Theorem ghmeql 14047

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.

Step	Hyp	Ref	Expression
1		ghmmhm 14033	. . 3 |- ( F e. ( S GrpHom T ) -> F e. ( S MndHom T ) )
2		ghmmhm 14033	. . 3 |- ( G e. ( S GrpHom T ) -> G e. ( S MndHom T ) )
3		mhmeql 13776	. . 3 |- ( ( F e. ( S MndHom T ) /\ G e. ( S MndHom T ) ) -> dom ( F i^i G ) e. (SubMnd ` S ) )
4	1, 2, 3	syl2an 289	. 2 |- ( ( F e. ( S GrpHom T ) /\ G e. ( S GrpHom T ) ) -> dom ( F i^i G ) e. (SubMnd ` S ) )
5		fveq2 5690	. . . . . . . 8 |- ( y = ( ( invg ` S ) ` x ) -> ( F ` y ) = ( F ` ( ( invg ` S ) ` x ) ) )
6		fveq2 5690	. . . . . . . 8 |- ( y = ( ( invg ` S ) ` x ) -> ( G ` y ) = ( G ` ( ( invg ` S ) ` x ) ) )
7	5, 6	eqeq12d 2253	. . . . . . 7 |- ( y = ( ( invg ` S ) ` x ) -> ( ( F ` y ) = ( G ` y ) <-> ( F ` ( ( invg ` S ) ` x ) ) = ( G ` ( ( invg ` S ) ` x ) ) ) )
8		ghmgrp1 14025	. . . . . . . . . 10 |- ( F e. ( S GrpHom T ) -> S e. Grp )
9	8	adantr 276	. . . . . . . . 9 |- ( ( F e. ( S GrpHom T ) /\ G e. ( S GrpHom T ) ) -> S e. Grp )
10	9	adantr 276	. . . . . . . 8 |- ( ( ( F e. ( S GrpHom T ) /\ G e. ( S GrpHom T ) ) /\ ( x e. ( Base ` S ) /\ ( F ` x ) = ( G ` x ) ) ) -> S e. Grp )
11		simprl 535	. . . . . . . 8 |- ( ( ( F e. ( S GrpHom T ) /\ G e. ( S GrpHom T ) ) /\ ( x e. ( Base ` S ) /\ ( F ` x ) = ( G ` x ) ) ) -> x e. ( Base ` S ) )
12		eqid 2238	. . . . . . . . 9 |- ( Base ` S ) = ( Base ` S )
13		eqid 2238	. . . . . . . . 9 |- ( invg ` S ) = ( invg ` S )
14	12, 13	grpinvcl 13830	. . . . . . . 8 |- ( ( S e. Grp /\ x e. ( Base ` S ) ) -> ( ( invg ` S ) ` x ) e. ( Base ` S ) )
15	10, 11, 14	syl2anc 415	. . . . . . 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 ) )
16		simprr 537	. . . . . . . . 9 |- ( ( ( F e. ( S GrpHom T ) /\ G e. ( S GrpHom T ) ) /\ ( x e. ( Base ` S ) /\ ( F ` x ) = ( G ` x ) ) ) -> ( F ` x ) = ( G ` x ) )
17	16	fveq2d 5694	. . . . . . . 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 ) ) )
18		eqid 2238	. . . . . . . . . 10 |- ( invg ` T ) = ( invg ` T )
19	12, 13, 18	ghminv 14030	. . . . . . . . 9 |- ( ( F e. ( S GrpHom T ) /\ x e. ( Base ` S ) ) -> ( F ` ( ( invg ` S ) ` x ) ) = ( ( invg ` T ) ` ( F ` x ) ) )
20	19	ad2ant2r 513	. . . . . . . 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 ) ) )
21	12, 13, 18	ghminv 14030	. . . . . . . . 9 |- ( ( G e. ( S GrpHom T ) /\ x e. ( Base ` S ) ) -> ( G ` ( ( invg ` S ) ` x ) ) = ( ( invg ` T ) ` ( G ` x ) ) )
22	21	ad2ant2lr 514	. . . . . . . 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 ) ) )
23	17, 20, 22	3eqtr4d 2281	. . . . . . 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 ) ) )
24	7, 15, 23	elrabd 2984	. . . . . 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 ) } )
25	24	expr 375	. . . . 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 ) } ) )
26	25	ralrimiva 2623	. . . 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 ) } ) )
27		fveq2 5690	. . . . . 6 |- ( y = x -> ( F ` y ) = ( F ` x ) )
28		fveq2 5690	. . . . . 6 |- ( y = x -> ( G ` y ) = ( G ` x ) )
29	27, 28	eqeq12d 2253	. . . . 5 |- ( y = x -> ( ( F ` y ) = ( G ` y ) <-> ( F ` x ) = ( G ` x ) ) )
30	29	ralrab 2987	. . . 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 ) } ) )
31	26, 30	sylibr 134	. . 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 ) } )
32		eqid 2238	. . . . . . . 8 |- ( Base ` T ) = ( Base ` T )
33	12, 32	ghmf 14027	. . . . . . 7 |- ( F e. ( S GrpHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
34	33	adantr 276	. . . . . 6 |- ( ( F e. ( S GrpHom T ) /\ G e. ( S GrpHom T ) ) -> F : ( Base ` S ) --> ( Base ` T ) )
35	34	ffnd 5529	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ G e. ( S GrpHom T ) ) -> F Fn ( Base ` S ) )
36	12, 32	ghmf 14027	. . . . . . 7 |- ( G e. ( S GrpHom T ) -> G : ( Base ` S ) --> ( Base ` T ) )
37	36	adantl 277	. . . . . 6 |- ( ( F e. ( S GrpHom T ) /\ G e. ( S GrpHom T ) ) -> G : ( Base ` S ) --> ( Base ` T ) )
38	37	ffnd 5529	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ G e. ( S GrpHom T ) ) -> G Fn ( Base ` S ) )
39		fndmin 5807	. . . . 5 |- ( ( F Fn ( Base ` S ) /\ G Fn ( Base ` S ) ) -> dom ( F i^i G ) = { y e. ( Base ` S ) | ( F ` y ) = ( G ` y ) } )
40	35, 38, 39	syl2anc 415	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ G e. ( S GrpHom T ) ) -> dom ( F i^i G ) = { y e. ( Base ` S ) | ( F ` y ) = ( G ` y ) } )
41		eleq2 2302	. . . . 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 ) } ) )
42	41	raleqbi1dv 2761	. . . 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 ) } ) )
43	40, 42	syl 14	. . 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 ) } ) )
44	31, 43	mpbird 167	. 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 ) )
45	13	issubg3 13972	. . 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 ) ) ) )
46	9, 45	syl 14	. 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 ) ) ) )
47	4, 44, 46	mpbir2and 957	1 |- ( ( F e. ( S GrpHom T ) /\ G e. ( S GrpHom T ) ) -> dom ( F i^i G ) e. (SubGrp ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   i^i cin 3219   dom cdm 4769   Fn wfn 5367   -->wf 5368   ` cfv 5372  (class class class)co 6075   Basecbs 13330   MndHom cmhm 13741  SubMndcsubmnd 13742   Grpcgrp 13782   invgcminusg 13783  SubGrpcsubg 13947   GrpHom cghm 14020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-map 6914  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-mhm 13743  df-submnd 13744  df-grp 13785  df-minusg 13786  df-subg 13950  df-ghm 14021

This theorem is referenced by:  rhmeql  14531