Theorem ghmeql 13877

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 13863	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇))
2		ghmmhm 13863	. . 3 ⊢ (𝐺 ∈ (𝑆 GrpHom 𝑇) → 𝐺 ∈ (𝑆 MndHom 𝑇))
3		mhmeql 13598	. . 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 5642	. . . . . . . 8 ⊢ (𝑦 = ((invg‘𝑆)‘𝑥) → (𝐹‘𝑦) = (𝐹‘((invg‘𝑆)‘𝑥)))
6		fveq2 5642	. . . . . . . 8 ⊢ (𝑦 = ((invg‘𝑆)‘𝑥) → (𝐺‘𝑦) = (𝐺‘((invg‘𝑆)‘𝑥)))
7	5, 6	eqeq12d 2245	. . . . . . 7 ⊢ (𝑦 = ((invg‘𝑆)‘𝑥) → ((𝐹‘𝑦) = (𝐺‘𝑦) ↔ (𝐹‘((invg‘𝑆)‘𝑥)) = (𝐺‘((invg‘𝑆)‘𝑥))))
8		ghmgrp1 13855	. . . . . . . . . 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 531	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → 𝑥 ∈ (Base‘𝑆))
12		eqid 2230	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
13		eqid 2230	. . . . . . . . 9 ⊢ (invg‘𝑆) = (invg‘𝑆)
14	12, 13	grpinvcl 13654	. . . . . . . 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 533	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → (𝐹‘𝑥) = (𝐺‘𝑥))
17	16	fveq2d 5646	. . . . . . . 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 2230	. . . . . . . . . 10 ⊢ (invg‘𝑇) = (invg‘𝑇)
19	12, 13, 18	ghminv 13860	. . . . . . . . 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 13860	. . . . . . . . 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 2273	. . . . . . 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 2963	. . . . . 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 2604	. . . 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 5642	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
28		fveq2 5642	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝐺‘𝑦) = (𝐺‘𝑥))
29	27, 28	eqeq12d 2245	. . . . 5 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑦) = (𝐺‘𝑦) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
30	29	ralrab 2966	. . . 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 2230	. . . . . . . 8 ⊢ (Base‘𝑇) = (Base‘𝑇)
33	12, 32	ghmf 13857	. . . . . . 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 5485	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝐹 Fn (Base‘𝑆))
36	12, 32	ghmf 13857	. . . . . . 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 5485	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝐺 Fn (Base‘𝑆))
39		fndmin 5757	. . . . 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 2294	. . . . 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 2741	. . . 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 13802	. . 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 952	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubGrp‘𝑆))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2201  ∀wral 2509  {crab 2513   ∩ cin 3198  dom cdm 4727   Fn wfn 5323  ⟶wf 5324  ‘cfv 5328  (class class class)co 6023  Basecbs 13105   MndHom cmhm 13563  SubMndcsubmnd 13564  Grpcgrp 13606  inv_gcminusg 13607  SubGrpcsubg 13777   GrpHom cghm 13850

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-addcom 8137  ax-addass 8139  ax-i2m1 8142  ax-0lt1 8143  ax-0id 8145  ax-rnegex 8146  ax-pre-ltirr 8149  ax-pre-ltadd 8153

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-map 6824  df-pnf 8221  df-mnf 8222  df-ltxr 8224  df-inn 9149  df-2 9207  df-ndx 13108  df-slot 13109  df-base 13111  df-sets 13112  df-iress 13113  df-plusg 13196  df-0g 13364  df-mgm 13462  df-sgrp 13508  df-mnd 13523  df-mhm 13565  df-submnd 13566  df-grp 13609  df-minusg 13610  df-subg 13780  df-ghm 13851

This theorem is referenced by:  rhmeql  14288