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Theorem ghmeql 15020
Description: The equalizer of two group homomorphisms is a subgroup. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
Assertion
Ref Expression
ghmeql  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  dom  ( F  i^i  G )  e.  (SubGrp `  S )
)

Proof of Theorem ghmeql
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghmmhm 15008 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  F  e.  ( S MndHom  T ) )
2 ghmmhm 15008 . . 3  |-  ( G  e.  ( S  GrpHom  T )  ->  G  e.  ( S MndHom  T ) )
3 mhmeql 14756 . . 3  |-  ( ( F  e.  ( S MndHom  T )  /\  G  e.  ( S MndHom  T ) )  ->  dom  ( F  i^i  G )  e.  (SubMnd `  S )
)
41, 2, 3syl2an 464 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  dom  ( F  i^i  G )  e.  (SubMnd `  S )
)
5 ghmgrp1 15000 . . . . . . . . . 10  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
65adantr 452 . . . . . . . . 9  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  S  e.  Grp )
76adantr 452 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  /\  ( x  e.  ( Base `  S )  /\  ( F `  x )  =  ( G `  x ) ) )  ->  S  e.  Grp )
8 simprl 733 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  /\  ( x  e.  ( Base `  S )  /\  ( F `  x )  =  ( G `  x ) ) )  ->  x  e.  (
Base `  S )
)
9 eqid 2435 . . . . . . . . 9  |-  ( Base `  S )  =  (
Base `  S )
10 eqid 2435 . . . . . . . . 9  |-  ( inv g `  S )  =  ( inv g `  S )
119, 10grpinvcl 14842 . . . . . . . 8  |-  ( ( S  e.  Grp  /\  x  e.  ( Base `  S ) )  -> 
( ( inv g `  S ) `  x
)  e.  ( Base `  S ) )
127, 8, 11syl2anc 643 . . . . . . 7  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  /\  ( x  e.  ( Base `  S )  /\  ( F `  x )  =  ( G `  x ) ) )  ->  ( ( inv g `  S ) `
 x )  e.  ( Base `  S
) )
13 simprr 734 . . . . . . . . 9  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  /\  ( x  e.  ( Base `  S )  /\  ( F `  x )  =  ( G `  x ) ) )  ->  ( F `  x )  =  ( G `  x ) )
1413fveq2d 5724 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  /\  ( x  e.  ( Base `  S )  /\  ( F `  x )  =  ( G `  x ) ) )  ->  ( ( inv g `  T ) `
 ( F `  x ) )  =  ( ( inv g `  T ) `  ( G `  x )
) )
15 eqid 2435 . . . . . . . . . 10  |-  ( inv g `  T )  =  ( inv g `  T )
169, 10, 15ghminv 15005 . . . . . . . . 9  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  x  e.  ( Base `  S
) )  ->  ( F `  ( ( inv g `  S ) `
 x ) )  =  ( ( inv g `  T ) `
 ( F `  x ) ) )
1716ad2ant2r 728 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  /\  ( x  e.  ( Base `  S )  /\  ( F `  x )  =  ( G `  x ) ) )  ->  ( F `  ( ( inv g `  S ) `  x
) )  =  ( ( inv g `  T ) `  ( F `  x )
) )
189, 10, 15ghminv 15005 . . . . . . . . 9  |-  ( ( G  e.  ( S 
GrpHom  T )  /\  x  e.  ( Base `  S
) )  ->  ( G `  ( ( inv g `  S ) `
 x ) )  =  ( ( inv g `  T ) `
 ( G `  x ) ) )
1918ad2ant2lr 729 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  /\  ( x  e.  ( Base `  S )  /\  ( F `  x )  =  ( G `  x ) ) )  ->  ( G `  ( ( inv g `  S ) `  x
) )  =  ( ( inv g `  T ) `  ( G `  x )
) )
2014, 17, 193eqtr4d 2477 . . . . . . 7  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  /\  ( x  e.  ( Base `  S )  /\  ( F `  x )  =  ( G `  x ) ) )  ->  ( F `  ( ( inv g `  S ) `  x
) )  =  ( G `  ( ( inv g `  S
) `  x )
) )
21 fveq2 5720 . . . . . . . . 9  |-  ( y  =  ( ( inv g `  S ) `
 x )  -> 
( F `  y
)  =  ( F `
 ( ( inv g `  S ) `
 x ) ) )
22 fveq2 5720 . . . . . . . . 9  |-  ( y  =  ( ( inv g `  S ) `
 x )  -> 
( G `  y
)  =  ( G `
 ( ( inv g `  S ) `
 x ) ) )
2321, 22eqeq12d 2449 . . . . . . . 8  |-  ( y  =  ( ( inv g `  S ) `
 x )  -> 
( ( F `  y )  =  ( G `  y )  <-> 
( F `  (
( inv g `  S ) `  x
) )  =  ( G `  ( ( inv g `  S
) `  x )
) ) )
2423elrab 3084 . . . . . . 7  |-  ( ( ( inv g `  S ) `  x
)  e.  { y  e.  ( Base `  S
)  |  ( F `
 y )  =  ( G `  y
) }  <->  ( (
( inv g `  S ) `  x
)  e.  ( Base `  S )  /\  ( F `  ( ( inv g `  S ) `
 x ) )  =  ( G `  ( ( inv g `  S ) `  x
) ) ) )
2512, 20, 24sylanbrc 646 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  /\  ( x  e.  ( Base `  S )  /\  ( F `  x )  =  ( G `  x ) ) )  ->  ( ( inv g `  S ) `
 x )  e. 
{ y  e.  (
Base `  S )  |  ( F `  y )  =  ( G `  y ) } )
2625expr 599 . . . . 5  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  /\  x  e.  ( Base `  S ) )  -> 
( ( F `  x )  =  ( G `  x )  ->  ( ( inv g `  S ) `
 x )  e. 
{ y  e.  (
Base `  S )  |  ( F `  y )  =  ( G `  y ) } ) )
2726ralrimiva 2781 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  A. x  e.  ( Base `  S
) ( ( F `
 x )  =  ( G `  x
)  ->  ( ( inv g `  S ) `
 x )  e. 
{ y  e.  (
Base `  S )  |  ( F `  y )  =  ( G `  y ) } ) )
28 fveq2 5720 . . . . . 6  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
29 fveq2 5720 . . . . . 6  |-  ( y  =  x  ->  ( G `  y )  =  ( G `  x ) )
3028, 29eqeq12d 2449 . . . . 5  |-  ( y  =  x  ->  (
( F `  y
)  =  ( G `
 y )  <->  ( F `  x )  =  ( G `  x ) ) )
3130ralrab 3088 . . . 4  |-  ( A. x  e.  { y  e.  ( Base `  S
)  |  ( F `
 y )  =  ( G `  y
) }  ( ( inv g `  S
) `  x )  e.  { y  e.  (
Base `  S )  |  ( F `  y )  =  ( G `  y ) }  <->  A. x  e.  (
Base `  S )
( ( F `  x )  =  ( G `  x )  ->  ( ( inv g `  S ) `
 x )  e. 
{ y  e.  (
Base `  S )  |  ( F `  y )  =  ( G `  y ) } ) )
3227, 31sylibr 204 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  A. x  e.  { y  e.  (
Base `  S )  |  ( F `  y )  =  ( G `  y ) }  ( ( inv g `  S ) `
 x )  e. 
{ y  e.  (
Base `  S )  |  ( F `  y )  =  ( G `  y ) } )
33 eqid 2435 . . . . . . . 8  |-  ( Base `  T )  =  (
Base `  T )
349, 33ghmf 15002 . . . . . . 7  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
3534adantr 452 . . . . . 6  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  F :
( Base `  S ) --> ( Base `  T )
)
36 ffn 5583 . . . . . 6  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  F  Fn  ( Base `  S )
)
3735, 36syl 16 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  F  Fn  ( Base `  S )
)
389, 33ghmf 15002 . . . . . . 7  |-  ( G  e.  ( S  GrpHom  T )  ->  G :
( Base `  S ) --> ( Base `  T )
)
3938adantl 453 . . . . . 6  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  G :
( Base `  S ) --> ( Base `  T )
)
40 ffn 5583 . . . . . 6  |-  ( G : ( Base `  S
) --> ( Base `  T
)  ->  G  Fn  ( Base `  S )
)
4139, 40syl 16 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  G  Fn  ( Base `  S )
)
42 fndmin 5829 . . . . 5  |-  ( ( F  Fn  ( Base `  S )  /\  G  Fn  ( Base `  S
) )  ->  dom  ( F  i^i  G )  =  { y  e.  ( Base `  S
)  |  ( F `
 y )  =  ( G `  y
) } )
4337, 41, 42syl2anc 643 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  dom  ( F  i^i  G )  =  { y  e.  (
Base `  S )  |  ( F `  y )  =  ( G `  y ) } )
44 eleq2 2496 . . . . 5  |-  ( dom  ( F  i^i  G
)  =  { y  e.  ( Base `  S
)  |  ( F `
 y )  =  ( G `  y
) }  ->  (
( ( inv g `  S ) `  x
)  e.  dom  ( F  i^i  G )  <->  ( ( inv g `  S ) `
 x )  e. 
{ y  e.  (
Base `  S )  |  ( F `  y )  =  ( G `  y ) } ) )
4544raleqbi1dv 2904 . . . 4  |-  ( dom  ( F  i^i  G
)  =  { y  e.  ( Base `  S
)  |  ( F `
 y )  =  ( G `  y
) }  ->  ( A. x  e.  dom  ( F  i^i  G ) ( ( inv g `  S ) `  x
)  e.  dom  ( F  i^i  G )  <->  A. x  e.  { y  e.  (
Base `  S )  |  ( F `  y )  =  ( G `  y ) }  ( ( inv g `  S ) `
 x )  e. 
{ y  e.  (
Base `  S )  |  ( F `  y )  =  ( G `  y ) } ) )
4643, 45syl 16 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  ( A. x  e.  dom  ( F  i^i  G ) ( ( inv g `  S ) `  x
)  e.  dom  ( F  i^i  G )  <->  A. x  e.  { y  e.  (
Base `  S )  |  ( F `  y )  =  ( G `  y ) }  ( ( inv g `  S ) `
 x )  e. 
{ y  e.  (
Base `  S )  |  ( F `  y )  =  ( G `  y ) } ) )
4732, 46mpbird 224 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  A. x  e.  dom  ( F  i^i  G ) ( ( inv g `  S ) `
 x )  e. 
dom  ( F  i^i  G ) )
4810issubg3 14952 . . 3  |-  ( S  e.  Grp  ->  ( dom  ( F  i^i  G
)  e.  (SubGrp `  S )  <->  ( dom  ( F  i^i  G )  e.  (SubMnd `  S
)  /\  A. x  e.  dom  ( F  i^i  G ) ( ( inv g `  S ) `
 x )  e. 
dom  ( F  i^i  G ) ) ) )
496, 48syl 16 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  ( dom  ( F  i^i  G )  e.  (SubGrp `  S
)  <->  ( dom  ( F  i^i  G )  e.  (SubMnd `  S )  /\  A. x  e.  dom  ( F  i^i  G ) ( ( inv g `  S ) `  x
)  e.  dom  ( F  i^i  G ) ) ) )
504, 47, 49mpbir2and 889 1  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  dom  ( F  i^i  G )  e.  (SubGrp `  S )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   {crab 2701    i^i cin 3311   dom cdm 4870    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   Basecbs 13461   Grpcgrp 14677   inv gcminusg 14678   MndHom cmhm 14728  SubMndcsubmnd 14729  SubGrpcsubg 14930    GrpHom cghm 14995
This theorem is referenced by:  rhmeql  15890  lmhmeql  16123
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-iun 4087  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-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-0g 13719  df-mnd 14682  df-mhm 14730  df-submnd 14731  df-grp 14804  df-minusg 14805  df-subg 14933  df-ghm 14996
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