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Theorem ghmpreima 16090
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 5356 . . 3  |-  ( `' F " V ) 
C_  dom  F
2 eqid 2467 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
3 eqid 2467 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
42, 3ghmf 16073 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
54adantr 465 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
6 fdm 5734 . . . 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 3552 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( `' F " V )  C_  ( Base `  S )
)
9 ghmgrp1 16071 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
109adantr 465 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  S  e.  Grp )
11 eqid 2467 . . . . . 6  |-  ( 0g
`  S )  =  ( 0g `  S
)
122, 11grpidcl 15885 . . . . 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 2467 . . . . . . 7  |-  ( 0g
`  T )  =  ( 0g `  T
)
1511, 14ghmid 16075 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
1615adantr 465 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
1714subg0cl 16011 . . . . . 6  |-  ( V  e.  (SubGrp `  T
)  ->  ( 0g `  T )  e.  V
)
1817adantl 466 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( 0g `  T )  e.  V
)
1916, 18eqeltrd 2555 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( F `  ( 0g `  S
) )  e.  V
)
20 ffn 5730 . . . . . 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 6000 . . . . 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 920 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  V  e.  (SubGrp `  T )
)  ->  ( 0g `  S )  e.  ( `' F " V ) )
25 ne0i 3791 . . 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 6000 . . . . 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 6000 . . . . . . . . . 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 465 . . . . . . . 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 725 . . . . . . . . . . 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 761 . . . . . . . . . . 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 763 . . . . . . . . . . 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 2467 . . . . . . . . . . . 12  |-  ( +g  `  S )  =  ( +g  `  S )
362, 35grpcl 15870 . . . . . . . . . . 11  |-  ( ( S  e.  Grp  /\  a  e.  ( Base `  S )  /\  b  e.  ( Base `  S
) )  ->  (
a ( +g  `  S
) b )  e.  ( Base `  S
) )
3732, 33, 34, 36syl3anc 1228 . . . . . . . . . 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 753 . . . . . . . . . . . 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 2467 . . . . . . . . . . . . 13  |-  ( +g  `  T )  =  ( +g  `  T )
402, 35, 39ghmlin 16074 . . . . . . . . . . . 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 1228 . . . . . . . . . . 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 754 . . . . . . . . . . . 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 762 . . . . . . . . . . . 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 764 . . . . . . . . . . . 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 16013 . . . . . . . . . . . 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 1228 . . . . . . . . . . 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 2555 . . . . . . . . . 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 6000 . . . . . . . . . . . 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 465 . . . . . . . . . 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 920 . . . . . . . . 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 615 . . . . . . . 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 2876 . . . . . 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 465 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
a  e.  ( Base `  S )  /\  ( F `  a )  e.  V ) )  ->  S  e.  Grp )
56 simprl 755 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  V  e.  (SubGrp `  T
) )  /\  (
a  e.  ( Base `  S )  /\  ( F `  a )  e.  V ) )  -> 
a  e.  ( Base `  S ) )
57 eqid 2467 . . . . . . . . 9  |-  ( invg `  S )  =  ( invg `  S )
582, 57grpinvcl 15902 . . . . . . . 8  |-  ( ( S  e.  Grp  /\  a  e.  ( Base `  S ) )  -> 
( ( invg `  S ) `  a
)  e.  ( Base `  S ) )
5955, 56, 58syl2anc 661 . . . . . . 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 2467 . . . . . . . . . 10  |-  ( invg `  T )  =  ( invg `  T )
612, 57, 60ghminv 16076 . . . . . . . . 9  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  a  e.  ( Base `  S
) )  ->  ( F `  ( ( invg `  S ) `
 a ) )  =  ( ( invg `  T ) `
 ( F `  a ) ) )
6261ad2ant2r 746 . . . . . . . 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 16012 . . . . . . . . 9  |-  ( ( V  e.  (SubGrp `  T )  /\  ( F `  a )  e.  V )  ->  (
( invg `  T ) `  ( F `  a )
)  e.  V )
6463ad2ant2l 745 . . . . . . . 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 2555 . . . . . . 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 6000 . . . . . . . . 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 465 . . . . . . 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 920 . . . . . 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 532 . . . . 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 2876 . 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 16018 . . 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 1179 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814    C_ wss 3476   (/)c0 3785   `'ccnv 4998   dom cdm 4999   "cima 5002    Fn wfn 5582   -->wf 5583   ` cfv 5587  (class class class)co 6283   Basecbs 14489   +g cplusg 14554   0gc0g 14694   Grpcgrp 15726   invgcminusg 15727  SubGrpcsubg 15997    GrpHom cghm 16066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-nn 10536  df-2 10593  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-0g 14696  df-mnd 15731  df-grp 15864  df-minusg 15865  df-subg 16000  df-ghm 16067
This theorem is referenced by:  ghmnsgpreima  16093  subggim  16116  gicsubgen  16128  lmhmpreima  17489
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