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Theorem ghmeql 15762
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 15750 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  F  e.  ( S MndHom  T ) )
2 ghmmhm 15750 . . 3  |-  ( G  e.  ( S  GrpHom  T )  ->  G  e.  ( S MndHom  T ) )
3 mhmeql 15487 . . 3  |-  ( ( F  e.  ( S MndHom  T )  /\  G  e.  ( S MndHom  T ) )  ->  dom  ( F  i^i  G )  e.  (SubMnd `  S )
)
41, 2, 3syl2an 474 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  dom  ( F  i^i  G )  e.  (SubMnd `  S )
)
5 ghmgrp1 15742 . . . . . . . . . 10  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
65adantr 462 . . . . . . . . 9  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  S  e.  Grp )
76adantr 462 . . . . . . . 8  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  /\  ( x  e.  ( Base `  S )  /\  ( F `  x )  =  ( G `  x ) ) )  ->  S  e.  Grp )
8 simprl 750 . . . . . . . 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 2441 . . . . . . . . 9  |-  ( Base `  S )  =  (
Base `  S )
10 eqid 2441 . . . . . . . . 9  |-  ( invg `  S )  =  ( invg `  S )
119, 10grpinvcl 15576 . . . . . . . 8  |-  ( ( S  e.  Grp  /\  x  e.  ( Base `  S ) )  -> 
( ( invg `  S ) `  x
)  e.  ( Base `  S ) )
127, 8, 11syl2anc 656 . . . . . . 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 751 . . . . . . . . 9  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  /\  ( x  e.  ( Base `  S )  /\  ( F `  x )  =  ( G `  x ) ) )  ->  ( F `  x )  =  ( G `  x ) )
1413fveq2d 5692 . . . . . . . 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 2441 . . . . . . . . . 10  |-  ( invg `  T )  =  ( invg `  T )
169, 10, 15ghminv 15747 . . . . . . . . 9  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  x  e.  ( Base `  S
) )  ->  ( F `  ( ( invg `  S ) `
 x ) )  =  ( ( invg `  T ) `
 ( F `  x ) ) )
1716ad2ant2r 741 . . . . . . . 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 15747 . . . . . . . . 9  |-  ( ( G  e.  ( S 
GrpHom  T )  /\  x  e.  ( Base `  S
) )  ->  ( G `  ( ( invg `  S ) `
 x ) )  =  ( ( invg `  T ) `
 ( G `  x ) ) )
1918ad2ant2lr 742 . . . . . . . 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 2483 . . . . . . 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 5688 . . . . . . . . 9  |-  ( y  =  ( ( invg `  S ) `
 x )  -> 
( F `  y
)  =  ( F `
 ( ( invg `  S ) `
 x ) ) )
22 fveq2 5688 . . . . . . . . 9  |-  ( y  =  ( ( invg `  S ) `
 x )  -> 
( G `  y
)  =  ( G `
 ( ( invg `  S ) `
 x ) ) )
2321, 22eqeq12d 2455 . . . . . . . 8  |-  ( y  =  ( ( invg `  S ) `
 x )  -> 
( ( F `  y )  =  ( G `  y )  <-> 
( F `  (
( invg `  S ) `  x
) )  =  ( G `  ( ( invg `  S
) `  x )
) ) )
2423elrab 3114 . . . . . . 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 659 . . . . . 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 612 . . . . 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 2797 . . . 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 5688 . . . . . 6  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
29 fveq2 5688 . . . . . 6  |-  ( y  =  x  ->  ( G `  y )  =  ( G `  x ) )
3028, 29eqeq12d 2455 . . . . 5  |-  ( y  =  x  ->  (
( F `  y
)  =  ( G `
 y )  <->  ( F `  x )  =  ( G `  x ) ) )
3130ralrab 3118 . . . 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 2441 . . . . . . . 8  |-  ( Base `  T )  =  (
Base `  T )
349, 33ghmf 15744 . . . . . . 7  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
3534adantr 462 . . . . . 6  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  F :
( Base `  S ) --> ( Base `  T )
)
36 ffn 5556 . . . . . 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 15744 . . . . . . 7  |-  ( G  e.  ( S  GrpHom  T )  ->  G :
( Base `  S ) --> ( Base `  T )
)
3938adantl 463 . . . . . 6  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  G :
( Base `  S ) --> ( Base `  T )
)
40 ffn 5556 . . . . . 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 5807 . . . . 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 656 . . . 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 2502 . . . . 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 2923 . . . 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 15692 . . 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 908 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 1364    e. wcel 1761   A.wral 2713   {crab 2717    i^i cin 3324   dom cdm 4836    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   Basecbs 14170   Grpcgrp 15406   invgcminusg 15407   MndHom cmhm 15458  SubMndcsubmnd 15459  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 2261  df-mo 2262  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-map 7212  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-mhm 15460  df-submnd 15461  df-grp 15538  df-minusg 15539  df-subg 15671  df-ghm 15738
This theorem is referenced by:  rhmeql  16875  lmhmeql  17114
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