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Theorem ghmeql 14754
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 14742 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  F  e.  ( S MndHom  T ) )
2 ghmmhm 14742 . . 3  |-  ( G  e.  ( S  GrpHom  T )  ->  G  e.  ( S MndHom  T ) )
3 mhmeql 14490 . . 3  |-  ( ( F  e.  ( S MndHom  T )  /\  G  e.  ( S MndHom  T ) )  ->  dom  ( F  i^i  G )  e.  (SubMnd `  S )
)
41, 2, 3syl2an 463 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  dom  ( F  i^i  G )  e.  (SubMnd `  S )
)
5 ghmgrp1 14734 . . . . . . . . . 10  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
65adantr 451 . . . . . . . . 9  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  S  e.  Grp )
76adantr 451 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  /\  ( x  e.  ( Base `  S )  /\  ( F `  x )  =  ( G `  x ) ) )  ->  S  e.  Grp )
8 simprl 732 . . . . . . . 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 2316 . . . . . . . . 9  |-  ( Base `  S )  =  (
Base `  S )
10 eqid 2316 . . . . . . . . 9  |-  ( inv g `  S )  =  ( inv g `  S )
119, 10grpinvcl 14576 . . . . . . . 8  |-  ( ( S  e.  Grp  /\  x  e.  ( Base `  S ) )  -> 
( ( inv g `  S ) `  x
)  e.  ( Base `  S ) )
127, 8, 11syl2anc 642 . . . . . . 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 733 . . . . . . . . 9  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  /\  ( x  e.  ( Base `  S )  /\  ( F `  x )  =  ( G `  x ) ) )  ->  ( F `  x )  =  ( G `  x ) )
1413fveq2d 5567 . . . . . . . 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 2316 . . . . . . . . . 10  |-  ( inv g `  T )  =  ( inv g `  T )
169, 10, 15ghminv 14739 . . . . . . . . 9  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  x  e.  ( Base `  S
) )  ->  ( F `  ( ( inv g `  S ) `
 x ) )  =  ( ( inv g `  T ) `
 ( F `  x ) ) )
1716ad2ant2r 727 . . . . . . . 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 14739 . . . . . . . . 9  |-  ( ( G  e.  ( S 
GrpHom  T )  /\  x  e.  ( Base `  S
) )  ->  ( G `  ( ( inv g `  S ) `
 x ) )  =  ( ( inv g `  T ) `
 ( G `  x ) ) )
1918ad2ant2lr 728 . . . . . . . 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 2358 . . . . . . 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 5563 . . . . . . . . 9  |-  ( y  =  ( ( inv g `  S ) `
 x )  -> 
( F `  y
)  =  ( F `
 ( ( inv g `  S ) `
 x ) ) )
22 fveq2 5563 . . . . . . . . 9  |-  ( y  =  ( ( inv g `  S ) `
 x )  -> 
( G `  y
)  =  ( G `
 ( ( inv g `  S ) `
 x ) ) )
2321, 22eqeq12d 2330 . . . . . . . 8  |-  ( y  =  ( ( inv g `  S ) `
 x )  -> 
( ( F `  y )  =  ( G `  y )  <-> 
( F `  (
( inv g `  S ) `  x
) )  =  ( G `  ( ( inv g `  S
) `  x )
) ) )
2423elrab 2957 . . . . . . 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 645 . . . . . 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 598 . . . . 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 2660 . . . 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 5563 . . . . . 6  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
29 fveq2 5563 . . . . . 6  |-  ( y  =  x  ->  ( G `  y )  =  ( G `  x ) )
3028, 29eqeq12d 2330 . . . . 5  |-  ( y  =  x  ->  (
( F `  y
)  =  ( G `
 y )  <->  ( F `  x )  =  ( G `  x ) ) )
3130ralrab 2961 . . . 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 203 . . 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 2316 . . . . . . . 8  |-  ( Base `  T )  =  (
Base `  T )
349, 33ghmf 14736 . . . . . . 7  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
3534adantr 451 . . . . . 6  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  F :
( Base `  S ) --> ( Base `  T )
)
36 ffn 5427 . . . . . 6  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  F  Fn  ( Base `  S )
)
3735, 36syl 15 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  F  Fn  ( Base `  S )
)
389, 33ghmf 14736 . . . . . . 7  |-  ( G  e.  ( S  GrpHom  T )  ->  G :
( Base `  S ) --> ( Base `  T )
)
3938adantl 452 . . . . . 6  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  G :
( Base `  S ) --> ( Base `  T )
)
40 ffn 5427 . . . . . 6  |-  ( G : ( Base `  S
) --> ( Base `  T
)  ->  G  Fn  ( Base `  S )
)
4139, 40syl 15 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  G  Fn  ( Base `  S )
)
42 fndmin 5670 . . . . 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 642 . . . 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 2377 . . . . 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 2778 . . . 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 15 . . 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 223 . 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 14686 . . 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 15 . 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 888 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 176    /\ wa 358    = wceq 1633    e. wcel 1701   A.wral 2577   {crab 2581    i^i cin 3185   dom cdm 4726    Fn wfn 5287   -->wf 5288   ` cfv 5292  (class class class)co 5900   Basecbs 13195   Grpcgrp 14411   inv gcminusg 14412   MndHom cmhm 14462  SubMndcsubmnd 14463  SubGrpcsubg 14664    GrpHom cghm 14729
This theorem is referenced by:  rhmeql  15624  lmhmeql  15861
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-0g 13453  df-mnd 14416  df-mhm 14464  df-submnd 14465  df-grp 14538  df-minusg 14539  df-subg 14667  df-ghm 14730
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