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Theorem dprdres 17654
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 17630 . . . 4  |-  ( G dom DProd  S  ->  G  e. 
Grp )
31, 2syl 17 . . 3  |-  ( ph  ->  G  e.  Grp )
4 dprdres.2 . . . . 5  |-  ( ph  ->  dom  S  =  I )
51, 4dprdf2 17632 . . . 4  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
6 dprdres.3 . . . 4  |-  ( ph  ->  A  C_  I )
75, 6fssresd 5748 . . 3  |-  ( ph  ->  ( S  |`  A ) : A --> (SubGrp `  G ) )
81ad2antrr 731 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  G dom DProd  S )
94ad2antrr 731 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  dom  S  =  I )
106ad2antrr 731 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  A  C_  I
)
11 simplr 761 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  x  e.  A
)
1210, 11sseldd 3432 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  x  e.  I
)
13 eldifi 3554 . . . . . . . . . 10  |-  ( y  e.  ( A  \  { x } )  ->  y  e.  A
)
1413adantl 468 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  y  e.  A
)
1510, 14sseldd 3432 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  y  e.  I
)
16 eldifsni 4097 . . . . . . . . . 10  |-  ( y  e.  ( A  \  { x } )  ->  y  =/=  x
)
1716adantl 468 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  y  =/=  x
)
1817necomd 2678 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  x  =/=  y
)
19 eqid 2450 . . . . . . . 8  |-  (Cntz `  G )  =  (Cntz `  G )
208, 9, 12, 15, 18, 19dprdcntz 17633 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  ( S `  x )  C_  (
(Cntz `  G ) `  ( S `  y
) ) )
21 fvres 5877 . . . . . . . 8  |-  ( x  e.  A  ->  (
( S  |`  A ) `
 x )  =  ( S `  x
) )
2211, 21syl 17 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  ( ( S  |`  A ) `  x
)  =  ( S `
 x ) )
23 fvres 5877 . . . . . . . . 9  |-  ( y  e.  A  ->  (
( S  |`  A ) `
 y )  =  ( S `  y
) )
2414, 23syl 17 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  ( ( S  |`  A ) `  y
)  =  ( S `
 y ) )
2524fveq2d 5867 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  ( (Cntz `  G ) `  (
( S  |`  A ) `
 y ) )  =  ( (Cntz `  G ) `  ( S `  y )
) )
2620, 22, 253sstr4d 3474 . . . . . 6  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  ( ( S  |`  A ) `  x
)  C_  ( (Cntz `  G ) `  (
( S  |`  A ) `
 y ) ) )
2726ralrimiva 2801 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  A. y  e.  ( A  \  {
x } ) ( ( S  |`  A ) `
 x )  C_  ( (Cntz `  G ) `  ( ( S  |`  A ) `  y
) ) )
2821adantl 468 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
( S  |`  A ) `
 x )  =  ( S `  x
) )
2928ineq1d 3632 . . . . . 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 2450 . . . . . . . . . . . . 13  |-  ( Base `  G )  =  (
Base `  G )
3130subgacs 16845 . . . . . . . . . . . 12  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
32 acsmre 15551 . . . . . . . . . . . 12  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
333, 31, 323syl 18 . . . . . . . . . . 11  |-  ( ph  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G ) ) )
3433adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
35 eqid 2450 . . . . . . . . . 10  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
36 resss 5127 . . . . . . . . . . . . 13  |-  ( S  |`  A )  C_  S
37 imass1 5202 . . . . . . . . . . . . 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 467 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  A )  ->  A  C_  I )
40 ssdif 3567 . . . . . . . . . . . . 13  |-  ( A 
C_  I  ->  ( A  \  { x }
)  C_  ( I  \  { x } ) )
41 imass2 5203 . . . . . . . . . . . . 13  |-  ( ( A  \  { x } )  C_  (
I  \  { x } )  ->  ( S " ( A  \  { x } ) )  C_  ( S " ( I  \  {
x } ) ) )
4239, 40, 413syl 18 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  A )  ->  ( S " ( A  \  { x } ) )  C_  ( S " ( I  \  {
x } ) ) )
4338, 42syl5ss 3442 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  (
( S  |`  A )
" ( A  \  { x } ) )  C_  ( S " ( I  \  {
x } ) ) )
4443unissd 4221 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  U. (
( S  |`  A )
" ( A  \  { x } ) )  C_  U. ( S " ( I  \  { x } ) ) )
45 imassrn 5178 . . . . . . . . . . . 12  |-  ( S
" ( I  \  { x } ) )  C_  ran  S
46 frn 5733 . . . . . . . . . . . . . . 15  |-  ( S : I --> (SubGrp `  G )  ->  ran  S 
C_  (SubGrp `  G )
)
475, 46syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  S  C_  (SubGrp `  G ) )
4830subgss 16811 . . . . . . . . . . . . . . . 16  |-  ( x  e.  (SubGrp `  G
)  ->  x  C_  ( Base `  G ) )
49 selpw 3957 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ~P ( Base `  G )  <->  x  C_  ( Base `  G ) )
5048, 49sylibr 216 . . . . . . . . . . . . . . 15  |-  ( x  e.  (SubGrp `  G
)  ->  x  e.  ~P ( Base `  G
) )
5150ssriv 3435 . . . . . . . . . . . . . 14  |-  (SubGrp `  G )  C_  ~P ( Base `  G )
5247, 51syl6ss 3443 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  S  C_  ~P ( Base `  G )
)
5352adantr 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  A )  ->  ran  S 
C_  ~P ( Base `  G
) )
5445, 53syl5ss 3442 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  ( S " ( I  \  { x } ) )  C_  ~P ( Base `  G ) )
55 sspwuni 4366 . . . . . . . . . . 11  |-  ( ( S " ( I 
\  { x }
) )  C_  ~P ( Base `  G )  <->  U. ( S " (
I  \  { x } ) )  C_  ( Base `  G )
)
5654, 55sylib 200 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  U. ( S " ( I  \  { x } ) )  C_  ( Base `  G ) )
5734, 35, 44, 56mrcssd 15523 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) )  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) ) )
58 sslin 3657 . . . . . . . . 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 17 . . . . . . . 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 467 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  G dom DProd  S )
614adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  dom  S  =  I )
626sselda 3431 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  I )
63 eqid 2450 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
6460, 61, 62, 63, 35dprddisj 17634 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )  =  { ( 0g `  G ) } )
6559, 64sseqtrd 3467 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )  C_  { ( 0g `  G
) } )
665ffvelrnda 6020 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
6762, 66syldan 473 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  ( S `  x )  e.  (SubGrp `  G )
)
6863subg0cl 16818 . . . . . . . . . 10  |-  ( ( S `  x )  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  ( S `  x ) )
6967, 68syl 17 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  ( 0g `  G )  e.  ( S `  x
) )
7044, 56sstrd 3441 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  U. (
( S  |`  A )
" ( A  \  { x } ) )  C_  ( Base `  G ) )
7135mrccl 15510 . . . . . . . . . . 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 666 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) )  e.  (SubGrp `  G )
)
7363subg0cl 16818 . . . . . . . . . 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 17 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  ( 0g `  G )  e.  ( (mrCls `  (SubGrp `  G ) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) ) )
7569, 74elind 3617 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  ( 0g `  G )  e.  ( ( S `  x )  i^i  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) ) ) )
7675snssd 4116 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  { ( 0g `  G ) }  C_  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) ) ) )
7765, 76eqssd 3448 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )  =  { ( 0g `  G ) } )
7829, 77eqtrd 2484 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
( ( S  |`  A ) `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )  =  { ( 0g `  G ) } )
7927, 78jca 535 . . . 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 2801 . . 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 17626 . . . . 5  |-  ( ph  ->  I  e.  _V )
8281, 6ssexd 4549 . . . 4  |-  ( ph  ->  A  e.  _V )
83 fdm 5731 . . . . 5  |-  ( ( S  |`  A ) : A --> (SubGrp `  G )  ->  dom  ( S  |`  A )  =  A )
847, 83syl 17 . . . 4  |-  ( ph  ->  dom  ( S  |`  A )  =  A )
8519, 63, 35dmdprd 17623 . . . 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 666 . . 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 1190 . 2  |-  ( ph  ->  G dom DProd  ( S  |`  A ) )
88 rnss 5062 . . . . . 6  |-  ( ( S  |`  A )  C_  S  ->  ran  ( S  |`  A )  C_  ran  S )
89 uniss 4218 . . . . . 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 4366 . . . . 5  |-  ( ran 
S  C_  ~P ( Base `  G )  <->  U. ran  S  C_  ( Base `  G
) )
9352, 92sylib 200 . . . 4  |-  ( ph  ->  U. ran  S  C_  ( Base `  G )
)
9433, 35, 91, 93mrcssd 15523 . . 3  |-  ( ph  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  ( S  |`  A ) )  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ran  S ) )
9535dprdspan 17653 . . . 4  |-  ( G dom DProd  ( S  |`  A )  ->  ( G DProd  ( S  |`  A ) )  =  ( (mrCls `  (SubGrp `  G )
) `  U. ran  ( S  |`  A ) ) )
9687, 95syl 17 . . 3  |-  ( ph  ->  ( G DProd  ( S  |`  A ) )  =  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  ( S  |`  A ) ) )
9735dprdspan 17653 . . . 4  |-  ( G dom DProd  S  ->  ( G DProd 
S )  =  ( (mrCls `  (SubGrp `  G
) ) `  U. ran  S ) )
981, 97syl 17 . . 3  |-  ( ph  ->  ( G DProd  S )  =  ( (mrCls `  (SubGrp `  G ) ) `
 U. ran  S
) )
9994, 96, 983sstr4d 3474 . 2  |-  ( ph  ->  ( G DProd  ( S  |`  A ) )  C_  ( G DProd  S ) )
10087, 99jca 535 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 188    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886    =/= wne 2621   A.wral 2736   _Vcvv 3044    \ cdif 3400    i^i cin 3402    C_ wss 3403   ~Pcpw 3950   {csn 3967   U.cuni 4197   class class class wbr 4401   dom cdm 4833   ran crn 4834    |` cres 4835   "cima 4836   -->wf 5577   ` cfv 5581  (class class class)co 6288   Basecbs 15114   0gc0g 15331  Moorecmre 15481  mrClscmrc 15482  ACScacs 15484   Grpcgrp 16662  SubGrpcsubg 16804  Cntzccntz 16962   DProd cdprd 17618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-tpos 6970  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-map 7471  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-oi 8022  df-card 8370  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-n0 10867  df-z 10935  df-uz 11157  df-fz 11782  df-fzo 11913  df-seq 12211  df-hash 12513  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-0g 15333  df-gsum 15334  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-mhm 16575  df-submnd 16576  df-grp 16666  df-minusg 16667  df-sbg 16668  df-mulg 16669  df-subg 16807  df-ghm 16874  df-gim 16916  df-cntz 16964  df-oppg 16990  df-cmn 17425  df-dprd 17620
This theorem is referenced by:  dprdf1  17659  dprdcntz2  17664  dprddisj2  17665  dprd2dlem1  17667  dprd2da  17668  dmdprdsplit  17673  dprdsplit  17674  dpjf  17683  dpjidcl  17684  dpjlid  17687  dpjghm  17689  ablfac1eulem  17698  ablfac1eu  17699
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