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Theorem clssubg 17843
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 2316 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
31, 2tgptopon 17817 . . . . . 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 16720 . . . . 5  |-  ( J  e.  (TopOn `  ( Base `  G ) )  ->  J  e.  Top )
64, 5syl 15 . . . 4  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  J  e.  Top )
72subgss 14671 . . . . . 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 16721 . . . . . 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 3248 . . . 4  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  S  C_  U. J
)
12 eqid 2316 . . . . 5  |-  U. J  =  U. J
1312clsss3 16852 . . . 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 3249 . 2  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  J ) `  S )  C_  ( Base `  G ) )
1612sscls 16849 . . . 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 2316 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
1918subg0cl 14678 . . . . 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 3495 . . . 4  |-  ( ( 0g `  G )  e.  S  ->  S  =/=  (/) )
2220, 21syl 15 . . 3  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  S  =/=  (/) )
23 ssn0 3521 . . 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 5903 . . . 4  |-  ( x ( -g `  G
) y )  =  ( ( -g `  G
) `  <. x ,  y >. )
26 opelxpi 4758 . . . . . . 7  |-  ( ( x  e.  ( ( cls `  J ) `
 S )  /\  y  e.  ( ( cls `  J ) `  S ) )  ->  <. x ,  y >.  e.  ( ( ( cls `  J ) `  S
)  X.  ( ( cls `  J ) `
 S ) ) )
27 txcls 17355 . . . . . . . . . 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 17342 . . . . . . . . . . . . 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 16720 . . . . . . . . . . . 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 5070 . . . . . . . . . . . . 13  |-  ( `' ( -g `  G
) " S ) 
C_  dom  ( -g `  G )
34 tgpgrp 17813 . . . . . . . . . . . . . . . 16  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
3534adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  G  e.  Grp )
36 eqid 2316 . . . . . . . . . . . . . . . 16  |-  ( -g `  G )  =  (
-g `  G )
372, 36grpsubf 14594 . . . . . . . . . . . . . . 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 5431 . . . . . . . . . . . . . 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 3260 . . . . . . . . . . . 12  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( `' ( -g `  G )
" S )  C_  ( ( Base `  G
)  X.  ( Base `  G ) ) )
42 toponuni 16721 . . . . . . . . . . . . 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 3248 . . . . . . . . . . 11  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( `' ( -g `  G )
" S )  C_  U. ( J  tX  J
) )
4536subgsubcl 14681 . . . . . . . . . . . . . . . 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 2669 . . . . . . . . . . . . . 14  |-  ( S  e.  (SubGrp `  G
)  ->  A. x  e.  S  A. y  e.  S  ( x
( -g `  G ) y )  e.  S
)
48 fveq2 5563 . . . . . . . . . . . . . . . . 17  |-  ( z  =  <. x ,  y
>.  ->  ( ( -g `  G ) `  z
)  =  ( (
-g `  G ) `  <. x ,  y
>. ) )
4948, 25syl6eqr 2366 . . . . . . . . . . . . . . . 16  |-  ( z  =  <. x ,  y
>.  ->  ( ( -g `  G ) `  z
)  =  ( x ( -g `  G
) y ) )
5049eleq1d 2382 . . . . . . . . . . . . . . 15  |-  ( z  =  <. x ,  y
>.  ->  ( ( (
-g `  G ) `  z )  e.  S  <->  ( x ( -g `  G
) y )  e.  S ) )
5150ralxp 4864 . . . . . . . . . . . . . 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 5429 . . . . . . . . . . . . . 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 4829 . . . . . . . . . . . . . . 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 3249 . . . . . . . . . . . . 13  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( S  X.  S )  C_  dom  ( -g `  G ) )
59 funimass5 5680 . . . . . . . . . . . . 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 2316 . . . . . . . . . . . 12  |-  U. ( J  tX  J )  = 
U. ( J  tX  J )
6362clsss 16847 . . . . . . . . . . 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 17825 . . . . . . . . . . . 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 17055 . . . . . . . . . . 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 3223 . . . . . . . . 9  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  ( J  tX  J ) ) `  ( S  X.  S
) )  C_  ( `' ( -g `  G
) " ( ( cls `  J ) `
 S ) ) )
7028, 69eqsstr3d 3247 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( (
( cls `  J
) `  S )  X.  ( ( cls `  J
) `  S )
)  C_  ( `' ( -g `  G )
" ( ( cls `  J ) `  S
) ) )
7170sselda 3214 . . . . . . 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 5427 . . . . . . 7  |-  ( (
-g `  G ) : ( ( Base `  G )  X.  ( Base `  G ) ) --> ( Base `  G
)  ->  ( -g `  G )  Fn  (
( Base `  G )  X.  ( Base `  G
) ) )
76 elpreima 5683 . . . . . . 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 2400 . . 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 2669 . 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 14687 . . 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 1633    e. wcel 1701    =/= wne 2479   A.wral 2577    C_ wss 3186   (/)c0 3489   <.cop 3677   U.cuni 3864    X. cxp 4724   `'ccnv 4725   dom cdm 4726   "cima 4729   Fun wfun 5286    Fn wfn 5287   -->wf 5288   ` cfv 5292  (class class class)co 5900   Basecbs 13195   TopOpenctopn 13375   0gc0g 13449   Grpcgrp 14411   -gcsg 14414  SubGrpcsubg 14664   Topctop 16687  TopOnctopon 16688   clsccl 16811    Cn ccn 17010    tX ctx 17311   TopGrpctgp 17806
This theorem is referenced by:  clsnsg  17844  tgptsmscls  17884
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-topgen 13393  df-0g 13453  df-mnd 14416  df-plusf 14417  df-grp 14538  df-minusg 14539  df-sbg 14540  df-subg 14667  df-top 16692  df-bases 16694  df-topon 16695  df-topsp 16696  df-cld 16812  df-ntr 16813  df-cls 16814  df-cn 17013  df-tx 17313  df-tmd 17807  df-tgp 17808
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