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Theorem ghmquskerco 19507

Description: In the case of theorem ghmqusker 19510, 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 2760	. . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻)
4	2, 3	ghmf 19443	. . . 4 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐹:𝐵⟶(Base‘𝐻))
5	1, 4	syl 18	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐻))
6	5	ffnd 6715	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐵)
7	1	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
8	7	imaexd 7932	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)) ∈ V)
9	8	uniexd 7763	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)) ∈ V)
10	9	ralrimiva 3154	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)) ∈ V)
11		eqid 2760	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾))) = (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)))
12	11	fnmpt 6684	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)) ∈ V → (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾))) Fn 𝐵)
13	10, 12	syl 18	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾))) Fn 𝐵)
14		ovex 7457	. . . . . . . 8 ⊢ (𝐺 ~_QG 𝐾) ∈ V
15	14	ecelqsi 8798	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → [𝑥](𝐺 ~_QG 𝐾) ∈ (𝐵 / (𝐺 ~_QG 𝐾)))
16	15	adantl 487	. . . . . 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 7459	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ~_QG 𝐾) ∈ V)
21		reldmghm 19438	. . . . . . . . . . 11 ⊢ Rel dom GrpHom
22	21	ovrcl 7465	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (𝐺 ∈ V ∧ 𝐻 ∈ V))
23	22	simpld 500	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ V)
24	1, 23	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ V)
25	18, 19, 20, 24	qusbas 17726	. . . . . . 7 ⊢ (𝜑 → (𝐵 / (𝐺 ~_QG 𝐾)) = (Base‘𝑄))
26	25	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐵 / (𝐺 ~_QG 𝐾)) = (Base‘𝑄))
27	16, 26	eleqtrd 2862	. . . . 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 6050	. . . . . 6 ⊢ (𝑞 = [𝑥](𝐺 ~_QG 𝐾) → (𝐹 “ 𝑞) = (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)))
33	32	unieqd 4879	. . . . 5 ⊢ (𝑞 = [𝑥](𝐺 ~_QG 𝐾) → ∪ (𝐹 “ 𝑞) = ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)))
34	27, 29, 31, 33	fmptco 7135	. . . 4 ⊢ (𝜑 → (𝐽 ∘ 𝐿) = (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾))))
35	34	fneq1d 6637	. . 3 ⊢ (𝜑 → ((𝐽 ∘ 𝐿) Fn 𝐵 ↔ (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾))) Fn 𝐵))
36	13, 35	mpbird 260	. 2 ⊢ (𝜑 → (𝐽 ∘ 𝐿) Fn 𝐵)
37		ecexg 8729	. . . . . 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 6687	. . . 4 ⊢ 𝐿 Fn 𝐵
40		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
41		fvco2 6987	. . . 4 ⊢ ((𝐿 Fn 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝐽 ∘ 𝐿)‘𝑥) = (𝐽‘(𝐿‘𝑥)))
42	39, 40, 41	sylancr 599	. . 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 7012	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐿‘𝑥) = [𝑥](𝐺 ~_QG 𝐾))
45	44	fveq2d 6894	. . 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 2870	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (Base‘𝐺))
49	46, 7, 47, 17, 30, 48	ghmquskerlem1 19506	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐽‘[𝑥](𝐺 ~_QG 𝐾)) = (𝐹‘𝑥))
50	42, 45, 49	3eqtrrd 2800	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = ((𝐽 ∘ 𝐿)‘𝑥))
51	6, 36, 50	eqfnfvd 7037	1 ⊢ (𝜑 → 𝐹 = (𝐽 ∘ 𝐿))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ∀wral 3076 Vcvv 3450 {csn 4584 ∪ cuni 4866 ↦ cmpt 5185 ^◡ccnv 5650 “ cima 5654 ∘ ccom 5655 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7424 [cec 8723 / cqs 8724 Basecbs 17396 0_gc0g 17619 /_s cqus 17686 ~_QG cqg 19341 GrpHom cghm 19436

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7882 df-1st 8005 df-2nd 8006 df-frecs 8307 df-wrecs 8338 df-recs 8387 df-rdg 8426 df-1o 8484 df-er 8725 df-ec 8727 df-qs 8731 df-map 8857 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-sup 9441 df-inf 9442 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-nn 12343 df-2 12412 df-3 12413 df-4 12414 df-5 12415 df-6 12416 df-7 12417 df-8 12418 df-9 12419 df-n0 12614 df-z 12701 df-dec 12822 df-uz 12973 df-fz 13647 df-struct 17334 df-sets 17351 df-slot 17369 df-ndx 17381 df-base 17397 df-ress 17418 df-plusg 17450 df-mulr 17451 df-sca 17453 df-vsca 17454 df-ip 17455 df-tset 17456 df-ple 17457 df-ds 17459 df-0g 17621 df-imas 17689 df-qus 17690 df-mgm 18825 df-sgrp 18917 df-mnd 18933 df-submnd 18988 df-grp 19156 df-minusg 19157 df-sbg 19158 df-subg 19342 df-nsg 19343 df-eqg 19344 df-ghm 19437

This theorem is used by: algextdeglem4 34402 aks6d1c6lem5 43364

W3C validator