Theorem ghmeql 13397

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 13383	. . 3 |- ( F e. ( S GrpHom T ) -> F e. ( S MndHom T ) )
2		ghmmhm 13383	. . 3 |- ( G e. ( S GrpHom T ) -> G e. ( S MndHom T ) )
3		mhmeql 13124	. . 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 5558	. . . . . . . 8 |- ( y = ( ( invg ` S ) ` x ) -> ( F ` y ) = ( F ` ( ( invg ` S ) ` x ) ) )
6		fveq2 5558	. . . . . . . 8 |- ( y = ( ( invg ` S ) ` x ) -> ( G ` y ) = ( G ` ( ( invg ` S ) ` x ) ) )
7	5, 6	eqeq12d 2211	. . . . . . 7 |- ( y = ( ( invg ` S ) ` x ) -> ( ( F ` y ) = ( G ` y ) <-> ( F ` ( ( invg ` S ) ` x ) ) = ( G ` ( ( invg ` S ) ` x ) ) ) )
8		ghmgrp1 13375	. . . . . . . . . 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 2196	. . . . . . . . 9 |- ( Base ` S ) = ( Base ` S )
13		eqid 2196	. . . . . . . . 9 |- ( invg ` S ) = ( invg ` S )
14	12, 13	grpinvcl 13180	. . . . . . . 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 5562	. . . . . . . 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 2196	. . . . . . . . . 10 |- ( invg ` T ) = ( invg ` T )
19	12, 13, 18	ghminv 13380	. . . . . . . . 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 13380	. . . . . . . . 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 2239	. . . . . . 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 2922	. . . . . 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 2570	. . . 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 5558	. . . . . 6 |- ( y = x -> ( F ` y ) = ( F ` x ) )
28		fveq2 5558	. . . . . 6 |- ( y = x -> ( G ` y ) = ( G ` x ) )
29	27, 28	eqeq12d 2211	. . . . 5 |- ( y = x -> ( ( F ` y ) = ( G ` y ) <-> ( F ` x ) = ( G ` x ) ) )
30	29	ralrab 2925	. . . 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 2196	. . . . . . . 8 |- ( Base ` T ) = ( Base ` T )
33	12, 32	ghmf 13377	. . . . . . 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 5408	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ G e. ( S GrpHom T ) ) -> F Fn ( Base ` S ) )
36	12, 32	ghmf 13377	. . . . . . 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 5408	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ G e. ( S GrpHom T ) ) -> G Fn ( Base ` S ) )
39		fndmin 5669	. . . . 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 2260	. . . . 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 2705	. . . 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 13322	. . 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 2167   A.wral 2475   {crab 2479   i^i cin 3156   dom cdm 4663   Fn wfn 5253   -->wf 5254   ` cfv 5258  (class class class)co 5922   Basecbs 12678   MndHom cmhm 13089  SubMndcsubmnd 13090   Grpcgrp 13132   invgcminusg 13133  SubGrpcsubg 13297   GrpHom cghm 13370

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-map 6709  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-mhm 13091  df-submnd 13092  df-grp 13135  df-minusg 13136  df-subg 13300  df-ghm 13371

This theorem is referenced by:  rhmeql  13806