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Theorem clssubg 21115
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 2423 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
31, 2tgptopon 21089 . . . . . 6  |-  ( G  e.  TopGrp  ->  J  e.  (TopOn `  ( Base `  G
) ) )
43adantr 467 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  J  e.  (TopOn `  ( Base `  G
) ) )
5 topontop 19933 . . . . 5  |-  ( J  e.  (TopOn `  ( Base `  G ) )  ->  J  e.  Top )
64, 5syl 17 . . . 4  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  J  e.  Top )
72subgss 16811 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
87adantl 468 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  S  C_  ( Base `  G ) )
9 toponuni 19934 . . . . . 6  |-  ( J  e.  (TopOn `  ( Base `  G ) )  ->  ( Base `  G
)  =  U. J
)
104, 9syl 17 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( Base `  G )  =  U. J )
118, 10sseqtrd 3501 . . . 4  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  S  C_  U. J
)
12 eqid 2423 . . . . 5  |-  U. J  =  U. J
1312clsss3 20066 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  ( ( cls `  J ) `  S
)  C_  U. J )
146, 11, 13syl2anc 666 . . 3  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  J ) `  S )  C_  U. J
)
1514, 10sseqtr4d 3502 . 2  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  J ) `  S )  C_  ( Base `  G ) )
1612sscls 20063 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  S  C_  (
( cls `  J
) `  S )
)
176, 11, 16syl2anc 666 . . 3  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  S  C_  (
( cls `  J
) `  S )
)
18 eqid 2423 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
1918subg0cl 16818 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  S
)
2019adantl 468 . . . 4  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( 0g `  G )  e.  S
)
21 ne0i 3768 . . . 4  |-  ( ( 0g `  G )  e.  S  ->  S  =/=  (/) )
2220, 21syl 17 . . 3  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  S  =/=  (/) )
23 ssn0 3796 . . 3  |-  ( ( S  C_  ( ( cls `  J ) `  S )  /\  S  =/=  (/) )  ->  (
( cls `  J
) `  S )  =/=  (/) )
2417, 22, 23syl2anc 666 . 2  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  J ) `  S )  =/=  (/) )
25 df-ov 6306 . . . 4  |-  ( x ( -g `  G
) y )  =  ( ( -g `  G
) `  <. x ,  y >. )
26 opelxpi 4883 . . . . . . 7  |-  ( ( x  e.  ( ( cls `  J ) `
 S )  /\  y  e.  ( ( cls `  J ) `  S ) )  ->  <. x ,  y >.  e.  ( ( ( cls `  J ) `  S
)  X.  ( ( cls `  J ) `
 S ) ) )
27 txcls 20611 . . . . . . . . . 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 1266 . . . . . . . . 9  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  ( J  tX  J ) ) `  ( S  X.  S
) )  =  ( ( ( cls `  J
) `  S )  X.  ( ( cls `  J
) `  S )
) )
29 txtopon 20598 . . . . . . . . . . . . 13  |-  ( ( J  e.  (TopOn `  ( Base `  G )
)  /\  J  e.  (TopOn `  ( Base `  G
) ) )  -> 
( J  tX  J
)  e.  (TopOn `  ( ( Base `  G
)  X.  ( Base `  G ) ) ) )
304, 4, 29syl2anc 666 . . . . . . . . . . . 12  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( J  tX  J )  e.  (TopOn `  ( ( Base `  G
)  X.  ( Base `  G ) ) ) )
31 topontop 19933 . . . . . . . . . . . 12  |-  ( ( J  tX  J )  e.  (TopOn `  (
( Base `  G )  X.  ( Base `  G
) ) )  -> 
( J  tX  J
)  e.  Top )
3230, 31syl 17 . . . . . . . . . . 11  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( J  tX  J )  e.  Top )
33 cnvimass 5205 . . . . . . . . . . . . 13  |-  ( `' ( -g `  G
) " S ) 
C_  dom  ( -g `  G )
34 tgpgrp 21085 . . . . . . . . . . . . . . . 16  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
3534adantr 467 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  G  e.  Grp )
36 eqid 2423 . . . . . . . . . . . . . . . 16  |-  ( -g `  G )  =  (
-g `  G )
372, 36grpsubf 16726 . . . . . . . . . . . . . . 15  |-  ( G  e.  Grp  ->  ( -g `  G ) : ( ( Base `  G
)  X.  ( Base `  G ) ) --> (
Base `  G )
)
3835, 37syl 17 . . . . . . . . . . . . . 14  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( -g `  G ) : ( ( Base `  G
)  X.  ( Base `  G ) ) --> (
Base `  G )
)
39 fdm 5748 . . . . . . . . . . . . . 14  |-  ( (
-g `  G ) : ( ( Base `  G )  X.  ( Base `  G ) ) --> ( Base `  G
)  ->  dom  ( -g `  G )  =  ( ( Base `  G
)  X.  ( Base `  G ) ) )
4038, 39syl 17 . . . . . . . . . . . . 13  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  dom  ( -g `  G )  =  ( ( Base `  G
)  X.  ( Base `  G ) ) )
4133, 40syl5sseq 3513 . . . . . . . . . . . 12  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( `' ( -g `  G )
" S )  C_  ( ( Base `  G
)  X.  ( Base `  G ) ) )
42 toponuni 19934 . . . . . . . . . . . . 13  |-  ( ( J  tX  J )  e.  (TopOn `  (
( Base `  G )  X.  ( Base `  G
) ) )  -> 
( ( Base `  G
)  X.  ( Base `  G ) )  = 
U. ( J  tX  J ) )
4330, 42syl 17 . . . . . . . . . . . 12  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( Base `  G )  X.  ( Base `  G
) )  =  U. ( J  tX  J ) )
4441, 43sseqtrd 3501 . . . . . . . . . . 11  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( `' ( -g `  G )
" S )  C_  U. ( J  tX  J
) )
4536subgsubcl 16821 . . . . . . . . . . . . . . . 16  |-  ( ( S  e.  (SubGrp `  G )  /\  x  e.  S  /\  y  e.  S )  ->  (
x ( -g `  G
) y )  e.  S )
46453expb 1207 . . . . . . . . . . . . . . 15  |-  ( ( S  e.  (SubGrp `  G )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x
( -g `  G ) y )  e.  S
)
4746ralrimivva 2847 . . . . . . . . . . . . . 14  |-  ( S  e.  (SubGrp `  G
)  ->  A. x  e.  S  A. y  e.  S  ( x
( -g `  G ) y )  e.  S
)
48 fveq2 5879 . . . . . . . . . . . . . . . . 17  |-  ( z  =  <. x ,  y
>.  ->  ( ( -g `  G ) `  z
)  =  ( (
-g `  G ) `  <. x ,  y
>. ) )
4948, 25syl6eqr 2482 . . . . . . . . . . . . . . . 16  |-  ( z  =  <. x ,  y
>.  ->  ( ( -g `  G ) `  z
)  =  ( x ( -g `  G
) y ) )
5049eleq1d 2492 . . . . . . . . . . . . . . 15  |-  ( z  =  <. x ,  y
>.  ->  ( ( (
-g `  G ) `  z )  e.  S  <->  ( x ( -g `  G
) y )  e.  S ) )
5150ralxp 4993 . . . . . . . . . . . . . 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 216 . . . . . . . . . . . . 13  |-  ( S  e.  (SubGrp `  G
)  ->  A. z  e.  ( S  X.  S
) ( ( -g `  G ) `  z
)  e.  S )
5352adantl 468 . . . . . . . . . . . 12  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  A. z  e.  ( S  X.  S
) ( ( -g `  G ) `  z
)  e.  S )
54 ffun 5746 . . . . . . . . . . . . . 14  |-  ( (
-g `  G ) : ( ( Base `  G )  X.  ( Base `  G ) ) --> ( Base `  G
)  ->  Fun  ( -g `  G ) )
5538, 54syl 17 . . . . . . . . . . . . 13  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  Fun  ( -g `  G ) )
56 xpss12 4957 . . . . . . . . . . . . . . 15  |-  ( ( S  C_  ( Base `  G )  /\  S  C_  ( Base `  G
) )  ->  ( S  X.  S )  C_  ( ( Base `  G
)  X.  ( Base `  G ) ) )
578, 8, 56syl2anc 666 . . . . . . . . . . . . . 14  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( S  X.  S )  C_  (
( Base `  G )  X.  ( Base `  G
) ) )
5857, 40sseqtr4d 3502 . . . . . . . . . . . . 13  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( S  X.  S )  C_  dom  ( -g `  G ) )
59 funimass5 6012 . . . . . . . . . . . . 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 666 . . . . . . . . . . . 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 236 . . . . . . . . . . 11  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( S  X.  S )  C_  ( `' ( -g `  G
) " S ) )
62 eqid 2423 . . . . . . . . . . . 12  |-  U. ( J  tX  J )  = 
U. ( J  tX  J )
6362clsss 20061 . . . . . . . . . . 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 1265 . . . . . . . . . 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 21097 . . . . . . . . . . . 12  |-  ( G  e.  TopGrp  ->  ( -g `  G
)  e.  ( ( J  tX  J )  Cn  J ) )
6665adantr 467 . . . . . . . . . . 11  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( -g `  G )  e.  ( ( J  tX  J
)  Cn  J ) )
6712cncls2i 20278 . . . . . . . . . . 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 666 . . . . . . . . . 10  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  ( J  tX  J ) ) `  ( `' ( -g `  G
) " S ) )  C_  ( `' ( -g `  G )
" ( ( cls `  J ) `  S
) ) )
6964, 68sstrd 3475 . . . . . . . . 9  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  ( J  tX  J ) ) `  ( S  X.  S
) )  C_  ( `' ( -g `  G
) " ( ( cls `  J ) `
 S ) ) )
7028, 69eqsstr3d 3500 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( (
( cls `  J
) `  S )  X.  ( ( cls `  J
) `  S )
)  C_  ( `' ( -g `  G )
" ( ( cls `  J ) `  S
) ) )
7170sselda 3465 . . . . . . 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 477 . . . . . 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 731 . . . . . . 7  |-  ( ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  (
( cls `  J
) `  S )  /\  y  e.  (
( cls `  J
) `  S )
) )  ->  G  e.  Grp )
74 ffn 5744 . . . . . . 7  |-  ( (
-g `  G ) : ( ( Base `  G )  X.  ( Base `  G ) ) --> ( Base `  G
)  ->  ( -g `  G )  Fn  (
( Base `  G )  X.  ( Base `  G
) ) )
75 elpreima 6015 . . . . . . 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 19 . . . . . 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 214 . . . . 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 465 . . . 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 2515 . . 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 2847 . 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 16829 . . 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 17 . 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 1189 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 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869    =/= wne 2619   A.wral 2776    C_ wss 3437   (/)c0 3762   <.cop 4003   U.cuni 4217    X. cxp 4849   `'ccnv 4850   dom cdm 4851   "cima 4854   Fun wfun 5593    Fn wfn 5594   -->wf 5595   ` cfv 5599  (class class class)co 6303   Basecbs 15114   TopOpenctopn 15313   0gc0g 15331   Grpcgrp 16662   -gcsg 16664  SubGrpcsubg 16804   Topctop 19909  TopOnctopon 19910   clsccl 20025    Cn ccn 20232    tX ctx 20567   TopGrpctgp 21078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-iin 4300  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-er 7369  df-map 7480  df-en 7576  df-dom 7577  df-sdom 7578  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-nn 10612  df-2 10670  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-0g 15333  df-topgen 15335  df-plusf 16480  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-grp 16666  df-minusg 16667  df-sbg 16668  df-subg 16807  df-top 19913  df-bases 19914  df-topon 19915  df-topsp 19916  df-cld 20026  df-ntr 20027  df-cls 20028  df-cn 20235  df-tx 20569  df-tmd 21079  df-tgp 21080
This theorem is referenced by:  clsnsg  21116  tgptsmscls  21156
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