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Theorem ghmeql 15873
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 15861 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  F  e.  ( S MndHom  T ) )
2 ghmmhm 15861 . . 3  |-  ( G  e.  ( S  GrpHom  T )  ->  G  e.  ( S MndHom  T ) )
3 mhmeql 15596 . . 3  |-  ( ( F  e.  ( S MndHom  T )  /\  G  e.  ( S MndHom  T ) )  ->  dom  ( F  i^i  G )  e.  (SubMnd `  S )
)
41, 2, 3syl2an 477 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  dom  ( F  i^i  G )  e.  (SubMnd `  S )
)
5 ghmgrp1 15853 . . . . . . . . . 10  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
65adantr 465 . . . . . . . . 9  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  S  e.  Grp )
76adantr 465 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  /\  ( x  e.  ( Base `  S )  /\  ( F `  x )  =  ( G `  x ) ) )  ->  S  e.  Grp )
8 simprl 755 . . . . . . . 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 2451 . . . . . . . . 9  |-  ( Base `  S )  =  (
Base `  S )
10 eqid 2451 . . . . . . . . 9  |-  ( invg `  S )  =  ( invg `  S )
119, 10grpinvcl 15687 . . . . . . . 8  |-  ( ( S  e.  Grp  /\  x  e.  ( Base `  S ) )  -> 
( ( invg `  S ) `  x
)  e.  ( Base `  S ) )
127, 8, 11syl2anc 661 . . . . . . 7  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  /\  ( x  e.  ( Base `  S )  /\  ( F `  x )  =  ( G `  x ) ) )  ->  ( ( invg `  S ) `
 x )  e.  ( Base `  S
) )
13 simprr 756 . . . . . . . . 9  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  /\  ( x  e.  ( Base `  S )  /\  ( F `  x )  =  ( G `  x ) ) )  ->  ( F `  x )  =  ( G `  x ) )
1413fveq2d 5795 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  /\  ( x  e.  ( Base `  S )  /\  ( F `  x )  =  ( G `  x ) ) )  ->  ( ( invg `  T ) `
 ( F `  x ) )  =  ( ( invg `  T ) `  ( G `  x )
) )
15 eqid 2451 . . . . . . . . . 10  |-  ( invg `  T )  =  ( invg `  T )
169, 10, 15ghminv 15858 . . . . . . . . 9  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  x  e.  ( Base `  S
) )  ->  ( F `  ( ( invg `  S ) `
 x ) )  =  ( ( invg `  T ) `
 ( F `  x ) ) )
1716ad2ant2r 746 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  /\  ( x  e.  ( Base `  S )  /\  ( F `  x )  =  ( G `  x ) ) )  ->  ( F `  ( ( invg `  S ) `  x
) )  =  ( ( invg `  T ) `  ( F `  x )
) )
189, 10, 15ghminv 15858 . . . . . . . . 9  |-  ( ( G  e.  ( S 
GrpHom  T )  /\  x  e.  ( Base `  S
) )  ->  ( G `  ( ( invg `  S ) `
 x ) )  =  ( ( invg `  T ) `
 ( G `  x ) ) )
1918ad2ant2lr 747 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  /\  ( x  e.  ( Base `  S )  /\  ( F `  x )  =  ( G `  x ) ) )  ->  ( G `  ( ( invg `  S ) `  x
) )  =  ( ( invg `  T ) `  ( G `  x )
) )
2014, 17, 193eqtr4d 2502 . . . . . . 7  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  /\  ( x  e.  ( Base `  S )  /\  ( F `  x )  =  ( G `  x ) ) )  ->  ( F `  ( ( invg `  S ) `  x
) )  =  ( G `  ( ( invg `  S
) `  x )
) )
21 fveq2 5791 . . . . . . . . 9  |-  ( y  =  ( ( invg `  S ) `
 x )  -> 
( F `  y
)  =  ( F `
 ( ( invg `  S ) `
 x ) ) )
22 fveq2 5791 . . . . . . . . 9  |-  ( y  =  ( ( invg `  S ) `
 x )  -> 
( G `  y
)  =  ( G `
 ( ( invg `  S ) `
 x ) ) )
2321, 22eqeq12d 2473 . . . . . . . 8  |-  ( y  =  ( ( invg `  S ) `
 x )  -> 
( ( F `  y )  =  ( G `  y )  <-> 
( F `  (
( invg `  S ) `  x
) )  =  ( G `  ( ( invg `  S
) `  x )
) ) )
2423elrab 3216 . . . . . . 7  |-  ( ( ( invg `  S ) `  x
)  e.  { y  e.  ( Base `  S
)  |  ( F `
 y )  =  ( G `  y
) }  <->  ( (
( invg `  S ) `  x
)  e.  ( Base `  S )  /\  ( F `  ( ( invg `  S ) `
 x ) )  =  ( G `  ( ( invg `  S ) `  x
) ) ) )
2512, 20, 24sylanbrc 664 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  /\  ( x  e.  ( Base `  S )  /\  ( F `  x )  =  ( G `  x ) ) )  ->  ( ( invg `  S ) `
 x )  e. 
{ y  e.  (
Base `  S )  |  ( F `  y )  =  ( G `  y ) } )
2625expr 615 . . . . 5  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  /\  x  e.  ( Base `  S ) )  -> 
( ( F `  x )  =  ( G `  x )  ->  ( ( invg `  S ) `
 x )  e. 
{ y  e.  (
Base `  S )  |  ( F `  y )  =  ( G `  y ) } ) )
2726ralrimiva 2822 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  A. x  e.  ( Base `  S
) ( ( F `
 x )  =  ( G `  x
)  ->  ( ( invg `  S ) `
 x )  e. 
{ y  e.  (
Base `  S )  |  ( F `  y )  =  ( G `  y ) } ) )
28 fveq2 5791 . . . . . 6  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
29 fveq2 5791 . . . . . 6  |-  ( y  =  x  ->  ( G `  y )  =  ( G `  x ) )
3028, 29eqeq12d 2473 . . . . 5  |-  ( y  =  x  ->  (
( F `  y
)  =  ( G `
 y )  <->  ( F `  x )  =  ( G `  x ) ) )
3130ralrab 3220 . . . 4  |-  ( A. x  e.  { y  e.  ( Base `  S
)  |  ( F `
 y )  =  ( G `  y
) }  ( ( invg `  S
) `  x )  e.  { y  e.  (
Base `  S )  |  ( F `  y )  =  ( G `  y ) }  <->  A. x  e.  (
Base `  S )
( ( F `  x )  =  ( G `  x )  ->  ( ( invg `  S ) `
 x )  e. 
{ y  e.  (
Base `  S )  |  ( F `  y )  =  ( G `  y ) } ) )
3227, 31sylibr 212 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  A. x  e.  { y  e.  (
Base `  S )  |  ( F `  y )  =  ( G `  y ) }  ( ( invg `  S ) `
 x )  e. 
{ y  e.  (
Base `  S )  |  ( F `  y )  =  ( G `  y ) } )
33 eqid 2451 . . . . . . . 8  |-  ( Base `  T )  =  (
Base `  T )
349, 33ghmf 15855 . . . . . . 7  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
3534adantr 465 . . . . . 6  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  F :
( Base `  S ) --> ( Base `  T )
)
36 ffn 5659 . . . . . 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 15855 . . . . . . 7  |-  ( G  e.  ( S  GrpHom  T )  ->  G :
( Base `  S ) --> ( Base `  T )
)
3938adantl 466 . . . . . 6  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  G :
( Base `  S ) --> ( Base `  T )
)
40 ffn 5659 . . . . . 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 5911 . . . . 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 661 . . . 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 2524 . . . . 5  |-  ( dom  ( F  i^i  G
)  =  { y  e.  ( Base `  S
)  |  ( F `
 y )  =  ( G `  y
) }  ->  (
( ( invg `  S ) `  x
)  e.  dom  ( F  i^i  G )  <->  ( ( invg `  S ) `
 x )  e. 
{ y  e.  (
Base `  S )  |  ( F `  y )  =  ( G `  y ) } ) )
4544raleqbi1dv 3023 . . . 4  |-  ( dom  ( F  i^i  G
)  =  { y  e.  ( Base `  S
)  |  ( F `
 y )  =  ( G `  y
) }  ->  ( A. x  e.  dom  ( F  i^i  G ) ( ( invg `  S ) `  x
)  e.  dom  ( F  i^i  G )  <->  A. x  e.  { y  e.  (
Base `  S )  |  ( F `  y )  =  ( G `  y ) }  ( ( invg `  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 ) ( ( invg `  S ) `  x
)  e.  dom  ( F  i^i  G )  <->  A. x  e.  { y  e.  (
Base `  S )  |  ( F `  y )  =  ( G `  y ) }  ( ( invg `  S ) `
 x )  e. 
{ y  e.  (
Base `  S )  |  ( F `  y )  =  ( G `  y ) } ) )
4732, 46mpbird 232 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  A. x  e.  dom  ( F  i^i  G ) ( ( invg `  S ) `
 x )  e. 
dom  ( F  i^i  G ) )
4810issubg3 15803 . . 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 ) ( ( invg `  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 ) ( ( invg `  S ) `  x
)  e.  dom  ( F  i^i  G ) ) ) )
504, 47, 49mpbir2and 913 1  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  dom  ( F  i^i  G )  e.  (SubGrp `  S )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   {crab 2799    i^i cin 3427   dom cdm 4940    Fn wfn 5513   -->wf 5514   ` cfv 5518  (class class class)co 6192   Basecbs 14278   Grpcgrp 15514   invgcminusg 15515   MndHom cmhm 15566  SubMndcsubmnd 15567  SubGrpcsubg 15779    GrpHom cghm 15848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-recs 6934  df-rdg 6968  df-er 7203  df-map 7318  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-ress 14285  df-plusg 14355  df-0g 14484  df-mnd 15519  df-mhm 15568  df-submnd 15569  df-grp 15649  df-minusg 15650  df-subg 15782  df-ghm 15849
This theorem is referenced by:  rhmeql  17003  lmhmeql  17244
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