ghmquskerco - Mathbox for Thierry Arnoux

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Theorem ghmquskerco 32998

Description: In the case of theorem ghmqusker 33001, 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 2724	. . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻)
4	2, 3	ghmf 19135	. . . 4 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐹:𝐵⟶(Base‘𝐻))
5	1, 4	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐻))
6	5	ffnd 6708	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐵)
7	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
8	7	imaexd 32373	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)) ∈ V)
9	8	uniexd 7725	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)) ∈ V)
10	9	ralrimiva 3138	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)) ∈ V)
11		eqid 2724	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾))) = (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)))
12	11	fnmpt 6680	. . . 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 7434	. . . . . . . 8 ⊢ (𝐺 ~_QG 𝐾) ∈ V
15	14	ecelqsi 8763	. . . . . . 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 7436	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ~_QG 𝐾) ∈ V)
21		reldmghm 19130	. . . . . . . . . . 11 ⊢ Rel dom GrpHom
22	21	ovrcl 7442	. . . . . . . . . 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 17490	. . . . . . 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 2827	. . . . 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 6045	. . . . . 6 ⊢ (𝑞 = [𝑥](𝐺 ~_QG 𝐾) → (𝐹 “ 𝑞) = (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)))
33	32	unieqd 4912	. . . . 5 ⊢ (𝑞 = [𝑥](𝐺 ~_QG 𝐾) → ∪ (𝐹 “ 𝑞) = ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)))
34	27, 29, 31, 33	fmptco 7119	. . . 4 ⊢ (𝜑 → (𝐽 ∘ 𝐿) = (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾))))
35	34	fneq1d 6632	. . 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 8703	. . . . . 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 6683	. . . 4 ⊢ 𝐿 Fn 𝐵
40		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
41		fvco2 6978	. . . 4 ⊢ ((𝐿 Fn 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝐽 ∘ 𝐿)‘𝑥) = (𝐽‘(𝐿‘𝑥)))
42	39, 40, 41	sylancr 586	. . 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 7001	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐿‘𝑥) = [𝑥](𝐺 ~_QG 𝐾))
45	44	fveq2d 6885	. . 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 2835	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (Base‘𝐺))
49	46, 7, 47, 17, 30, 48	ghmquskerlem1 32997	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐽‘[𝑥](𝐺 ~_QG 𝐾)) = (𝐹‘𝑥))
50	42, 45, 49	3eqtrrd 2769	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = ((𝐽 ∘ 𝐿)‘𝑥))
51	6, 36, 50	eqfnfvd 7025	1 ⊢ (𝜑 → 𝐹 = (𝐽 ∘ 𝐿))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3053 Vcvv 3466 {csn 4620 ∪ cuni 4899 ↦ cmpt 5221 ^◡ccnv 5665 “ cima 5669 ∘ ccom 5670 Fn wfn 6528 ⟶wf 6529 ‘cfv 6533 (class class class)co 7401 [cec 8697 / cqs 8698 Basecbs 17143 0_gc0g 17384 /_s cqus 17450 ~_QG cqg 19039 GrpHom cghm 19128

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-ec 8701 df-qs 8705 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-0g 17386 df-imas 17453 df-qus 17454 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18704 df-grp 18856 df-minusg 18857 df-sbg 18858 df-subg 19040 df-nsg 19041 df-eqg 19042 df-ghm 19129

This theorem is referenced by: algextdeglem4 33256

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