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Theorem clssubg 18130
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 2435 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
31, 2tgptopon 18104 . . . . . 6  |-  ( G  e.  TopGrp  ->  J  e.  (TopOn `  ( Base `  G
) ) )
43adantr 452 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  J  e.  (TopOn `  ( Base `  G
) ) )
5 topontop 16983 . . . . 5  |-  ( J  e.  (TopOn `  ( Base `  G ) )  ->  J  e.  Top )
64, 5syl 16 . . . 4  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  J  e.  Top )
72subgss 14937 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
87adantl 453 . . . . 5  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  S  C_  ( Base `  G ) )
9 toponuni 16984 . . . . . 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 3376 . . . 4  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  S  C_  U. J
)
12 eqid 2435 . . . . 5  |-  U. J  =  U. J
1312clsss3 17115 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  ( ( cls `  J ) `  S
)  C_  U. J )
146, 11, 13syl2anc 643 . . 3  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  J ) `  S )  C_  U. J
)
1514, 10sseqtr4d 3377 . 2  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  J ) `  S )  C_  ( Base `  G ) )
1612sscls 17112 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  S  C_  (
( cls `  J
) `  S )
)
176, 11, 16syl2anc 643 . . 3  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  S  C_  (
( cls `  J
) `  S )
)
18 eqid 2435 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
1918subg0cl 14944 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  S
)
2019adantl 453 . . . 4  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( 0g `  G )  e.  S
)
21 ne0i 3626 . . . 4  |-  ( ( 0g `  G )  e.  S  ->  S  =/=  (/) )
2220, 21syl 16 . . 3  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  S  =/=  (/) )
23 ssn0 3652 . . 3  |-  ( ( S  C_  ( ( cls `  J ) `  S )  /\  S  =/=  (/) )  ->  (
( cls `  J
) `  S )  =/=  (/) )
2417, 22, 23syl2anc 643 . 2  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  J ) `  S )  =/=  (/) )
25 df-ov 6076 . . . 4  |-  ( x ( -g `  G
) y )  =  ( ( -g `  G
) `  <. x ,  y >. )
26 opelxpi 4902 . . . . . . 7  |-  ( ( x  e.  ( ( cls `  J ) `
 S )  /\  y  e.  ( ( cls `  J ) `  S ) )  ->  <. x ,  y >.  e.  ( ( ( cls `  J ) `  S
)  X.  ( ( cls `  J ) `
 S ) ) )
27 txcls 17628 . . . . . . . . . 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 1185 . . . . . . . . 9  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  ( J  tX  J ) ) `  ( S  X.  S
) )  =  ( ( ( cls `  J
) `  S )  X.  ( ( cls `  J
) `  S )
) )
29 txtopon 17615 . . . . . . . . . . . . 13  |-  ( ( J  e.  (TopOn `  ( Base `  G )
)  /\  J  e.  (TopOn `  ( Base `  G
) ) )  -> 
( J  tX  J
)  e.  (TopOn `  ( ( Base `  G
)  X.  ( Base `  G ) ) ) )
304, 4, 29syl2anc 643 . . . . . . . . . . . 12  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( J  tX  J )  e.  (TopOn `  ( ( Base `  G
)  X.  ( Base `  G ) ) ) )
31 topontop 16983 . . . . . . . . . . . 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 5216 . . . . . . . . . . . . 13  |-  ( `' ( -g `  G
) " S ) 
C_  dom  ( -g `  G )
34 tgpgrp 18100 . . . . . . . . . . . . . . . 16  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
3534adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  G  e.  Grp )
36 eqid 2435 . . . . . . . . . . . . . . . 16  |-  ( -g `  G )  =  (
-g `  G )
372, 36grpsubf 14860 . . . . . . . . . . . . . . 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 5587 . . . . . . . . . . . . . 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 3388 . . . . . . . . . . . 12  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( `' ( -g `  G )
" S )  C_  ( ( Base `  G
)  X.  ( Base `  G ) ) )
42 toponuni 16984 . . . . . . . . . . . . 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 3376 . . . . . . . . . . 11  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( `' ( -g `  G )
" S )  C_  U. ( J  tX  J
) )
4536subgsubcl 14947 . . . . . . . . . . . . . . . 16  |-  ( ( S  e.  (SubGrp `  G )  /\  x  e.  S  /\  y  e.  S )  ->  (
x ( -g `  G
) y )  e.  S )
46453expb 1154 . . . . . . . . . . . . . . 15  |-  ( ( S  e.  (SubGrp `  G )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x
( -g `  G ) y )  e.  S
)
4746ralrimivva 2790 . . . . . . . . . . . . . 14  |-  ( S  e.  (SubGrp `  G
)  ->  A. x  e.  S  A. y  e.  S  ( x
( -g `  G ) y )  e.  S
)
48 fveq2 5720 . . . . . . . . . . . . . . . . 17  |-  ( z  =  <. x ,  y
>.  ->  ( ( -g `  G ) `  z
)  =  ( (
-g `  G ) `  <. x ,  y
>. ) )
4948, 25syl6eqr 2485 . . . . . . . . . . . . . . . 16  |-  ( z  =  <. x ,  y
>.  ->  ( ( -g `  G ) `  z
)  =  ( x ( -g `  G
) y ) )
5049eleq1d 2501 . . . . . . . . . . . . . . 15  |-  ( z  =  <. x ,  y
>.  ->  ( ( (
-g `  G ) `  z )  e.  S  <->  ( x ( -g `  G
) y )  e.  S ) )
5150ralxp 5008 . . . . . . . . . . . . . 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 204 . . . . . . . . . . . . 13  |-  ( S  e.  (SubGrp `  G
)  ->  A. z  e.  ( S  X.  S
) ( ( -g `  G ) `  z
)  e.  S )
5352adantl 453 . . . . . . . . . . . 12  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  A. z  e.  ( S  X.  S
) ( ( -g `  G ) `  z
)  e.  S )
54 ffun 5585 . . . . . . . . . . . . . 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 4973 . . . . . . . . . . . . . . 15  |-  ( ( S  C_  ( Base `  G )  /\  S  C_  ( Base `  G
) )  ->  ( S  X.  S )  C_  ( ( Base `  G
)  X.  ( Base `  G ) ) )
578, 8, 56syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( S  X.  S )  C_  (
( Base `  G )  X.  ( Base `  G
) ) )
5857, 40sseqtr4d 3377 . . . . . . . . . . . . 13  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( S  X.  S )  C_  dom  ( -g `  G ) )
59 funimass5 5839 . . . . . . . . . . . . 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 643 . . . . . . . . . . . 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 224 . . . . . . . . . . 11  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( S  X.  S )  C_  ( `' ( -g `  G
) " S ) )
62 eqid 2435 . . . . . . . . . . . 12  |-  U. ( J  tX  J )  = 
U. ( J  tX  J )
6362clsss 17110 . . . . . . . . . . 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 1184 . . . . . . . . . 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 18112 . . . . . . . . . . . 12  |-  ( G  e.  TopGrp  ->  ( -g `  G
)  e.  ( ( J  tX  J )  Cn  J ) )
6665adantr 452 . . . . . . . . . . 11  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( -g `  G )  e.  ( ( J  tX  J
)  Cn  J ) )
6712cncls2i 17326 . . . . . . . . . . 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 643 . . . . . . . . . 10  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  ( J  tX  J ) ) `  ( `' ( -g `  G
) " S ) )  C_  ( `' ( -g `  G )
" ( ( cls `  J ) `  S
) ) )
6964, 68sstrd 3350 . . . . . . . . 9  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( ( cls `  ( J  tX  J ) ) `  ( S  X.  S
) )  C_  ( `' ( -g `  G
) " ( ( cls `  J ) `
 S ) ) )
7028, 69eqsstr3d 3375 . . . . . . . 8  |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )
)  ->  ( (
( cls `  J
) `  S )  X.  ( ( cls `  J
) `  S )
)  C_  ( `' ( -g `  G )
" ( ( cls `  J ) `  S
) ) )
7170sselda 3340 . . . . . . 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 461 . . . . . 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 707 . . . . . . . 8  |-  ( ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  (
( cls `  J
) `  S )  /\  y  e.  (
( cls `  J
) `  S )
) )  ->  G  e.  Grp )
7473, 37syl 16 . . . . . . 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 5583 . . . . . . 7  |-  ( (
-g `  G ) : ( ( Base `  G )  X.  ( Base `  G ) ) --> ( Base `  G
)  ->  ( -g `  G )  Fn  (
( Base `  G )  X.  ( Base `  G
) ) )
76 elpreima 5842 . . . . . . 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 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 )
) ) )
7872, 77mpbid 202 . . . . 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 450 . . . 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 2519 . . 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 2790 . 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 14953 . . 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 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
) ) ) )
8415, 24, 81, 83mpbir3and 1137 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697    C_ wss 3312   (/)c0 3620   <.cop 3809   U.cuni 4007    X. cxp 4868   `'ccnv 4869   dom cdm 4870   "cima 4873   Fun wfun 5440    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   Basecbs 13461   TopOpenctopn 13641   0gc0g 13715   Grpcgrp 14677   -gcsg 14680  SubGrpcsubg 14930   Topctop 16950  TopOnctopon 16951   clsccl 17074    Cn ccn 17280    tX ctx 17584   TopGrpctgp 18093
This theorem is referenced by:  clsnsg  18131  tgptsmscls  18171
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-topgen 13659  df-0g 13719  df-mnd 14682  df-plusf 14683  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-ntr 17076  df-cls 17077  df-cn 17283  df-tx 17586  df-tmd 18094  df-tgp 18095
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