Theorem ghmeql 13819

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 13805	. . 3 |- ( F e. ( S GrpHom T ) -> F e. ( S MndHom T ) )
2		ghmmhm 13805	. . 3 |- ( G e. ( S GrpHom T ) -> G e. ( S MndHom T ) )
3		mhmeql 13540	. . 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 5629	. . . . . . . 8 |- ( y = ( ( invg ` S ) ` x ) -> ( F ` y ) = ( F ` ( ( invg ` S ) ` x ) ) )
6		fveq2 5629	. . . . . . . 8 |- ( y = ( ( invg ` S ) ` x ) -> ( G ` y ) = ( G ` ( ( invg ` S ) ` x ) ) )
7	5, 6	eqeq12d 2244	. . . . . . 7 |- ( y = ( ( invg ` S ) ` x ) -> ( ( F ` y ) = ( G ` y ) <-> ( F ` ( ( invg ` S ) ` x ) ) = ( G ` ( ( invg ` S ) ` x ) ) ) )
8		ghmgrp1 13797	. . . . . . . . . 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 2229	. . . . . . . . 9 |- ( Base ` S ) = ( Base ` S )
13		eqid 2229	. . . . . . . . 9 |- ( invg ` S ) = ( invg ` S )
14	12, 13	grpinvcl 13596	. . . . . . . 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 5633	. . . . . . . 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 2229	. . . . . . . . . 10 |- ( invg ` T ) = ( invg ` T )
19	12, 13, 18	ghminv 13802	. . . . . . . . 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 13802	. . . . . . . . 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 2272	. . . . . . 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 2961	. . . . . 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 2603	. . . 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 5629	. . . . . 6 |- ( y = x -> ( F ` y ) = ( F ` x ) )
28		fveq2 5629	. . . . . 6 |- ( y = x -> ( G ` y ) = ( G ` x ) )
29	27, 28	eqeq12d 2244	. . . . 5 |- ( y = x -> ( ( F ` y ) = ( G ` y ) <-> ( F ` x ) = ( G ` x ) ) )
30	29	ralrab 2964	. . . 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 2229	. . . . . . . 8 |- ( Base ` T ) = ( Base ` T )
33	12, 32	ghmf 13799	. . . . . . 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 5474	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ G e. ( S GrpHom T ) ) -> F Fn ( Base ` S ) )
36	12, 32	ghmf 13799	. . . . . . 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 5474	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ G e. ( S GrpHom T ) ) -> G Fn ( Base ` S ) )
39		fndmin 5744	. . . . 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 2293	. . . . 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 2740	. . . 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 13744	. . 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 950	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 1395   e. wcel 2200   A.wral 2508   {crab 2512   i^i cin 3196   dom cdm 4719   Fn wfn 5313   -->wf 5314   ` cfv 5318  (class class class)co 6007   Basecbs 13047   MndHom cmhm 13505  SubMndcsubmnd 13506   Grpcgrp 13548   invgcminusg 13549  SubGrpcsubg 13719   GrpHom cghm 13792

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-pre-ltirr 8122  ax-pre-ltadd 8126

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-map 6805  df-pnf 8194  df-mnf 8195  df-ltxr 8197  df-inn 9122  df-2 9180  df-ndx 13050  df-slot 13051  df-base 13053  df-sets 13054  df-iress 13055  df-plusg 13138  df-0g 13306  df-mgm 13404  df-sgrp 13450  df-mnd 13465  df-mhm 13507  df-submnd 13508  df-grp 13551  df-minusg 13552  df-subg 13722  df-ghm 13793

This theorem is referenced by:  rhmeql  14229