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Theorem clssubg 20342
Description: The closure of a subgroup in a topological group is a subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
Hypothesis
Ref Expression
subgntr.h  |-  J  =  ( TopOpen `  G )
Assertion
Ref Expression
clssubg  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  J ) `  S )  e.  (SubGrp `  G ) )

Proof of Theorem clssubg
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subgntr.h . . . . . . 7  |-  J  =  ( TopOpen `  G )
2 eqid 2467 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
31, 2tgptopon 20316 . . . . . 6  |-  ( G  e.  TopGrp  ->  J  e.  (TopOn `  ( Base `  G
) ) )
43adantr 465 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  J  e.  (TopOn `  ( Base `  G
) ) )
5 topontop 19194 . . . . 5  |-  ( J  e.  (TopOn `  ( Base `  G ) )  ->  J  e.  Top )
64, 5syl 16 . . . 4  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  J  e.  Top )
72subgss 15997 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
87adantl 466 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  S  C_  ( Base `  G ) )
9 toponuni 19195 . . . . . 6  |-  ( J  e.  (TopOn `  ( Base `  G ) )  ->  ( Base `  G
)  =  U. J
)
104, 9syl 16 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( Base `  G )  =  U. J )
118, 10sseqtrd 3540 . . . 4  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  S  C_  U. J
)
12 eqid 2467 . . . . 5  |-  U. J  =  U. J
1312clsss3 19326 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  ( ( cls `  J ) `  S
)  C_  U. J )
146, 11, 13syl2anc 661 . . 3  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  J ) `  S )  C_  U. J
)
1514, 10sseqtr4d 3541 . 2  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  J ) `  S )  C_  ( Base `  G ) )
1612sscls 19323 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  S  C_  (
( cls `  J
) `  S )
)
176, 11, 16syl2anc 661 . . 3  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  S  C_  (
( cls `  J
) `  S )
)
18 eqid 2467 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
1918subg0cl 16004 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  S
)
2019adantl 466 . . . 4  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( 0g `  G )  e.  S
)
21 ne0i 3791 . . . 4  |-  ( ( 0g `  G )  e.  S  ->  S  =/=  (/) )
2220, 21syl 16 . . 3  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  S  =/=  (/) )
23 ssn0 3818 . . 3  |-  ( ( S  C_  ( ( cls `  J ) `  S )  /\  S  =/=  (/) )  ->  (
( cls `  J
) `  S )  =/=  (/) )
2417, 22, 23syl2anc 661 . 2  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  J ) `  S )  =/=  (/) )
25 df-ov 6285 . . . 4  |-  ( x ( -g `  G
) y )  =  ( ( -g `  G
) `  <. x ,  y >. )
26 opelxpi 5030 . . . . . . 7  |-  ( ( x  e.  ( ( cls `  J ) `
 S )  /\  y  e.  ( ( cls `  J ) `  S ) )  ->  <. x ,  y >.  e.  ( ( ( cls `  J ) `  S
)  X.  ( ( cls `  J ) `
 S ) ) )
27 txcls 19840 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  ( Base `  G
) )  /\  J  e.  (TopOn `  ( Base `  G ) ) )  /\  ( S  C_  ( Base `  G )  /\  S  C_  ( Base `  G ) ) )  ->  ( ( cls `  ( J  tX  J
) ) `  ( S  X.  S ) )  =  ( ( ( cls `  J ) `
 S )  X.  ( ( cls `  J
) `  S )
) )
284, 4, 8, 8, 27syl22anc 1229 . . . . . . . . 9  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  ( J  tX  J ) ) `  ( S  X.  S
) )  =  ( ( ( cls `  J
) `  S )  X.  ( ( cls `  J
) `  S )
) )
29 txtopon 19827 . . . . . . . . . . . . 13  |-  ( ( J  e.  (TopOn `  ( Base `  G )
)  /\  J  e.  (TopOn `  ( Base `  G
) ) )  -> 
( J  tX  J
)  e.  (TopOn `  ( ( Base `  G
)  X.  ( Base `  G ) ) ) )
304, 4, 29syl2anc 661 . . . . . . . . . . . 12  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( J  tX  J )  e.  (TopOn `  ( ( Base `  G
)  X.  ( Base `  G ) ) ) )
31 topontop 19194 . . . . . . . . . . . 12  |-  ( ( J  tX  J )  e.  (TopOn `  (
( Base `  G )  X.  ( Base `  G
) ) )  -> 
( J  tX  J
)  e.  Top )
3230, 31syl 16 . . . . . . . . . . 11  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( J  tX  J )  e.  Top )
33 cnvimass 5355 . . . . . . . . . . . . 13  |-  ( `' ( -g `  G
) " S ) 
C_  dom  ( -g `  G )
34 tgpgrp 20312 . . . . . . . . . . . . . . . 16  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
3534adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  G  e.  Grp )
36 eqid 2467 . . . . . . . . . . . . . . . 16  |-  ( -g `  G )  =  (
-g `  G )
372, 36grpsubf 15918 . . . . . . . . . . . . . . 15  |-  ( G  e.  Grp  ->  ( -g `  G ) : ( ( Base `  G
)  X.  ( Base `  G ) ) --> (
Base `  G )
)
3835, 37syl 16 . . . . . . . . . . . . . 14  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( -g `  G ) : ( ( Base `  G
)  X.  ( Base `  G ) ) --> (
Base `  G )
)
39 fdm 5733 . . . . . . . . . . . . . 14  |-  ( (
-g `  G ) : ( ( Base `  G )  X.  ( Base `  G ) ) --> ( Base `  G
)  ->  dom  ( -g `  G )  =  ( ( Base `  G
)  X.  ( Base `  G ) ) )
4038, 39syl 16 . . . . . . . . . . . . 13  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  dom  ( -g `  G )  =  ( ( Base `  G
)  X.  ( Base `  G ) ) )
4133, 40syl5sseq 3552 . . . . . . . . . . . 12  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( `' ( -g `  G )
" S )  C_  ( ( Base `  G
)  X.  ( Base `  G ) ) )
42 toponuni 19195 . . . . . . . . . . . . 13  |-  ( ( J  tX  J )  e.  (TopOn `  (
( Base `  G )  X.  ( Base `  G
) ) )  -> 
( ( Base `  G
)  X.  ( Base `  G ) )  = 
U. ( J  tX  J ) )
4330, 42syl 16 . . . . . . . . . . . 12  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( Base `  G )  X.  ( Base `  G
) )  =  U. ( J  tX  J ) )
4441, 43sseqtrd 3540 . . . . . . . . . . 11  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( `' ( -g `  G )
" S )  C_  U. ( J  tX  J
) )
4536subgsubcl 16007 . . . . . . . . . . . . . . . 16  |-  ( ( S  e.  (SubGrp `  G )  /\  x  e.  S  /\  y  e.  S )  ->  (
x ( -g `  G
) y )  e.  S )
46453expb 1197 . . . . . . . . . . . . . . 15  |-  ( ( S  e.  (SubGrp `  G )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x
( -g `  G ) y )  e.  S
)
4746ralrimivva 2885 . . . . . . . . . . . . . 14  |-  ( S  e.  (SubGrp `  G
)  ->  A. x  e.  S  A. y  e.  S  ( x
( -g `  G ) y )  e.  S
)
48 fveq2 5864 . . . . . . . . . . . . . . . . 17  |-  ( z  =  <. x ,  y
>.  ->  ( ( -g `  G ) `  z
)  =  ( (
-g `  G ) `  <. x ,  y
>. ) )
4948, 25syl6eqr 2526 . . . . . . . . . . . . . . . 16  |-  ( z  =  <. x ,  y
>.  ->  ( ( -g `  G ) `  z
)  =  ( x ( -g `  G
) y ) )
5049eleq1d 2536 . . . . . . . . . . . . . . 15  |-  ( z  =  <. x ,  y
>.  ->  ( ( (
-g `  G ) `  z )  e.  S  <->  ( x ( -g `  G
) y )  e.  S ) )
5150ralxp 5142 . . . . . . . . . . . . . 14  |-  ( A. z  e.  ( S  X.  S ) ( (
-g `  G ) `  z )  e.  S  <->  A. x  e.  S  A. y  e.  S  (
x ( -g `  G
) y )  e.  S )
5247, 51sylibr 212 . . . . . . . . . . . . 13  |-  ( S  e.  (SubGrp `  G
)  ->  A. z  e.  ( S  X.  S
) ( ( -g `  G ) `  z
)  e.  S )
5352adantl 466 . . . . . . . . . . . 12  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  A. z  e.  ( S  X.  S
) ( ( -g `  G ) `  z
)  e.  S )
54 ffun 5731 . . . . . . . . . . . . . 14  |-  ( (
-g `  G ) : ( ( Base `  G )  X.  ( Base `  G ) ) --> ( Base `  G
)  ->  Fun  ( -g `  G ) )
5538, 54syl 16 . . . . . . . . . . . . 13  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  Fun  ( -g `  G ) )
56 xpss12 5106 . . . . . . . . . . . . . . 15  |-  ( ( S  C_  ( Base `  G )  /\  S  C_  ( Base `  G
) )  ->  ( S  X.  S )  C_  ( ( Base `  G
)  X.  ( Base `  G ) ) )
578, 8, 56syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( S  X.  S )  C_  (
( Base `  G )  X.  ( Base `  G
) ) )
5857, 40sseqtr4d 3541 . . . . . . . . . . . . 13  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( S  X.  S )  C_  dom  ( -g `  G ) )
59 funimass5 5996 . . . . . . . . . . . . 13  |-  ( ( Fun  ( -g `  G
)  /\  ( S  X.  S )  C_  dom  ( -g `  G ) )  ->  ( ( S  X.  S )  C_  ( `' ( -g `  G
) " S )  <->  A. z  e.  ( S  X.  S ) ( ( -g `  G
) `  z )  e.  S ) )
6055, 58, 59syl2anc 661 . . . . . . . . . . . 12  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( S  X.  S )  C_  ( `' ( -g `  G
) " S )  <->  A. z  e.  ( S  X.  S ) ( ( -g `  G
) `  z )  e.  S ) )
6153, 60mpbird 232 . . . . . . . . . . 11  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( S  X.  S )  C_  ( `' ( -g `  G
) " S ) )
62 eqid 2467 . . . . . . . . . . . 12  |-  U. ( J  tX  J )  = 
U. ( J  tX  J )
6362clsss 19321 . . . . . . . . . . 11  |-  ( ( ( J  tX  J
)  e.  Top  /\  ( `' ( -g `  G
) " S ) 
C_  U. ( J  tX  J )  /\  ( S  X.  S )  C_  ( `' ( -g `  G
) " S ) )  ->  ( ( cls `  ( J  tX  J ) ) `  ( S  X.  S
) )  C_  (
( cls `  ( J  tX  J ) ) `
 ( `' (
-g `  G ) " S ) ) )
6432, 44, 61, 63syl3anc 1228 . . . . . . . . . 10  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  ( J  tX  J ) ) `  ( S  X.  S
) )  C_  (
( cls `  ( J  tX  J ) ) `
 ( `' (
-g `  G ) " S ) ) )
651, 36tgpsubcn 20324 . . . . . . . . . . . 12  |-  ( G  e.  TopGrp  ->  ( -g `  G
)  e.  ( ( J  tX  J )  Cn  J ) )
6665adantr 465 . . . . . . . . . . 11  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( -g `  G )  e.  ( ( J  tX  J
)  Cn  J ) )
6712cncls2i 19537 . . . . . . . . . . 11  |-  ( ( ( -g `  G
)  e.  ( ( J  tX  J )  Cn  J )  /\  S  C_  U. J )  ->  ( ( cls `  ( J  tX  J
) ) `  ( `' ( -g `  G
) " S ) )  C_  ( `' ( -g `  G )
" ( ( cls `  J ) `  S
) ) )
6866, 11, 67syl2anc 661 . . . . . . . . . 10  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  ( J  tX  J ) ) `  ( `' ( -g `  G
) " S ) )  C_  ( `' ( -g `  G )
" ( ( cls `  J ) `  S
) ) )
6964, 68sstrd 3514 . . . . . . . . 9  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  ( J  tX  J ) ) `  ( S  X.  S
) )  C_  ( `' ( -g `  G
) " ( ( cls `  J ) `
 S ) ) )
7028, 69eqsstr3d 3539 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( (
( cls `  J
) `  S )  X.  ( ( cls `  J
) `  S )
)  C_  ( `' ( -g `  G )
" ( ( cls `  J ) `  S
) ) )
7170sselda 3504 . . . . . . 7  |-  ( ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G ) )  /\  <.
x ,  y >.  e.  ( ( ( cls `  J ) `  S
)  X.  ( ( cls `  J ) `
 S ) ) )  ->  <. x ,  y >.  e.  ( `' ( -g `  G
) " ( ( cls `  J ) `
 S ) ) )
7226, 71sylan2 474 . . . . . 6  |-  ( ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  (
( cls `  J
) `  S )  /\  y  e.  (
( cls `  J
) `  S )
) )  ->  <. x ,  y >.  e.  ( `' ( -g `  G
) " ( ( cls `  J ) `
 S ) ) )
7334ad2antrr 725 . . . . . . 7  |-  ( ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  (
( cls `  J
) `  S )  /\  y  e.  (
( cls `  J
) `  S )
) )  ->  G  e.  Grp )
74 ffn 5729 . . . . . . 7  |-  ( (
-g `  G ) : ( ( Base `  G )  X.  ( Base `  G ) ) --> ( Base `  G
)  ->  ( -g `  G )  Fn  (
( Base `  G )  X.  ( Base `  G
) ) )
75 elpreima 5999 . . . . . . 7  |-  ( (
-g `  G )  Fn  ( ( Base `  G
)  X.  ( Base `  G ) )  -> 
( <. x ,  y
>.  e.  ( `' (
-g `  G ) " ( ( cls `  J ) `  S
) )  <->  ( <. x ,  y >.  e.  ( ( Base `  G
)  X.  ( Base `  G ) )  /\  ( ( -g `  G
) `  <. x ,  y >. )  e.  ( ( cls `  J
) `  S )
) ) )
7673, 37, 74, 754syl 21 . . . . . 6  |-  ( ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  (
( cls `  J
) `  S )  /\  y  e.  (
( cls `  J
) `  S )
) )  ->  ( <. x ,  y >.  e.  ( `' ( -g `  G ) " (
( cls `  J
) `  S )
)  <->  ( <. x ,  y >.  e.  ( ( Base `  G
)  X.  ( Base `  G ) )  /\  ( ( -g `  G
) `  <. x ,  y >. )  e.  ( ( cls `  J
) `  S )
) ) )
7772, 76mpbid 210 . . . . 5  |-  ( ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  (
( cls `  J
) `  S )  /\  y  e.  (
( cls `  J
) `  S )
) )  ->  ( <. x ,  y >.  e.  ( ( Base `  G
)  X.  ( Base `  G ) )  /\  ( ( -g `  G
) `  <. x ,  y >. )  e.  ( ( cls `  J
) `  S )
) )
7877simprd 463 . . . 4  |-  ( ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  (
( cls `  J
) `  S )  /\  y  e.  (
( cls `  J
) `  S )
) )  ->  (
( -g `  G ) `
 <. x ,  y
>. )  e.  (
( cls `  J
) `  S )
)
7925, 78syl5eqel 2559 . . 3  |-  ( ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  (
( cls `  J
) `  S )  /\  y  e.  (
( cls `  J
) `  S )
) )  ->  (
x ( -g `  G
) y )  e.  ( ( cls `  J
) `  S )
)
8079ralrimivva 2885 . 2  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  A. x  e.  ( ( cls `  J
) `  S ) A. y  e.  (
( cls `  J
) `  S )
( x ( -g `  G ) y )  e.  ( ( cls `  J ) `  S
) )
812, 36issubg4 16015 . . 3  |-  ( G  e.  Grp  ->  (
( ( cls `  J
) `  S )  e.  (SubGrp `  G )  <->  ( ( ( cls `  J
) `  S )  C_  ( Base `  G
)  /\  ( ( cls `  J ) `  S )  =/=  (/)  /\  A. x  e.  ( ( cls `  J ) `  S ) A. y  e.  ( ( cls `  J
) `  S )
( x ( -g `  G ) y )  e.  ( ( cls `  J ) `  S
) ) ) )
8235, 81syl 16 . 2  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( (
( cls `  J
) `  S )  e.  (SubGrp `  G )  <->  ( ( ( cls `  J
) `  S )  C_  ( Base `  G
)  /\  ( ( cls `  J ) `  S )  =/=  (/)  /\  A. x  e.  ( ( cls `  J ) `  S ) A. y  e.  ( ( cls `  J
) `  S )
( x ( -g `  G ) y )  e.  ( ( cls `  J ) `  S
) ) ) )
8315, 24, 80, 82mpbir3and 1179 1  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  J ) `  S )  e.  (SubGrp `  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814    C_ wss 3476   (/)c0 3785   <.cop 4033   U.cuni 4245    X. cxp 4997   `'ccnv 4998   dom cdm 4999   "cima 5002   Fun wfun 5580    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   Basecbs 14486   TopOpenctopn 14673   0gc0g 14691   Grpcgrp 15723   -gcsg 15726  SubGrpcsubg 15990   Topctop 19161  TopOnctopon 19162   clsccl 19285    Cn ccn 19491    tX ctx 19796   TopGrpctgp 20305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-0g 14693  df-topgen 14695  df-mnd 15728  df-plusf 15729  df-grp 15858  df-minusg 15859  df-sbg 15860  df-subg 15993  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cld 19286  df-ntr 19287  df-cls 19288  df-cn 19494  df-tx 19798  df-tmd 20306  df-tgp 20307
This theorem is referenced by:  clsnsg  20343  tgptsmscls  20387
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