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Theorem ghmpreima 15029
Description: The inverse image of a subgroup under a homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Assertion
Ref Expression
ghmpreima  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( `' F " V )  e.  (SubGrp `  S )
)

Proof of Theorem ghmpreima
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnvimass 5226 . . 3  |-  ( `' F " V ) 
C_  dom  F
2 eqid 2438 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
3 eqid 2438 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
42, 3ghmf 15012 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
54adantr 453 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
6 fdm 5597 . . . 4  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  dom  F  =  ( Base `  S
) )
75, 6syl 16 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  dom  F  =  ( Base `  S
) )
81, 7syl5sseq 3398 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( `' F " V )  C_  ( Base `  S )
)
9 ghmgrp1 15010 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
109adantr 453 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  S  e.  Grp )
11 eqid 2438 . . . . . 6  |-  ( 0g
`  S )  =  ( 0g `  S
)
122, 11grpidcl 14835 . . . . 5  |-  ( S  e.  Grp  ->  ( 0g `  S )  e.  ( Base `  S
) )
1310, 12syl 16 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( 0g `  S )  e.  (
Base `  S )
)
14 eqid 2438 . . . . . . 7  |-  ( 0g
`  T )  =  ( 0g `  T
)
1511, 14ghmid 15014 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
1615adantr 453 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
1714subg0cl 14954 . . . . . 6  |-  ( V  e.  (SubGrp `  T
)  ->  ( 0g `  T )  e.  V
)
1817adantl 454 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( 0g `  T )  e.  V
)
1916, 18eqeltrd 2512 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( F `  ( 0g `  S
) )  e.  V
)
20 ffn 5593 . . . . . 6  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  F  Fn  ( Base `  S )
)
215, 20syl 16 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  F  Fn  ( Base `  S )
)
22 elpreima 5852 . . . . 5  |-  ( F  Fn  ( Base `  S
)  ->  ( ( 0g `  S )  e.  ( `' F " V )  <->  ( ( 0g `  S )  e.  ( Base `  S
)  /\  ( F `  ( 0g `  S
) )  e.  V
) ) )
2321, 22syl 16 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( ( 0g `  S )  e.  ( `' F " V )  <->  ( ( 0g `  S )  e.  ( Base `  S
)  /\  ( F `  ( 0g `  S
) )  e.  V
) ) )
2413, 19, 23mpbir2and 890 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( 0g `  S )  e.  ( `' F " V ) )
25 ne0i 3636 . . 3  |-  ( ( 0g `  S )  e.  ( `' F " V )  ->  ( `' F " V )  =/=  (/) )
2624, 25syl 16 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( `' F " V )  =/=  (/) )
27 elpreima 5852 . . . . 5  |-  ( F  Fn  ( Base `  S
)  ->  ( a  e.  ( `' F " V )  <->  ( a  e.  ( Base `  S
)  /\  ( F `  a )  e.  V
) ) )
2821, 27syl 16 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( a  e.  ( `' F " V )  <->  ( a  e.  ( Base `  S
)  /\  ( F `  a )  e.  V
) ) )
29 elpreima 5852 . . . . . . . . . 10  |-  ( F  Fn  ( Base `  S
)  ->  ( b  e.  ( `' F " V )  <->  ( b  e.  ( Base `  S
)  /\  ( F `  b )  e.  V
) ) )
3021, 29syl 16 . . . . . . . . 9  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( b  e.  ( `' F " V )  <->  ( b  e.  ( Base `  S
)  /\  ( F `  b )  e.  V
) ) )
3130adantr 453 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
a  e.  ( Base `  S )  /\  ( F `  a )  e.  V ) )  -> 
( b  e.  ( `' F " V )  <-> 
( b  e.  (
Base `  S )  /\  ( F `  b
)  e.  V ) ) )
329ad2antrr 708 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
( a  e.  (
Base `  S )  /\  ( F `  a
)  e.  V )  /\  ( b  e.  ( Base `  S
)  /\  ( F `  b )  e.  V
) ) )  ->  S  e.  Grp )
33 simprll 740 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
( a  e.  (
Base `  S )  /\  ( F `  a
)  e.  V )  /\  ( b  e.  ( Base `  S
)  /\  ( F `  b )  e.  V
) ) )  -> 
a  e.  ( Base `  S ) )
34 simprrl 742 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
( a  e.  (
Base `  S )  /\  ( F `  a
)  e.  V )  /\  ( b  e.  ( Base `  S
)  /\  ( F `  b )  e.  V
) ) )  -> 
b  e.  ( Base `  S ) )
35 eqid 2438 . . . . . . . . . . . 12  |-  ( +g  `  S )  =  ( +g  `  S )
362, 35grpcl 14820 . . . . . . . . . . 11  |-  ( ( S  e.  Grp  /\  a  e.  ( Base `  S )  /\  b  e.  ( Base `  S
) )  ->  (
a ( +g  `  S
) b )  e.  ( Base `  S
) )
3732, 33, 34, 36syl3anc 1185 . . . . . . . . . 10  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
( a  e.  (
Base `  S )  /\  ( F `  a
)  e.  V )  /\  ( b  e.  ( Base `  S
)  /\  ( F `  b )  e.  V
) ) )  -> 
( a ( +g  `  S ) b )  e.  ( Base `  S
) )
38 simpll 732 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
( a  e.  (
Base `  S )  /\  ( F `  a
)  e.  V )  /\  ( b  e.  ( Base `  S
)  /\  ( F `  b )  e.  V
) ) )  ->  F  e.  ( S  GrpHom  T ) )
39 eqid 2438 . . . . . . . . . . . . 13  |-  ( +g  `  T )  =  ( +g  `  T )
402, 35, 39ghmlin 15013 . . . . . . . . . . . 12  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  a  e.  ( Base `  S
)  /\  b  e.  ( Base `  S )
)  ->  ( F `  ( a ( +g  `  S ) b ) )  =  ( ( F `  a ) ( +g  `  T
) ( F `  b ) ) )
4138, 33, 34, 40syl3anc 1185 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
( a  e.  (
Base `  S )  /\  ( F `  a
)  e.  V )  /\  ( b  e.  ( Base `  S
)  /\  ( F `  b )  e.  V
) ) )  -> 
( F `  (
a ( +g  `  S
) b ) )  =  ( ( F `
 a ) ( +g  `  T ) ( F `  b
) ) )
42 simplr 733 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
( a  e.  (
Base `  S )  /\  ( F `  a
)  e.  V )  /\  ( b  e.  ( Base `  S
)  /\  ( F `  b )  e.  V
) ) )  ->  V  e.  (SubGrp `  T
) )
43 simprlr 741 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
( a  e.  (
Base `  S )  /\  ( F `  a
)  e.  V )  /\  ( b  e.  ( Base `  S
)  /\  ( F `  b )  e.  V
) ) )  -> 
( F `  a
)  e.  V )
44 simprrr 743 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
( a  e.  (
Base `  S )  /\  ( F `  a
)  e.  V )  /\  ( b  e.  ( Base `  S
)  /\  ( F `  b )  e.  V
) ) )  -> 
( F `  b
)  e.  V )
4539subgcl 14956 . . . . . . . . . . . 12  |-  ( ( V  e.  (SubGrp `  T )  /\  ( F `  a )  e.  V  /\  ( F `  b )  e.  V )  ->  (
( F `  a
) ( +g  `  T
) ( F `  b ) )  e.  V )
4642, 43, 44, 45syl3anc 1185 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
( a  e.  (
Base `  S )  /\  ( F `  a
)  e.  V )  /\  ( b  e.  ( Base `  S
)  /\  ( F `  b )  e.  V
) ) )  -> 
( ( F `  a ) ( +g  `  T ) ( F `
 b ) )  e.  V )
4741, 46eqeltrd 2512 . . . . . . . . . 10  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
( a  e.  (
Base `  S )  /\  ( F `  a
)  e.  V )  /\  ( b  e.  ( Base `  S
)  /\  ( F `  b )  e.  V
) ) )  -> 
( F `  (
a ( +g  `  S
) b ) )  e.  V )
48 elpreima 5852 . . . . . . . . . . . 12  |-  ( F  Fn  ( Base `  S
)  ->  ( (
a ( +g  `  S
) b )  e.  ( `' F " V )  <->  ( (
a ( +g  `  S
) b )  e.  ( Base `  S
)  /\  ( F `  ( a ( +g  `  S ) b ) )  e.  V ) ) )
4921, 48syl 16 . . . . . . . . . . 11  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( (
a ( +g  `  S
) b )  e.  ( `' F " V )  <->  ( (
a ( +g  `  S
) b )  e.  ( Base `  S
)  /\  ( F `  ( a ( +g  `  S ) b ) )  e.  V ) ) )
5049adantr 453 . . . . . . . . . 10  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
( a  e.  (
Base `  S )  /\  ( F `  a
)  e.  V )  /\  ( b  e.  ( Base `  S
)  /\  ( F `  b )  e.  V
) ) )  -> 
( ( a ( +g  `  S ) b )  e.  ( `' F " V )  <-> 
( ( a ( +g  `  S ) b )  e.  (
Base `  S )  /\  ( F `  (
a ( +g  `  S
) b ) )  e.  V ) ) )
5137, 47, 50mpbir2and 890 . . . . . . . . 9  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
( a  e.  (
Base `  S )  /\  ( F `  a
)  e.  V )  /\  ( b  e.  ( Base `  S
)  /\  ( F `  b )  e.  V
) ) )  -> 
( a ( +g  `  S ) b )  e.  ( `' F " V ) )
5251expr 600 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
a  e.  ( Base `  S )  /\  ( F `  a )  e.  V ) )  -> 
( ( b  e.  ( Base `  S
)  /\  ( F `  b )  e.  V
)  ->  ( a
( +g  `  S ) b )  e.  ( `' F " V ) ) )
5331, 52sylbid 208 . . . . . . 7  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
a  e.  ( Base `  S )  /\  ( F `  a )  e.  V ) )  -> 
( b  e.  ( `' F " V )  ->  ( a ( +g  `  S ) b )  e.  ( `' F " V ) ) )
5453ralrimiv 2790 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
a  e.  ( Base `  S )  /\  ( F `  a )  e.  V ) )  ->  A. b  e.  ( `' F " V ) ( a ( +g  `  S ) b )  e.  ( `' F " V ) )
5510adantr 453 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
a  e.  ( Base `  S )  /\  ( F `  a )  e.  V ) )  ->  S  e.  Grp )
56 simprl 734 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
a  e.  ( Base `  S )  /\  ( F `  a )  e.  V ) )  -> 
a  e.  ( Base `  S ) )
57 eqid 2438 . . . . . . . . 9  |-  ( inv g `  S )  =  ( inv g `  S )
582, 57grpinvcl 14852 . . . . . . . 8  |-  ( ( S  e.  Grp  /\  a  e.  ( Base `  S ) )  -> 
( ( inv g `  S ) `  a
)  e.  ( Base `  S ) )
5955, 56, 58syl2anc 644 . . . . . . 7  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
a  e.  ( Base `  S )  /\  ( F `  a )  e.  V ) )  -> 
( ( inv g `  S ) `  a
)  e.  ( Base `  S ) )
60 eqid 2438 . . . . . . . . . 10  |-  ( inv g `  T )  =  ( inv g `  T )
612, 57, 60ghminv 15015 . . . . . . . . 9  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  a  e.  ( Base `  S
) )  ->  ( F `  ( ( inv g `  S ) `
 a ) )  =  ( ( inv g `  T ) `
 ( F `  a ) ) )
6261ad2ant2r 729 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
a  e.  ( Base `  S )  /\  ( F `  a )  e.  V ) )  -> 
( F `  (
( inv g `  S ) `  a
) )  =  ( ( inv g `  T ) `  ( F `  a )
) )
6360subginvcl 14955 . . . . . . . . 9  |-  ( ( V  e.  (SubGrp `  T )  /\  ( F `  a )  e.  V )  ->  (
( inv g `  T ) `  ( F `  a )
)  e.  V )
6463ad2ant2l 728 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
a  e.  ( Base `  S )  /\  ( F `  a )  e.  V ) )  -> 
( ( inv g `  T ) `  ( F `  a )
)  e.  V )
6562, 64eqeltrd 2512 . . . . . . 7  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
a  e.  ( Base `  S )  /\  ( F `  a )  e.  V ) )  -> 
( F `  (
( inv g `  S ) `  a
) )  e.  V
)
66 elpreima 5852 . . . . . . . . 9  |-  ( F  Fn  ( Base `  S
)  ->  ( (
( inv g `  S ) `  a
)  e.  ( `' F " V )  <-> 
( ( ( inv g `  S ) `
 a )  e.  ( Base `  S
)  /\  ( F `  ( ( inv g `  S ) `  a
) )  e.  V
) ) )
6721, 66syl 16 . . . . . . . 8  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( (
( inv g `  S ) `  a
)  e.  ( `' F " V )  <-> 
( ( ( inv g `  S ) `
 a )  e.  ( Base `  S
)  /\  ( F `  ( ( inv g `  S ) `  a
) )  e.  V
) ) )
6867adantr 453 . . . . . . 7  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
a  e.  ( Base `  S )  /\  ( F `  a )  e.  V ) )  -> 
( ( ( inv g `  S ) `
 a )  e.  ( `' F " V )  <->  ( (
( inv g `  S ) `  a
)  e.  ( Base `  S )  /\  ( F `  ( ( inv g `  S ) `
 a ) )  e.  V ) ) )
6959, 65, 68mpbir2and 890 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
a  e.  ( Base `  S )  /\  ( F `  a )  e.  V ) )  -> 
( ( inv g `  S ) `  a
)  e.  ( `' F " V ) )
7054, 69jca 520 . . . . 5  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
a  e.  ( Base `  S )  /\  ( F `  a )  e.  V ) )  -> 
( A. b  e.  ( `' F " V ) ( a ( +g  `  S
) b )  e.  ( `' F " V )  /\  (
( inv g `  S ) `  a
)  e.  ( `' F " V ) ) )
7170ex 425 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( (
a  e.  ( Base `  S )  /\  ( F `  a )  e.  V )  ->  ( A. b  e.  ( `' F " V ) ( a ( +g  `  S ) b )  e.  ( `' F " V )  /\  (
( inv g `  S ) `  a
)  e.  ( `' F " V ) ) ) )
7228, 71sylbid 208 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( a  e.  ( `' F " V )  ->  ( A. b  e.  ( `' F " V ) ( a ( +g  `  S ) b )  e.  ( `' F " V )  /\  (
( inv g `  S ) `  a
)  e.  ( `' F " V ) ) ) )
7372ralrimiv 2790 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  A. a  e.  ( `' F " V ) ( A. b  e.  ( `' F " V ) ( a ( +g  `  S
) b )  e.  ( `' F " V )  /\  (
( inv g `  S ) `  a
)  e.  ( `' F " V ) ) )
742, 35, 57issubg2 14961 . . 3  |-  ( S  e.  Grp  ->  (
( `' F " V )  e.  (SubGrp `  S )  <->  ( ( `' F " V ) 
C_  ( Base `  S
)  /\  ( `' F " V )  =/=  (/)  /\  A. a  e.  ( `' F " V ) ( A. b  e.  ( `' F " V ) ( a ( +g  `  S
) b )  e.  ( `' F " V )  /\  (
( inv g `  S ) `  a
)  e.  ( `' F " V ) ) ) ) )
7510, 74syl 16 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( ( `' F " V )  e.  (SubGrp `  S
)  <->  ( ( `' F " V ) 
C_  ( Base `  S
)  /\  ( `' F " V )  =/=  (/)  /\  A. a  e.  ( `' F " V ) ( A. b  e.  ( `' F " V ) ( a ( +g  `  S
) b )  e.  ( `' F " V )  /\  (
( inv g `  S ) `  a
)  e.  ( `' F " V ) ) ) ) )
768, 26, 73, 75mpbir3and 1138 1  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( `' F " V )  e.  (SubGrp `  S )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707    C_ wss 3322   (/)c0 3630   `'ccnv 4879   dom cdm 4880   "cima 4883    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531   0gc0g 13725   Grpcgrp 14687   inv gcminusg 14688  SubGrpcsubg 14940    GrpHom cghm 15005
This theorem is referenced by:  ghmnsgpreima  15032  subggim  15055  gicsubgen  15067  lmhmpreima  16126
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815  df-subg 14943  df-ghm 15006
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