MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ghmeql Structured version   Unicode version

Theorem ghmeql 16077
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 16065 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  F  e.  ( S MndHom  T ) )
2 ghmmhm 16065 . . 3  |-  ( G  e.  ( S  GrpHom  T )  ->  G  e.  ( S MndHom  T ) )
3 mhmeql 15798 . . 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 16057 . . . . . . . . . 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 2460 . . . . . . . . 9  |-  ( Base `  S )  =  (
Base `  S )
10 eqid 2460 . . . . . . . . 9  |-  ( invg `  S )  =  ( invg `  S )
119, 10grpinvcl 15889 . . . . . . . 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 5861 . . . . . . . 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 2460 . . . . . . . . . 10  |-  ( invg `  T )  =  ( invg `  T )
169, 10, 15ghminv 16062 . . . . . . . . 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 16062 . . . . . . . . 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 2511 . . . . . . 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 5857 . . . . . . . . 9  |-  ( y  =  ( ( invg `  S ) `
 x )  -> 
( F `  y
)  =  ( F `
 ( ( invg `  S ) `
 x ) ) )
22 fveq2 5857 . . . . . . . . 9  |-  ( y  =  ( ( invg `  S ) `
 x )  -> 
( G `  y
)  =  ( G `
 ( ( invg `  S ) `
 x ) ) )
2321, 22eqeq12d 2482 . . . . . . . 8  |-  ( y  =  ( ( invg `  S ) `
 x )  -> 
( ( F `  y )  =  ( G `  y )  <-> 
( F `  (
( invg `  S ) `  x
) )  =  ( G `  ( ( invg `  S
) `  x )
) ) )
2423elrab 3254 . . . . . . 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 2871 . . . 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 5857 . . . . . 6  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
29 fveq2 5857 . . . . . 6  |-  ( y  =  x  ->  ( G `  y )  =  ( G `  x ) )
3028, 29eqeq12d 2482 . . . . 5  |-  ( y  =  x  ->  (
( F `  y
)  =  ( G `
 y )  <->  ( F `  x )  =  ( G `  x ) ) )
3130ralrab 3258 . . . 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 2460 . . . . . . . 8  |-  ( Base `  T )  =  (
Base `  T )
349, 33ghmf 16059 . . . . . . 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 5722 . . . . . 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 16059 . . . . . . 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 5722 . . . . . 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 5979 . . . . 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 2533 . . . . 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 3059 . . . 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 16007 . . 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 915 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 1374    e. wcel 1762   A.wral 2807   {crab 2811    i^i cin 3468   dom cdm 4992    Fn wfn 5574   -->wf 5575   ` cfv 5579  (class class class)co 6275   Basecbs 14479   Grpcgrp 15716   invgcminusg 15717   MndHom cmhm 15768  SubMndcsubmnd 15769  SubGrpcsubg 15983    GrpHom cghm 16052
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-recs 7032  df-rdg 7066  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-0g 14686  df-mnd 15721  df-mhm 15770  df-submnd 15771  df-grp 15851  df-minusg 15852  df-subg 15986  df-ghm 16053
This theorem is referenced by:  rhmeql  17235  lmhmeql  17477
  Copyright terms: Public domain W3C validator