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Theorem ghmquskerco 19196

Description: In the case of theorem ghmqusker 19199, the composition of the natural homomorphism 𝐿 with the constructed homomorphism 𝐽 equals the original homomorphism 𝐹. One says that 𝐹 factors through 𝑄. (Proposed by Saveliy Skresanov, 15-Feb-2025.) (Contributed by Thierry Arnoux, 15-Feb-2025.)

**Hypotheses**
Ref	Expression
ghmqusker.1	⊢ 0 = (0_g‘𝐻)
ghmqusker.f	⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))
ghmqusker.k	⊢ 𝐾 = (^◡𝐹 “ { 0 })
ghmqusker.q	⊢ 𝑄 = (𝐺 /_s (𝐺 ~_QG 𝐾))
ghmqusker.j	⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))
ghmquskerco.b	⊢ 𝐵 = (Base‘𝐺)
ghmquskerco.l	⊢ 𝐿 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~_QG 𝐾))

**Assertion**
Ref	Expression
ghmquskerco	⊢ (𝜑 → 𝐹 = (𝐽 ∘ 𝐿))

Distinct variable groups: 𝑥, 0 𝐵,𝑞,𝑥 𝐹,𝑞,𝑥 𝐺,𝑞,𝑥 𝐻,𝑞,𝑥 𝐽,𝑞,𝑥 𝐾,𝑞,𝑥 𝑥,𝐿 𝑄,𝑞,𝑥 𝜑,𝑞,𝑥 Allowed substitution hints: 𝐿(𝑞) 0 (𝑞)

**Proof of Theorem ghmquskerco**
Step	Hyp	Ref	Expression
1		ghmqusker.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))
2		ghmquskerco.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
3		eqid 2731	. . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻)
4	2, 3	ghmf 19132	. . . 4 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐹:𝐵⟶(Base‘𝐻))
5	1, 4	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐻))
6	5	ffnd 6652	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐵)
7	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
8	7	imaexd 7846	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)) ∈ V)
9	8	uniexd 7675	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)) ∈ V)
10	9	ralrimiva 3124	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)) ∈ V)
11		eqid 2731	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾))) = (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)))
12	11	fnmpt 6621	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)) ∈ V → (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾))) Fn 𝐵)
13	10, 12	syl 17	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾))) Fn 𝐵)
14		ovex 7379	. . . . . . . 8 ⊢ (𝐺 ~_QG 𝐾) ∈ V
15	14	ecelqsi 8694	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → [𝑥](𝐺 ~_QG 𝐾) ∈ (𝐵 / (𝐺 ~_QG 𝐾)))
16	15	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → [𝑥](𝐺 ~_QG 𝐾) ∈ (𝐵 / (𝐺 ~_QG 𝐾)))
17		ghmqusker.q	. . . . . . . . 9 ⊢ 𝑄 = (𝐺 /_s (𝐺 ~_QG 𝐾))
18	17	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑄 = (𝐺 /_s (𝐺 ~_QG 𝐾)))
19	2	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
20		ovexd 7381	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ~_QG 𝐾) ∈ V)
21		reldmghm 19126	. . . . . . . . . . 11 ⊢ Rel dom GrpHom
22	21	ovrcl 7387	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (𝐺 ∈ V ∧ 𝐻 ∈ V))
23	22	simpld 494	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ V)
24	1, 23	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ V)
25	18, 19, 20, 24	qusbas 17449	. . . . . . 7 ⊢ (𝜑 → (𝐵 / (𝐺 ~_QG 𝐾)) = (Base‘𝑄))
26	25	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐵 / (𝐺 ~_QG 𝐾)) = (Base‘𝑄))
27	16, 26	eleqtrd 2833	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → [𝑥](𝐺 ~_QG 𝐾) ∈ (Base‘𝑄))
28		ghmquskerco.l	. . . . . 6 ⊢ 𝐿 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~_QG 𝐾))
29	28	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐿 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~_QG 𝐾)))
30		ghmqusker.j	. . . . . 6 ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))
31	30	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞)))
32		imaeq2 6004	. . . . . 6 ⊢ (𝑞 = [𝑥](𝐺 ~_QG 𝐾) → (𝐹 “ 𝑞) = (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)))
33	32	unieqd 4869	. . . . 5 ⊢ (𝑞 = [𝑥](𝐺 ~_QG 𝐾) → ∪ (𝐹 “ 𝑞) = ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)))
34	27, 29, 31, 33	fmptco 7062	. . . 4 ⊢ (𝜑 → (𝐽 ∘ 𝐿) = (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾))))
35	34	fneq1d 6574	. . 3 ⊢ (𝜑 → ((𝐽 ∘ 𝐿) Fn 𝐵 ↔ (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾))) Fn 𝐵))
36	13, 35	mpbird 257	. 2 ⊢ (𝜑 → (𝐽 ∘ 𝐿) Fn 𝐵)
37		ecexg 8626	. . . . . 6 ⊢ ((𝐺 ~_QG 𝐾) ∈ V → [𝑥](𝐺 ~_QG 𝐾) ∈ V)
38	14, 37	ax-mp 5	. . . . 5 ⊢ [𝑥](𝐺 ~_QG 𝐾) ∈ V
39	38, 28	fnmpti 6624	. . . 4 ⊢ 𝐿 Fn 𝐵
40		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
41		fvco2 6919	. . . 4 ⊢ ((𝐿 Fn 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝐽 ∘ 𝐿)‘𝑥) = (𝐽‘(𝐿‘𝑥)))
42	39, 40, 41	sylancr 587	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝐽 ∘ 𝐿)‘𝑥) = (𝐽‘(𝐿‘𝑥)))
43	38	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → [𝑥](𝐺 ~_QG 𝐾) ∈ V)
44	29, 43	fvmpt2d 6942	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐿‘𝑥) = [𝑥](𝐺 ~_QG 𝐾))
45	44	fveq2d 6826	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐽‘(𝐿‘𝑥)) = (𝐽‘[𝑥](𝐺 ~_QG 𝐾)))
46		ghmqusker.1	. . . 4 ⊢ 0 = (0_g‘𝐻)
47		ghmqusker.k	. . . 4 ⊢ 𝐾 = (^◡𝐹 “ { 0 })
48	40, 2	eleqtrdi 2841	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (Base‘𝐺))
49	46, 7, 47, 17, 30, 48	ghmquskerlem1 19195	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐽‘[𝑥](𝐺 ~_QG 𝐾)) = (𝐹‘𝑥))
50	42, 45, 49	3eqtrrd 2771	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = ((𝐽 ∘ 𝐿)‘𝑥))
51	6, 36, 50	eqfnfvd 6967	1 ⊢ (𝜑 → 𝐹 = (𝐽 ∘ 𝐿))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ∀wral 3047 Vcvv 3436 {csn 4573 ∪ cuni 4856 ↦ cmpt 5170 ^◡ccnv 5613 “ cima 5617 ∘ ccom 5618 Fn wfn 6476 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 [cec 8620 / cqs 8621 Basecbs 17120 0_gc0g 17343 /_s cqus 17409 ~_QG cqg 19035 GrpHom cghm 19124

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-ec 8624 df-qs 8628 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-sup 9326 df-inf 9327 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-fz 13408 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-0g 17345 df-imas 17412 df-qus 17413 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-subg 19036 df-nsg 19037 df-eqg 19038 df-ghm 19125

This theorem is referenced by: algextdeglem4 33733 aks6d1c6lem5 42218

W3C validator