  ( A  \  { x } )  ->  y  e.  A
)
1413adantl 464 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  y  e.  A
)
1510, 14sseldd 3490 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  y  e.  I
)
16 eldifsni 4142 . . . . . . . . . 10  |-  ( y  e.  ( A  \  { x } )  ->  y  =/=  x
)
1716adantl 464 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  y  =/=  x
)
1817necomd 2725 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  x  =/=  y
)
19 eqid 2454 . . . . . . . 8  |-  (Cntz `  G )  =  (Cntz `  G )
208, 9, 12, 15, 18, 19dprdcntz 17239 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  ( S `  x )  C_  (
(Cntz `  G ) `  ( S `  y
) ) )
21 fvres 5862 . . . . . . . 8  |-  ( x  e.  A  ->  (
( S  |`  A ) `
 x )  =  ( S `  x
) )
2211, 21syl 16 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  ( ( S  |`  A ) `  x
)  =  ( S `
 x ) )
23 fvres 5862 . . . . . . . . 9  |-  ( y  e.  A  ->  (
( S  |`  A ) `
 y )  =  ( S `  y
) )
2414, 23syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  ( ( S  |`  A ) `  y
)  =  ( S `
 y ) )
2524fveq2d 5852 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  ( (Cntz `  G ) `  (
( S  |`  A ) `
 y ) )  =  ( (Cntz `  G ) `  ( S `  y )
) )
2620, 22, 253sstr4d 3532 . . . . . 6  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  ( ( S  |`  A ) `  x
)  C_  ( (Cntz `  G ) `  (
( S  |`  A ) `
 y ) ) )
2726ralrimiva 2868 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  A. y  e.  ( A  \  {
x } ) ( ( S  |`  A ) `
 x )  C_  ( (Cntz `  G ) `  ( ( S  |`  A ) `  y
) ) )
2821adantl 464 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
( S  |`  A ) `
 x )  =  ( S `  x
) )
2928ineq1d 3685 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  (
( ( S  |`  A ) `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )  =  ( ( S `  x )  i^i  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) ) ) )
30 eqid 2454 . . . . . . . . . . . . 13  |-  ( Base `  G )  =  (
Base `  G )
3130subgacs 16438 . . . . . . . . . . . 12  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
32 acsmre 15144 . . . . . . . . . . . 12  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
333, 31, 323syl 20 . . . . . . . . . . 11  |-  ( ph  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G ) ) )
3433adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
35 eqid 2454 . . . . . . . . . 10  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
36 resss 5285 . . . . . . . . . . . . 13  |-  ( S  |`  A )  C_  S
37 imass1 5359 . . . . . . . . . . . . 13  |-  ( ( S  |`  A )  C_  S  ->  ( ( S  |`  A ) "
( A  \  {
x } ) ) 
C_  ( S "
( A  \  {
x } ) ) )
3836, 37ax-mp 5 . . . . . . . . . . . 12  |-  ( ( S  |`  A ) " ( A  \  { x } ) )  C_  ( S " ( A  \  {
x } ) )
396adantr 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  A )  ->  A  C_  I )
40 ssdif 3625 . . . . . . . . . . . . 13  |-  ( A 
C_  I  ->  ( A  \  { x }
)  C_  ( I  \  { x } ) )
41 imass2 5360 . . . . . . . . . . . . 13  |-  ( ( A  \  { x } )  C_  (
I  \  { x } )  ->  ( S " ( A  \  { x } ) )  C_  ( S " ( I  \  {
x } ) ) )
4239, 40, 413syl 20 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  A )  ->  ( S " ( A  \  { x } ) )  C_  ( S " ( I  \  {
x } ) ) )
4338, 42syl5ss 3500 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  (
( S  |`  A )
" ( A  \  { x } ) )  C_  ( S " ( I  \  {
x } ) ) )
4443unissd 4259 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  U. (
( S  |`  A )
" ( A  \  { x } ) )  C_  U. ( S " ( I  \  { x } ) ) )
45 imassrn 5336 . . . . . . . . . . . 12  |-  ( S
" ( I  \  { x } ) )  C_  ran  S
46 frn 5719 . . . . . . . . . . . . . . 15  |-  ( S : I --> (SubGrp `  G )  ->  ran  S 
C_  (SubGrp `  G )
)
475, 46syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  S  C_  (SubGrp `  G ) )
4830subgss 16404 . . . . . . . . . . . . . . . 16  |-  ( x  e.  (SubGrp `  G
)  ->  x  C_  ( Base `  G ) )
49 selpw 4006 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ~P ( Base `  G )  <->  x  C_  ( Base `  G ) )
5048, 49sylibr 212 . . . . . . . . . . . . . . 15  |-  ( x  e.  (SubGrp `  G
)  ->  x  e.  ~P ( Base `  G
) )
5150ssriv 3493 . . . . . . . . . . . . . 14  |-  (SubGrp `  G )  C_  ~P ( Base `  G )
5247, 51syl6ss 3501 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  S  C_  ~P ( Base `  G )
)
5352adantr 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  A )  ->  ran  S 
C_  ~P ( Base `  G
) )
5445, 53syl5ss 3500 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  ( S " ( I  \  { x } ) )  C_  ~P ( Base `  G ) )
55 sspwuni 4404 . . . . . . . . . . 11  |-  ( ( S " ( I 
\  { x }
) )  C_  ~P ( Base `  G )  <->  U. ( S " (
I  \  { x } ) )  C_  ( Base `  G )
)
5654, 55sylib 196 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  U. ( S " ( I  \  { x } ) )  C_  ( Base `  G ) )
5734, 35, 44, 56mrcssd 15116 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) )  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) ) )
58 sslin 3710 . . . . . . . . 9  |-  ( ( (mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) )  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) )  -> 
( ( S `  x )  i^i  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) ) ) 
C_  ( ( S `
 x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) ) )
5957, 58syl 16 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )  C_  ( ( S `  x )  i^i  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) ) ) )
601adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  G dom DProd  S )
614adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  dom  S  =  I )
626sselda 3489 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  I )
63 eqid 2454 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
6460, 61, 62, 63, 35dprddisj 17240 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )  =  { ( 0g `  G ) } )
6559, 64sseqtrd 3525 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )  C_  { ( 0g `  G
) } )
665ffvelrnda 6007 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
6762, 66syldan 468 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  ( S `  x )  e.  (SubGrp `  G )
)
6863subg0cl 16411 . . . . . . . . . 10  |-  ( ( S `  x )  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  ( S `  x ) )
6967, 68syl 16 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  ( 0g `  G )  e.  ( S `  x
) )
7044, 56sstrd 3499 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  U. (
( S  |`  A )
" ( A  \  { x } ) )  C_  ( Base `  G ) )
7135mrccl 15103 . . . . . . . . . . 11  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U. ( ( S  |`  A ) " ( A  \  { x }
) )  C_  ( Base `  G ) )  ->  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( ( S  |`  A ) " ( A  \  { x } ) ) )  e.  (SubGrp `  G ) )
7234, 70, 71syl2anc 659 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) )  e.  (SubGrp `  G )
)
7363subg0cl 16411 . . . . . . . . . 10  |-  ( ( (mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) )  e.  (SubGrp `  G )  ->  ( 0g `  G
)  e.  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )
7472, 73syl 16 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  ( 0g `  G )  e.  ( (mrCls `  (SubGrp `  G ) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) ) )
7569, 74elind 3674 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  ( 0g `  G )  e.  ( ( S `  x )  i^i  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) ) ) )
7675snssd 4161 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  { ( 0g `  G ) }  C_  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) ) ) )
7765, 76eqssd 3506 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )  =  { ( 0g `  G ) } )
7829, 77eqtrd 2495 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
( ( S  |`  A ) `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )  =  { ( 0g `  G ) } )
7927, 78jca 530 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( A. y  e.  ( A  \  { x }
) ( ( S  |`  A ) `  x
)  C_  ( (Cntz `  G ) `  (
( S  |`  A ) `
 y ) )  /\  ( ( ( S  |`  A ) `  x )  i^i  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) ) )  =  { ( 0g
`  G ) } ) )
8079ralrimiva 2868 . . 3  |-  ( ph  ->  A. x  e.  A  ( A. y  e.  ( A  \  { x } ) ( ( S  |`  A ) `  x )  C_  (
(Cntz `  G ) `  ( ( S  |`  A ) `  y
) )  /\  (
( ( S  |`  A ) `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) )
811, 4dprddomcld 17230 . . . . 5  |-  ( ph  ->  I  e.  _V )
8281, 6ssexd 4584 . . . 4  |-  ( ph  ->  A  e.  _V )
83 fdm 5717 . . . . 5  |-  ( ( S  |`  A ) : A --> (SubGrp `  G )  ->  dom  ( S  |`  A )  =  A )
847, 83syl 16 . . . 4  |-  ( ph  ->  dom  ( S  |`  A )  =  A )
8519, 63, 35dmdprd 17227 . . . 4  |-  ( ( A  e.  _V  /\  dom  ( S  |`  A )  =  A )  -> 
( G dom DProd  ( S  |`  A )  <->  ( G  e.  Grp  /\  ( S  |`  A ) : A --> (SubGrp `  G )  /\  A. x  e.  A  ( A. y  e.  ( A  \  { x } ) ( ( S  |`  A ) `  x )  C_  (
(Cntz `  G ) `  ( ( S  |`  A ) `  y
) )  /\  (
( ( S  |`  A ) `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) )
8682, 84, 85syl2anc 659 . . 3  |-  ( ph  ->  ( G dom DProd  ( S  |`  A )  <->  ( G  e.  Grp  /\  ( S  |`  A ) : A --> (SubGrp `  G )  /\  A. x  e.  A  ( A. y  e.  ( A  \  { x } ) ( ( S  |`  A ) `  x )  C_  (
(Cntz `  G ) `  ( ( S  |`  A ) `  y
) )  /\  (
( ( S  |`  A ) `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) )
873, 7, 80, 86mpbir3and 1177 . 2  |-  ( ph  ->  G dom DProd  ( S  |`  A ) )
88 rnss 5220 . . . . . 6  |-  ( ( S  |`  A )  C_  S  ->  ran  ( S  |`  A )  C_  ran  S )
89 uniss 4256 . . . . . 6  |-  ( ran  ( S  |`  A ) 
C_  ran  S  ->  U.
ran  ( S  |`  A )  C_  U. ran  S )
9036, 88, 89mp2b 10 . . . . 5  |-  U. ran  ( S  |`  A ) 
C_  U. ran  S
9190a1i 11 . . . 4  |-  ( ph  ->  U. ran  ( S  |`  A )  C_  U. ran  S )
92 sspwuni 4404 . . . . 5  |-  ( ran 
S  C_  ~P ( Base `  G )  <->  U. ran  S  C_  ( Base `  G
) )
9352, 92sylib 196 . . . 4  |-  ( ph  ->  U. ran  S  C_  ( Base `  G )
)
9433, 35, 91, 93mrcssd 15116 . . 3  |-  ( ph  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  ( S  |`  A ) )  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ran  S ) )
9535dprdspan 17272 . . . 4  |-  ( G dom DProd  ( S  |`  A )  ->  ( G DProd  ( S  |`  A ) )  =  ( (mrCls `  (SubGrp `  G )
) `  U. ran  ( S  |`  A ) ) )
9687, 95syl 16 . . 3  |-  ( ph  ->  ( G DProd  ( S  |`  A ) )  =  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  ( S  |`  A ) ) )
9735dprdspan 17272 . . . 4  |-  ( G dom DProd  S  ->  ( G DProd 
S )  =  ( (mrCls `  (SubGrp `  G
) ) `  U. ran  S ) )
981, 97syl 16 . . 3  |-  ( ph  ->  ( G DProd  S )  =  ( (mrCls `  (SubGrp `  G ) ) `
 U. ran  S
) )
9994, 96, 983sstr4d 3532 . 2  |-  ( ph  ->  ( G DProd  ( S  |`  A ) )  C_  ( G DProd  S ) )
10087, 99jca 530 1  |-  ( ph  ->  ( G dom DProd  ( S  |`  A )  /\  ( G DProd  ( S  |`  A ) )  C_  ( G DProd  S ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   _Vcvv 3106    \ cdif 3458    i^i cin 3460    C_ wss 3461   ~Pcpw 3999   {csn 4016   U.cuni 4235   class class class wbr 4439   dom cdm 4988   ran crn 4989    |` cres 4990   "cima 4991   -->wf 5566   ` cfv 5570  (class class class)co 6270   Basecbs 14719   0gc0g 14932  Moorecmre 15074  mrClscmrc 15075  ACScacs 15077   Grpcgrp 16255  SubGrpcsubg 16397  Cntzccntz 16555   DProd cdprd 17222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-tpos 6947  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12093  df-hash 12391  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-0g 14934  df-gsum 14935  df-mre 15078  df-mrc 15079  df-acs 15081  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-mhm 16168  df-submnd 16169  df-grp 16259  df-minusg 16260  df-sbg 16261  df-mulg 16262  df-subg 16400  df-ghm 16467  df-gim 16509  df-cntz 16557  df-oppg 16583  df-cmn 17002  df-dprd 17224
This theorem is referenced by:  dprdf1  17278  dprdcntz2  17284  dprddisj2  17285  dprddisj2OLD  17286  dprd2dlem1  17288  dprd2da  17289  dmdprdsplit  17294  dprdsplit  17295  dpjf  17304  dpjidcl  17305  dpjlid  17308  dpjghm  17310  dpjidclOLD  17312  ablfac1eulem  17321  ablfac1eu  17322
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