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Theorem ghmpreima 14704
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 5033 . . 3  |-  ( `' F " V ) 
C_  dom  F
2 eqid 2283 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
3 eqid 2283 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
42, 3ghmf 14687 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
54adantr 451 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
6 fdm 5393 . . . 4  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  dom  F  =  ( Base `  S
) )
75, 6syl 15 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  dom  F  =  ( Base `  S
) )
81, 7syl5sseq 3226 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( `' F " V )  C_  ( Base `  S )
)
9 ghmgrp1 14685 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
109adantr 451 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  S  e.  Grp )
11 eqid 2283 . . . . . 6  |-  ( 0g
`  S )  =  ( 0g `  S
)
122, 11grpidcl 14510 . . . . 5  |-  ( S  e.  Grp  ->  ( 0g `  S )  e.  ( Base `  S
) )
1310, 12syl 15 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( 0g `  S )  e.  (
Base `  S )
)
14 eqid 2283 . . . . . . 7  |-  ( 0g
`  T )  =  ( 0g `  T
)
1511, 14ghmid 14689 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
1615adantr 451 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
1714subg0cl 14629 . . . . . 6  |-  ( V  e.  (SubGrp `  T
)  ->  ( 0g `  T )  e.  V
)
1817adantl 452 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( 0g `  T )  e.  V
)
1916, 18eqeltrd 2357 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( F `  ( 0g `  S
) )  e.  V
)
20 ffn 5389 . . . . . 6  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  F  Fn  ( Base `  S )
)
215, 20syl 15 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  F  Fn  ( Base `  S )
)
22 elpreima 5645 . . . . 5  |-  ( F  Fn  ( Base `  S
)  ->  ( ( 0g `  S )  e.  ( `' F " V )  <->  ( ( 0g `  S )  e.  ( Base `  S
)  /\  ( F `  ( 0g `  S
) )  e.  V
) ) )
2321, 22syl 15 . . . 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 888 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( 0g `  S )  e.  ( `' F " V ) )
25 ne0i 3461 . . 3  |-  ( ( 0g `  S )  e.  ( `' F " V )  ->  ( `' F " V )  =/=  (/) )
2624, 25syl 15 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( `' F " V )  =/=  (/) )
27 elpreima 5645 . . . . 5  |-  ( F  Fn  ( Base `  S
)  ->  ( a  e.  ( `' F " V )  <->  ( a  e.  ( Base `  S
)  /\  ( F `  a )  e.  V
) ) )
2821, 27syl 15 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( a  e.  ( `' F " V )  <->  ( a  e.  ( Base `  S
)  /\  ( F `  a )  e.  V
) ) )
29 elpreima 5645 . . . . . . . . . 10  |-  ( F  Fn  ( Base `  S
)  ->  ( b  e.  ( `' F " V )  <->  ( b  e.  ( Base `  S
)  /\  ( F `  b )  e.  V
) ) )
3021, 29syl 15 . . . . . . . . 9  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( b  e.  ( `' F " V )  <->  ( b  e.  ( Base `  S
)  /\  ( F `  b )  e.  V
) ) )
3130adantr 451 . . . . . . . 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 706 . . . . . . . . . . 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 738 . . . . . . . . . . 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 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
) ) )  -> 
b  e.  ( Base `  S ) )
35 eqid 2283 . . . . . . . . . . . 12  |-  ( +g  `  S )  =  ( +g  `  S )
362, 35grpcl 14495 . . . . . . . . . . 11  |-  ( ( S  e.  Grp  /\  a  e.  ( Base `  S )  /\  b  e.  ( Base `  S
) )  ->  (
a ( +g  `  S
) b )  e.  ( Base `  S
) )
3732, 33, 34, 36syl3anc 1182 . . . . . . . . . 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 730 . . . . . . . . . . . 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 2283 . . . . . . . . . . . . 13  |-  ( +g  `  T )  =  ( +g  `  T )
402, 35, 39ghmlin 14688 . . . . . . . . . . . 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 1182 . . . . . . . . . . 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 731 . . . . . . . . . . . 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 739 . . . . . . . . . . . 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 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 `  b
)  e.  V )
4539subgcl 14631 . . . . . . . . . . . 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 1182 . . . . . . . . . . 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 2357 . . . . . . . . . 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 5645 . . . . . . . . . . . 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 15 . . . . . . . . . . 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 451 . . . . . . . . . 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 888 . . . . . . . . 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 598 . . . . . . . 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 206 . . . . . . 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 2625 . . . . . 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 451 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
a  e.  ( Base `  S )  /\  ( F `  a )  e.  V ) )  ->  S  e.  Grp )
56 simprl 732 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
a  e.  ( Base `  S )  /\  ( F `  a )  e.  V ) )  -> 
a  e.  ( Base `  S ) )
57 eqid 2283 . . . . . . . . 9  |-  ( inv g `  S )  =  ( inv g `  S )
582, 57grpinvcl 14527 . . . . . . . 8  |-  ( ( S  e.  Grp  /\  a  e.  ( Base `  S ) )  -> 
( ( inv g `  S ) `  a
)  e.  ( Base `  S ) )
5955, 56, 58syl2anc 642 . . . . . . 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 2283 . . . . . . . . . 10  |-  ( inv g `  T )  =  ( inv g `  T )
612, 57, 60ghminv 14690 . . . . . . . . 9  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  a  e.  ( Base `  S
) )  ->  ( F `  ( ( inv g `  S ) `
 a ) )  =  ( ( inv g `  T ) `
 ( F `  a ) ) )
6261ad2ant2r 727 . . . . . . . 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 14630 . . . . . . . . 9  |-  ( ( V  e.  (SubGrp `  T )  /\  ( F `  a )  e.  V )  ->  (
( inv g `  T ) `  ( F `  a )
)  e.  V )
6463ad2ant2l 726 . . . . . . . 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 2357 . . . . . . 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 5645 . . . . . . . . 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 15 . . . . . . . 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 451 . . . . . . 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 888 . . . . . 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 518 . . . . 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 423 . . . 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 206 . . 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 2625 . 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 14636 . . 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 15 . 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 1135 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    C_ wss 3152   (/)c0 3455   `'ccnv 4688   dom cdm 4689   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Grpcgrp 14362   inv gcminusg 14363  SubGrpcsubg 14615    GrpHom cghm 14680
This theorem is referenced by:  ghmnsgpreima  14707  subggim  14730  gicsubgen  14742  lmhmpreima  15805
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-iun 3907  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-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  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-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-subg 14618  df-ghm 14681
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