Theorem ghmeql 13475

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 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (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 13461	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇))
2		ghmmhm 13461	. . 3 ⊢ (𝐺 ∈ (𝑆 GrpHom 𝑇) → 𝐺 ∈ (𝑆 MndHom 𝑇))
3		mhmeql 13196	. . 3 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubMnd‘𝑆))
4	1, 2, 3	syl2an 289	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubMnd‘𝑆))
5		fveq2 5561	. . . . . . . 8 ⊢ (𝑦 = ((invg‘𝑆)‘𝑥) → (𝐹‘𝑦) = (𝐹‘((invg‘𝑆)‘𝑥)))
6		fveq2 5561	. . . . . . . 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 13453	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
9	8	adantr 276	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝑆 ∈ Grp)
10	9	adantr 276	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → 𝑆 ∈ Grp)
11		simprl 529	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → 𝑥 ∈ (Base‘𝑆))
12		eqid 2196	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
13		eqid 2196	. . . . . . . . 9 ⊢ (invg‘𝑆) = (invg‘𝑆)
14	12, 13	grpinvcl 13252	. . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑆)) → ((invg‘𝑆)‘𝑥) ∈ (Base‘𝑆))
15	10, 11, 14	syl2anc 411	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → ((invg‘𝑆)‘𝑥) ∈ (Base‘𝑆))
16		simprr 531	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → (𝐹‘𝑥) = (𝐺‘𝑥))
17	16	fveq2d 5565	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (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 13458	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐹‘((invg‘𝑆)‘𝑥)) = ((invg‘𝑇)‘(𝐹‘𝑥)))
20	19	ad2ant2r 509	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → (𝐹‘((invg‘𝑆)‘𝑥)) = ((invg‘𝑇)‘(𝐹‘𝑥)))
21	12, 13, 18	ghminv 13458	. . . . . . . . 9 ⊢ ((𝐺 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐺‘((invg‘𝑆)‘𝑥)) = ((invg‘𝑇)‘(𝐺‘𝑥)))
22	21	ad2ant2lr 510	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → (𝐺‘((invg‘𝑆)‘𝑥)) = ((invg‘𝑇)‘(𝐺‘𝑥)))
23	17, 20, 22	3eqtr4d 2239	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → (𝐹‘((invg‘𝑆)‘𝑥)) = (𝐺‘((invg‘𝑆)‘𝑥)))
24	7, 15, 23	elrabd 2922	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → ((invg‘𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)})
25	24	expr 375	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝐹‘𝑥) = (𝐺‘𝑥) → ((invg‘𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)}))
26	25	ralrimiva 2570	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → ∀𝑥 ∈ (Base‘𝑆)((𝐹‘𝑥) = (𝐺‘𝑥) → ((invg‘𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)}))
27		fveq2 5561	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
28		fveq2 5561	. . . . . 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 ⊢ (∀𝑥 ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)} ((invg‘𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)} ↔ ∀𝑥 ∈ (Base‘𝑆)((𝐹‘𝑥) = (𝐺‘𝑥) → ((invg‘𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)}))
31	26, 30	sylibr 134	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → ∀𝑥 ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)} ((invg‘𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)})
32		eqid 2196	. . . . . . . 8 ⊢ (Base‘𝑇) = (Base‘𝑇)
33	12, 32	ghmf 13455	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
34	33	adantr 276	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
35	34	ffnd 5411	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝐹 Fn (Base‘𝑆))
36	12, 32	ghmf 13455	. . . . . . 7 ⊢ (𝐺 ∈ (𝑆 GrpHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
37	36	adantl 277	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
38	37	ffnd 5411	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝐺 Fn (Base‘𝑆))
39		fndmin 5672	. . . . 5 ⊢ ((𝐹 Fn (Base‘𝑆) ∧ 𝐺 Fn (Base‘𝑆)) → dom (𝐹 ∩ 𝐺) = {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)})
40	35, 38, 39	syl2anc 411	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹 ∩ 𝐺) = {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)})
41		eleq2 2260	. . . . 5 ⊢ (dom (𝐹 ∩ 𝐺) = {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)} → (((invg‘𝑆)‘𝑥) ∈ dom (𝐹 ∩ 𝐺) ↔ ((invg‘𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)}))
42	41	raleqbi1dv 2705	. . . 4 ⊢ (dom (𝐹 ∩ 𝐺) = {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)} → (∀𝑥 ∈ dom (𝐹 ∩ 𝐺)((invg‘𝑆)‘𝑥) ∈ dom (𝐹 ∩ 𝐺) ↔ ∀𝑥 ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)} ((invg‘𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)}))
43	40, 42	syl 14	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (∀𝑥 ∈ dom (𝐹 ∩ 𝐺)((invg‘𝑆)‘𝑥) ∈ dom (𝐹 ∩ 𝐺) ↔ ∀𝑥 ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)} ((invg‘𝑆)‘𝑥) ∈ {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)}))
44	31, 43	mpbird 167	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → ∀𝑥 ∈ dom (𝐹 ∩ 𝐺)((invg‘𝑆)‘𝑥) ∈ dom (𝐹 ∩ 𝐺))
45	13	issubg3 13400	. . 3 ⊢ (𝑆 ∈ Grp → (dom (𝐹 ∩ 𝐺) ∈ (SubGrp‘𝑆) ↔ (dom (𝐹 ∩ 𝐺) ∈ (SubMnd‘𝑆) ∧ ∀𝑥 ∈ dom (𝐹 ∩ 𝐺)((invg‘𝑆)‘𝑥) ∈ dom (𝐹 ∩ 𝐺))))
46	9, 45	syl 14	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (dom (𝐹 ∩ 𝐺) ∈ (SubGrp‘𝑆) ↔ (dom (𝐹 ∩ 𝐺) ∈ (SubMnd‘𝑆) ∧ ∀𝑥 ∈ dom (𝐹 ∩ 𝐺)((invg‘𝑆)‘𝑥) ∈ dom (𝐹 ∩ 𝐺))))
47	4, 44, 46	mpbir2and 946	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubGrp‘𝑆))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  ∀wral 2475  {crab 2479   ∩ cin 3156  dom cdm 4664   Fn wfn 5254  ⟶wf 5255  ‘cfv 5259  (class class class)co 5925  Basecbs 12705   MndHom cmhm 13161  SubMndcsubmnd 13162  Grpcgrp 13204  inv_gcminusg 13205  SubGrpcsubg 13375   GrpHom cghm 13448

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 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-addcom 7998  ax-addass 8000  ax-i2m1 8003  ax-0lt1 8004  ax-0id 8006  ax-rnegex 8007  ax-pre-ltirr 8010  ax-pre-ltadd 8014

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 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-map 6718  df-pnf 8082  df-mnf 8083  df-ltxr 8085  df-inn 9010  df-2 9068  df-ndx 12708  df-slot 12709  df-base 12711  df-sets 12712  df-iress 12713  df-plusg 12795  df-0g 12962  df-mgm 13060  df-sgrp 13106  df-mnd 13121  df-mhm 13163  df-submnd 13164  df-grp 13207  df-minusg 13208  df-subg 13378  df-ghm 13449

This theorem is referenced by:  rhmeql  13884