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Theorem ghmpreima 15761
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 5186 . . 3  |-  ( `' F " V ) 
C_  dom  F
2 eqid 2441 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
3 eqid 2441 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
42, 3ghmf 15744 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
54adantr 462 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
6 fdm 5560 . . . 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 3401 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( `' F " V )  C_  ( Base `  S )
)
9 ghmgrp1 15742 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
109adantr 462 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  S  e.  Grp )
11 eqid 2441 . . . . . 6  |-  ( 0g
`  S )  =  ( 0g `  S
)
122, 11grpidcl 15559 . . . . 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 2441 . . . . . . 7  |-  ( 0g
`  T )  =  ( 0g `  T
)
1511, 14ghmid 15746 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
1615adantr 462 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
1714subg0cl 15682 . . . . . 6  |-  ( V  e.  (SubGrp `  T
)  ->  ( 0g `  T )  e.  V
)
1817adantl 463 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( 0g `  T )  e.  V
)
1916, 18eqeltrd 2515 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( F `  ( 0g `  S
) )  e.  V
)
20 ffn 5556 . . . . . 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 5820 . . . . 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 908 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( 0g `  S )  e.  ( `' F " V ) )
25 ne0i 3640 . . 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 5820 . . . . 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 5820 . . . . . . . . . 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 462 . . . . . . . 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 720 . . . . . . . . . . 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 756 . . . . . . . . . . 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 758 . . . . . . . . . . 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 2441 . . . . . . . . . . . 12  |-  ( +g  `  S )  =  ( +g  `  S )
362, 35grpcl 15544 . . . . . . . . . . 11  |-  ( ( S  e.  Grp  /\  a  e.  ( Base `  S )  /\  b  e.  ( Base `  S
) )  ->  (
a ( +g  `  S
) b )  e.  ( Base `  S
) )
3732, 33, 34, 36syl3anc 1213 . . . . . . . . . 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 748 . . . . . . . . . . . 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 2441 . . . . . . . . . . . . 13  |-  ( +g  `  T )  =  ( +g  `  T )
402, 35, 39ghmlin 15745 . . . . . . . . . . . 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 1213 . . . . . . . . . . 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 749 . . . . . . . . . . . 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 757 . . . . . . . . . . . 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 759 . . . . . . . . . . . 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 15684 . . . . . . . . . . . 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 1213 . . . . . . . . . . 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 2515 . . . . . . . . . 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 5820 . . . . . . . . . . . 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 462 . . . . . . . . . 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 908 . . . . . . . . 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 612 . . . . . . . 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 215 . . . . . . 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 2796 . . . . . 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 462 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
a  e.  ( Base `  S )  /\  ( F `  a )  e.  V ) )  ->  S  e.  Grp )
56 simprl 750 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
a  e.  ( Base `  S )  /\  ( F `  a )  e.  V ) )  -> 
a  e.  ( Base `  S ) )
57 eqid 2441 . . . . . . . . 9  |-  ( invg `  S )  =  ( invg `  S )
582, 57grpinvcl 15576 . . . . . . . 8  |-  ( ( S  e.  Grp  /\  a  e.  ( Base `  S ) )  -> 
( ( invg `  S ) `  a
)  e.  ( Base `  S ) )
5955, 56, 58syl2anc 656 . . . . . . 7  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
a  e.  ( Base `  S )  /\  ( F `  a )  e.  V ) )  -> 
( ( invg `  S ) `  a
)  e.  ( Base `  S ) )
60 eqid 2441 . . . . . . . . . 10  |-  ( invg `  T )  =  ( invg `  T )
612, 57, 60ghminv 15747 . . . . . . . . 9  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  a  e.  ( Base `  S
) )  ->  ( F `  ( ( invg `  S ) `
 a ) )  =  ( ( invg `  T ) `
 ( F `  a ) ) )
6261ad2ant2r 741 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
a  e.  ( Base `  S )  /\  ( F `  a )  e.  V ) )  -> 
( F `  (
( invg `  S ) `  a
) )  =  ( ( invg `  T ) `  ( F `  a )
) )
6360subginvcl 15683 . . . . . . . . 9  |-  ( ( V  e.  (SubGrp `  T )  /\  ( F `  a )  e.  V )  ->  (
( invg `  T ) `  ( F `  a )
)  e.  V )
6463ad2ant2l 740 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
a  e.  ( Base `  S )  /\  ( F `  a )  e.  V ) )  -> 
( ( invg `  T ) `  ( F `  a )
)  e.  V )
6562, 64eqeltrd 2515 . . . . . . 7  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
a  e.  ( Base `  S )  /\  ( F `  a )  e.  V ) )  -> 
( F `  (
( invg `  S ) `  a
) )  e.  V
)
66 elpreima 5820 . . . . . . . . 9  |-  ( F  Fn  ( Base `  S
)  ->  ( (
( invg `  S ) `  a
)  e.  ( `' F " V )  <-> 
( ( ( invg `  S ) `
 a )  e.  ( Base `  S
)  /\  ( F `  ( ( invg `  S ) `  a
) )  e.  V
) ) )
6721, 66syl 16 . . . . . . . 8  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( (
( invg `  S ) `  a
)  e.  ( `' F " V )  <-> 
( ( ( invg `  S ) `
 a )  e.  ( Base `  S
)  /\  ( F `  ( ( invg `  S ) `  a
) )  e.  V
) ) )
6867adantr 462 . . . . . . 7  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
a  e.  ( Base `  S )  /\  ( F `  a )  e.  V ) )  -> 
( ( ( invg `  S ) `
 a )  e.  ( `' F " V )  <->  ( (
( invg `  S ) `  a
)  e.  ( Base `  S )  /\  ( F `  ( ( invg `  S ) `
 a ) )  e.  V ) ) )
6959, 65, 68mpbir2and 908 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
a  e.  ( Base `  S )  /\  ( F `  a )  e.  V ) )  -> 
( ( invg `  S ) `  a
)  e.  ( `' F " V ) )
7054, 69jca 529 . . . . 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 )  /\  (
( invg `  S ) `  a
)  e.  ( `' F " V ) ) )
7170ex 434 . . . 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 )  /\  (
( invg `  S ) `  a
)  e.  ( `' F " V ) ) ) )
7228, 71sylbid 215 . . 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 )  /\  (
( invg `  S ) `  a
)  e.  ( `' F " V ) ) ) )
7372ralrimiv 2796 . 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 )  /\  (
( invg `  S ) `  a
)  e.  ( `' F " V ) ) )
742, 35, 57issubg2 15689 . . 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 )  /\  (
( invg `  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 )  /\  (
( invg `  S ) `  a
)  e.  ( `' F " V ) ) ) ) )
768, 26, 73, 75mpbir3and 1166 1  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( `' F " V )  e.  (SubGrp `  S )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713    C_ wss 3325   (/)c0 3634   `'ccnv 4835   dom cdm 4836   "cima 4839    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   Basecbs 14170   +g cplusg 14234   0gc0g 14374   Grpcgrp 15406   invgcminusg 15407  SubGrpcsubg 15668    GrpHom cghm 15737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-0g 14376  df-mnd 15411  df-grp 15538  df-minusg 15539  df-subg 15671  df-ghm 15738
This theorem is referenced by:  ghmnsgpreima  15764  subggim  15787  gicsubgen  15799  lmhmpreima  17107
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