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Theorem clssubg 17791
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 2283 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
31, 2tgptopon 17765 . . . . . 6  |-  ( G  e.  TopGrp  ->  J  e.  (TopOn `  ( Base `  G
) ) )
43adantr 451 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  J  e.  (TopOn `  ( Base `  G
) ) )
5 topontop 16664 . . . . 5  |-  ( J  e.  (TopOn `  ( Base `  G ) )  ->  J  e.  Top )
64, 5syl 15 . . . 4  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  J  e.  Top )
72subgss 14622 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
87adantl 452 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  S  C_  ( Base `  G ) )
9 toponuni 16665 . . . . . 6  |-  ( J  e.  (TopOn `  ( Base `  G ) )  ->  ( Base `  G
)  =  U. J
)
104, 9syl 15 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( Base `  G )  =  U. J )
118, 10sseqtrd 3214 . . . 4  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  S  C_  U. J
)
12 eqid 2283 . . . . 5  |-  U. J  =  U. J
1312clsss3 16796 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  ( ( cls `  J ) `  S
)  C_  U. J )
146, 11, 13syl2anc 642 . . 3  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  J ) `  S )  C_  U. J
)
1514, 10sseqtr4d 3215 . 2  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  J ) `  S )  C_  ( Base `  G ) )
1612sscls 16793 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  S  C_  (
( cls `  J
) `  S )
)
176, 11, 16syl2anc 642 . . 3  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  S  C_  (
( cls `  J
) `  S )
)
18 eqid 2283 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
1918subg0cl 14629 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  S
)
2019adantl 452 . . . 4  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( 0g `  G )  e.  S
)
21 ne0i 3461 . . . 4  |-  ( ( 0g `  G )  e.  S  ->  S  =/=  (/) )
2220, 21syl 15 . . 3  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  S  =/=  (/) )
23 ssn0 3487 . . 3  |-  ( ( S  C_  ( ( cls `  J ) `  S )  /\  S  =/=  (/) )  ->  (
( cls `  J
) `  S )  =/=  (/) )
2417, 22, 23syl2anc 642 . 2  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  J ) `  S )  =/=  (/) )
25 df-ov 5861 . . . 4  |-  ( x ( -g `  G
) y )  =  ( ( -g `  G
) `  <. x ,  y >. )
26 opelxpi 4721 . . . . . . 7  |-  ( ( x  e.  ( ( cls `  J ) `
 S )  /\  y  e.  ( ( cls `  J ) `  S ) )  ->  <. x ,  y >.  e.  ( ( ( cls `  J ) `  S
)  X.  ( ( cls `  J ) `
 S ) ) )
27 txcls 17299 . . . . . . . . . 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 1183 . . . . . . . . 9  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  ( J  tX  J ) ) `  ( S  X.  S
) )  =  ( ( ( cls `  J
) `  S )  X.  ( ( cls `  J
) `  S )
) )
29 txtopon 17286 . . . . . . . . . . . . 13  |-  ( ( J  e.  (TopOn `  ( Base `  G )
)  /\  J  e.  (TopOn `  ( Base `  G
) ) )  -> 
( J  tX  J
)  e.  (TopOn `  ( ( Base `  G
)  X.  ( Base `  G ) ) ) )
304, 4, 29syl2anc 642 . . . . . . . . . . . 12  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( J  tX  J )  e.  (TopOn `  ( ( Base `  G
)  X.  ( Base `  G ) ) ) )
31 topontop 16664 . . . . . . . . . . . 12  |-  ( ( J  tX  J )  e.  (TopOn `  (
( Base `  G )  X.  ( Base `  G
) ) )  -> 
( J  tX  J
)  e.  Top )
3230, 31syl 15 . . . . . . . . . . 11  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( J  tX  J )  e.  Top )
33 cnvimass 5033 . . . . . . . . . . . . 13  |-  ( `' ( -g `  G
) " S ) 
C_  dom  ( -g `  G )
34 tgpgrp 17761 . . . . . . . . . . . . . . . 16  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
3534adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  G  e.  Grp )
36 eqid 2283 . . . . . . . . . . . . . . . 16  |-  ( -g `  G )  =  (
-g `  G )
372, 36grpsubf 14545 . . . . . . . . . . . . . . 15  |-  ( G  e.  Grp  ->  ( -g `  G ) : ( ( Base `  G
)  X.  ( Base `  G ) ) --> (
Base `  G )
)
3835, 37syl 15 . . . . . . . . . . . . . 14  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( -g `  G ) : ( ( Base `  G
)  X.  ( Base `  G ) ) --> (
Base `  G )
)
39 fdm 5393 . . . . . . . . . . . . . 14  |-  ( (
-g `  G ) : ( ( Base `  G )  X.  ( Base `  G ) ) --> ( Base `  G
)  ->  dom  ( -g `  G )  =  ( ( Base `  G
)  X.  ( Base `  G ) ) )
4038, 39syl 15 . . . . . . . . . . . . 13  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  dom  ( -g `  G )  =  ( ( Base `  G
)  X.  ( Base `  G ) ) )
4133, 40syl5sseq 3226 . . . . . . . . . . . 12  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( `' ( -g `  G )
" S )  C_  ( ( Base `  G
)  X.  ( Base `  G ) ) )
42 toponuni 16665 . . . . . . . . . . . . 13  |-  ( ( J  tX  J )  e.  (TopOn `  (
( Base `  G )  X.  ( Base `  G
) ) )  -> 
( ( Base `  G
)  X.  ( Base `  G ) )  = 
U. ( J  tX  J ) )
4330, 42syl 15 . . . . . . . . . . . 12  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( Base `  G )  X.  ( Base `  G
) )  =  U. ( J  tX  J ) )
4441, 43sseqtrd 3214 . . . . . . . . . . 11  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( `' ( -g `  G )
" S )  C_  U. ( J  tX  J
) )
4536subgsubcl 14632 . . . . . . . . . . . . . . . 16  |-  ( ( S  e.  (SubGrp `  G )  /\  x  e.  S  /\  y  e.  S )  ->  (
x ( -g `  G
) y )  e.  S )
46453expb 1152 . . . . . . . . . . . . . . 15  |-  ( ( S  e.  (SubGrp `  G )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x
( -g `  G ) y )  e.  S
)
4746ralrimivva 2635 . . . . . . . . . . . . . 14  |-  ( S  e.  (SubGrp `  G
)  ->  A. x  e.  S  A. y  e.  S  ( x
( -g `  G ) y )  e.  S
)
48 fveq2 5525 . . . . . . . . . . . . . . . . 17  |-  ( z  =  <. x ,  y
>.  ->  ( ( -g `  G ) `  z
)  =  ( (
-g `  G ) `  <. x ,  y
>. ) )
4948, 25syl6eqr 2333 . . . . . . . . . . . . . . . 16  |-  ( z  =  <. x ,  y
>.  ->  ( ( -g `  G ) `  z
)  =  ( x ( -g `  G
) y ) )
5049eleq1d 2349 . . . . . . . . . . . . . . 15  |-  ( z  =  <. x ,  y
>.  ->  ( ( (
-g `  G ) `  z )  e.  S  <->  ( x ( -g `  G
) y )  e.  S ) )
5150ralxp 4827 . . . . . . . . . . . . . 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 203 . . . . . . . . . . . . 13  |-  ( S  e.  (SubGrp `  G
)  ->  A. z  e.  ( S  X.  S
) ( ( -g `  G ) `  z
)  e.  S )
5352adantl 452 . . . . . . . . . . . 12  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  A. z  e.  ( S  X.  S
) ( ( -g `  G ) `  z
)  e.  S )
54 ffun 5391 . . . . . . . . . . . . . 14  |-  ( (
-g `  G ) : ( ( Base `  G )  X.  ( Base `  G ) ) --> ( Base `  G
)  ->  Fun  ( -g `  G ) )
5538, 54syl 15 . . . . . . . . . . . . 13  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  Fun  ( -g `  G ) )
56 xpss12 4792 . . . . . . . . . . . . . . 15  |-  ( ( S  C_  ( Base `  G )  /\  S  C_  ( Base `  G
) )  ->  ( S  X.  S )  C_  ( ( Base `  G
)  X.  ( Base `  G ) ) )
578, 8, 56syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( S  X.  S )  C_  (
( Base `  G )  X.  ( Base `  G
) ) )
5857, 40sseqtr4d 3215 . . . . . . . . . . . . 13  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( S  X.  S )  C_  dom  ( -g `  G ) )
59 funimass5 5642 . . . . . . . . . . . . 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 642 . . . . . . . . . . . 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 223 . . . . . . . . . . 11  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( S  X.  S )  C_  ( `' ( -g `  G
) " S ) )
62 eqid 2283 . . . . . . . . . . . 12  |-  U. ( J  tX  J )  = 
U. ( J  tX  J )
6362clsss 16791 . . . . . . . . . . 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 1182 . . . . . . . . . 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 17773 . . . . . . . . . . . 12  |-  ( G  e.  TopGrp  ->  ( -g `  G
)  e.  ( ( J  tX  J )  Cn  J ) )
6665adantr 451 . . . . . . . . . . 11  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( -g `  G )  e.  ( ( J  tX  J
)  Cn  J ) )
6712cncls2i 16999 . . . . . . . . . . 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 642 . . . . . . . . . 10  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  ( J  tX  J ) ) `  ( `' ( -g `  G
) " S ) )  C_  ( `' ( -g `  G )
" ( ( cls `  J ) `  S
) ) )
6964, 68sstrd 3189 . . . . . . . . 9  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  ( J  tX  J ) ) `  ( S  X.  S
) )  C_  ( `' ( -g `  G
) " ( ( cls `  J ) `
 S ) ) )
7028, 69eqsstr3d 3213 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( (
( cls `  J
) `  S )  X.  ( ( cls `  J
) `  S )
)  C_  ( `' ( -g `  G )
" ( ( cls `  J ) `  S
) ) )
7170sselda 3180 . . . . . . 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 460 . . . . . 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 706 . . . . . . . 8  |-  ( ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  (
( cls `  J
) `  S )  /\  y  e.  (
( cls `  J
) `  S )
) )  ->  G  e.  Grp )
7473, 37syl 15 . . . . . . 7  |-  ( ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  (
( cls `  J
) `  S )  /\  y  e.  (
( cls `  J
) `  S )
) )  ->  ( -g `  G ) : ( ( Base `  G
)  X.  ( Base `  G ) ) --> (
Base `  G )
)
75 ffn 5389 . . . . . . 7  |-  ( (
-g `  G ) : ( ( Base `  G )  X.  ( Base `  G ) ) --> ( Base `  G
)  ->  ( -g `  G )  Fn  (
( Base `  G )  X.  ( Base `  G
) ) )
76 elpreima 5645 . . . . . . 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 )
) ) )
7774, 75, 763syl 18 . . . . . 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 )
) ) )
7872, 77mpbid 201 . . . . 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 )
) )
7978simprd 449 . . . 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 )
)
8025, 79syl5eqel 2367 . . 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 )
)
8180ralrimivva 2635 . 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
) )
822, 36issubg4 14638 . . 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
) ) ) )
8335, 82syl 15 . 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
) ) ) )
8415, 24, 81, 83mpbir3and 1135 1  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  J ) `  S )  e.  (SubGrp `  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    C_ wss 3152   (/)c0 3455   <.cop 3643   U.cuni 3827    X. cxp 4687   `'ccnv 4688   dom cdm 4689   "cima 4692   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   TopOpenctopn 13326   0gc0g 13400   Grpcgrp 14362   -gcsg 14365  SubGrpcsubg 14615   Topctop 16631  TopOnctopon 16632   clsccl 16755    Cn ccn 16954    tX ctx 17255   TopGrpctgp 17754
This theorem is referenced by:  clsnsg  17792  tgptsmscls  17832
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-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  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-topgen 13344  df-0g 13404  df-mnd 14367  df-plusf 14368  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-cn 16957  df-tx 17257  df-tmd 17755  df-tgp 17756
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