Theorem ghmeql 14123

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 14109	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇))
2		ghmmhm 14109	. . 3 ⊢ (𝐺 ∈ (𝑆 GrpHom 𝑇) → 𝐺 ∈ (𝑆 MndHom 𝑇))
3		mhmeql 13852	. . 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 5695	. . . . . . . 8 ⊢ (𝑦 = ((invg‘𝑆)‘𝑥) → (𝐹‘𝑦) = (𝐹‘((invg‘𝑆)‘𝑥)))
6		fveq2 5695	. . . . . . . 8 ⊢ (𝑦 = ((invg‘𝑆)‘𝑥) → (𝐺‘𝑦) = (𝐺‘((invg‘𝑆)‘𝑥)))
7	5, 6	eqeq12d 2253	. . . . . . 7 ⊢ (𝑦 = ((invg‘𝑆)‘𝑥) → ((𝐹‘𝑦) = (𝐺‘𝑦) ↔ (𝐹‘((invg‘𝑆)‘𝑥)) = (𝐺‘((invg‘𝑆)‘𝑥))))
8		ghmgrp1 14101	. . . . . . . . . 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 535	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → 𝑥 ∈ (Base‘𝑆))
12		eqid 2238	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
13		eqid 2238	. . . . . . . . 9 ⊢ (invg‘𝑆) = (invg‘𝑆)
14	12, 13	grpinvcl 13906	. . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑆)) → ((invg‘𝑆)‘𝑥) ∈ (Base‘𝑆))
15	10, 11, 14	syl2anc 415	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → ((invg‘𝑆)‘𝑥) ∈ (Base‘𝑆))
16		simprr 537	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → (𝐹‘𝑥) = (𝐺‘𝑥))
17	16	fveq2d 5699	. . . . . . . 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 2238	. . . . . . . . . 10 ⊢ (invg‘𝑇) = (invg‘𝑇)
19	12, 13, 18	ghminv 14106	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐹‘((invg‘𝑆)‘𝑥)) = ((invg‘𝑇)‘(𝐹‘𝑥)))
20	19	ad2ant2r 513	. . . . . . . 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 14106	. . . . . . . . 9 ⊢ ((𝐺 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐺‘((invg‘𝑆)‘𝑥)) = ((invg‘𝑇)‘(𝐺‘𝑥)))
22	21	ad2ant2lr 514	. . . . . . . 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 2281	. . . . . . 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 2984	. . . . . 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 2623	. . . 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 5695	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
28		fveq2 5695	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝐺‘𝑦) = (𝐺‘𝑥))
29	27, 28	eqeq12d 2253	. . . . 5 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑦) = (𝐺‘𝑦) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
30	29	ralrab 2987	. . . 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 2238	. . . . . . . 8 ⊢ (Base‘𝑇) = (Base‘𝑇)
33	12, 32	ghmf 14103	. . . . . . 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 5534	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝐹 Fn (Base‘𝑆))
36	12, 32	ghmf 14103	. . . . . . 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 5534	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 𝐺 Fn (Base‘𝑆))
39		fndmin 5816	. . . . 5 ⊢ ((𝐹 Fn (Base‘𝑆) ∧ 𝐺 Fn (Base‘𝑆)) → dom (𝐹 ∩ 𝐺) = {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)})
40	35, 38, 39	syl2anc 415	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹 ∩ 𝐺) = {𝑦 ∈ (Base‘𝑆) ∣ (𝐹‘𝑦) = (𝐺‘𝑦)})
41		eleq2 2302	. . . . 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 2761	. . . 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 14048	. . 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 957	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubGrp‘𝑆))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∀wral 2528  {crab 2532   ∩ cin 3219  dom cdm 4774   Fn wfn 5372  ⟶wf 5373  ‘cfv 5377  (class class class)co 6085  Basecbs 13404   MndHom cmhm 13817  SubMndcsubmnd 13818  Grpcgrp 13858  inv_gcminusg 13859  SubGrpcsubg 14023   GrpHom cghm 14096

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-addcom 8280  ax-addass 8282  ax-i2m1 8285  ax-0lt1 8286  ax-0id 8288  ax-rnegex 8289  ax-pre-ltirr 8292  ax-pre-ltadd 8296

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-map 6924  df-pnf 8363  df-mnf 8364  df-ltxr 8366  df-inn 9308  df-2 9366  df-ndx 13407  df-slot 13408  df-base 13410  df-sets 13411  df-iress 13412  df-plusg 13497  df-0g 13665  df-mgm 13729  df-sgrp 13770  df-mnd 13783  df-mhm 13819  df-submnd 13820  df-grp 13861  df-minusg 13862  df-subg 14026  df-ghm 14097

This theorem is used by:  rhmeql  14642