Theorem ghmeql 13847

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 13833	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇))
2		ghmmhm 13833	. . 3 ⊢ (𝐺 ∈ (𝑆 GrpHom 𝑇) → 𝐺 ∈ (𝑆 MndHom 𝑇))
3		mhmeql 13568	. . 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 5635	. . . . . . . 8 ⊢ (𝑦 = ((invg‘𝑆)‘𝑥) → (𝐹‘𝑦) = (𝐹‘((invg‘𝑆)‘𝑥)))
6		fveq2 5635	. . . . . . . 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 13825	. . . . . . . . . 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 2229	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
13		eqid 2229	. . . . . . . . 9 ⊢ (invg‘𝑆) = (invg‘𝑆)
14	12, 13	grpinvcl 13624	. . . . . . . 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 5639	. . . . . . . 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 2229	. . . . . . . . . 10 ⊢ (invg‘𝑇) = (invg‘𝑇)
19	12, 13, 18	ghminv 13830	. . . . . . . . 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 13830	. . . . . . . . 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 2272	. . . . . . 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 2962	. . . . . 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 2603	. . . 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 5635	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
28		fveq2 5635	. . . . . 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 2965	. . . 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 2229	. . . . . . . 8 ⊢ (Base‘𝑇) = (Base‘𝑇)
33	12, 32	ghmf 13827	. . . . . . 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 5480	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝐹 Fn (Base‘𝑆))
36	12, 32	ghmf 13827	. . . . . . 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 5480	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝐺 Fn (Base‘𝑆))
39		fndmin 5750	. . . . 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 2293	. . . . 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 2740	. . . 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 13772	. . 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 950	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubGrp‘𝑆))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  ∀wral 2508  {crab 2512   ∩ cin 3197  dom cdm 4723   Fn wfn 5319  ⟶wf 5320  ‘cfv 5324  (class class class)co 6013  Basecbs 13075   MndHom cmhm 13533  SubMndcsubmnd 13534  Grpcgrp 13576  inv_gcminusg 13577  SubGrpcsubg 13747   GrpHom cghm 13820

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 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-addass 8127  ax-i2m1 8130  ax-0lt1 8131  ax-0id 8133  ax-rnegex 8134  ax-pre-ltirr 8137  ax-pre-ltadd 8141

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-map 6814  df-pnf 8209  df-mnf 8210  df-ltxr 8212  df-inn 9137  df-2 9195  df-ndx 13078  df-slot 13079  df-base 13081  df-sets 13082  df-iress 13083  df-plusg 13166  df-0g 13334  df-mgm 13432  df-sgrp 13478  df-mnd 13493  df-mhm 13535  df-submnd 13536  df-grp 13579  df-minusg 13580  df-subg 13750  df-ghm 13821

This theorem is referenced by:  rhmeql  14257