Theorem ghmeql 13340

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 13326	. . 3 |- ( F e. ( S GrpHom T ) -> F e. ( S MndHom T ) )
2		ghmmhm 13326	. . 3 |- ( G e. ( S GrpHom T ) -> G e. ( S MndHom T ) )
3		mhmeql 13067	. . 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 5555	. . . . . . . 8 |- ( y = ( ( invg ` S ) ` x ) -> ( F ` y ) = ( F ` ( ( invg ` S ) ` x ) ) )
6		fveq2 5555	. . . . . . . 8 |- ( y = ( ( invg ` S ) ` x ) -> ( G ` y ) = ( G ` ( ( invg ` S ) ` x ) ) )
7	5, 6	eqeq12d 2208	. . . . . . 7 |- ( y = ( ( invg ` S ) ` x ) -> ( ( F ` y ) = ( G ` y ) <-> ( F ` ( ( invg ` S ) ` x ) ) = ( G ` ( ( invg ` S ) ` x ) ) ) )
8		ghmgrp1 13318	. . . . . . . . . 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 529	. . . . . . . 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 2193	. . . . . . . . 9 |- ( Base ` S ) = ( Base ` S )
13		eqid 2193	. . . . . . . . 9 |- ( invg ` S ) = ( invg ` S )
14	12, 13	grpinvcl 13123	. . . . . . . 8 |- ( ( S e. Grp /\ x e. ( Base ` S ) ) -> ( ( invg ` S ) ` x ) e. ( Base ` S ) )
15	10, 11, 14	syl2anc 411	. . . . . . 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 531	. . . . . . . . 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 5559	. . . . . . . 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 2193	. . . . . . . . . 10 |- ( invg ` T ) = ( invg ` T )
19	12, 13, 18	ghminv 13323	. . . . . . . . 9 |- ( ( F e. ( S GrpHom T ) /\ x e. ( Base ` S ) ) -> ( F ` ( ( invg ` S ) ` x ) ) = ( ( invg ` T ) ` ( F ` x ) ) )
20	19	ad2ant2r 509	. . . . . . . 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 13323	. . . . . . . . 9 |- ( ( G e. ( S GrpHom T ) /\ x e. ( Base ` S ) ) -> ( G ` ( ( invg ` S ) ` x ) ) = ( ( invg ` T ) ` ( G ` x ) ) )
22	21	ad2ant2lr 510	. . . . . . . 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 2236	. . . . . . 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 2919	. . . . . 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 2567	. . . 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 5555	. . . . . 6 |- ( y = x -> ( F ` y ) = ( F ` x ) )
28		fveq2 5555	. . . . . 6 |- ( y = x -> ( G ` y ) = ( G ` x ) )
29	27, 28	eqeq12d 2208	. . . . 5 |- ( y = x -> ( ( F ` y ) = ( G ` y ) <-> ( F ` x ) = ( G ` x ) ) )
30	29	ralrab 2922	. . . 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 2193	. . . . . . . 8 |- ( Base ` T ) = ( Base ` T )
33	12, 32	ghmf 13320	. . . . . . 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 5405	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ G e. ( S GrpHom T ) ) -> F Fn ( Base ` S ) )
36	12, 32	ghmf 13320	. . . . . . 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 5405	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ G e. ( S GrpHom T ) ) -> G Fn ( Base ` S ) )
39		fndmin 5666	. . . . 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 411	. . . 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 2257	. . . . 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 2702	. . . 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 13265	. . 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 946	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 1364   e. wcel 2164   A.wral 2472   {crab 2476   i^i cin 3153   dom cdm 4660   Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5919   Basecbs 12621   MndHom cmhm 13032  SubMndcsubmnd 13033   Grpcgrp 13075   invgcminusg 13076  SubGrpcsubg 13240   GrpHom cghm 13313

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-map 6706  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-iress 12629  df-plusg 12711  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-mhm 13034  df-submnd 13035  df-grp 13078  df-minusg 13079  df-subg 13243  df-ghm 13314

This theorem is referenced by:  rhmeql  13749