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Theorem ghmquskerco 19253

Description: In the case of theorem ghmqusker 19256, 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 2737	. . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻)
4	2, 3	ghmf 19189	. . . 4 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐹:𝐵⟶(Base‘𝐻))
5	1, 4	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐻))
6	5	ffnd 6664	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐵)
7	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
8	7	imaexd 7861	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)) ∈ V)
9	8	uniexd 7690	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)) ∈ V)
10	9	ralrimiva 3130	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)) ∈ V)
11		eqid 2737	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾))) = (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)))
12	11	fnmpt 6633	. . . 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 7394	. . . . . . . 8 ⊢ (𝐺 ~_QG 𝐾) ∈ V
15	14	ecelqsi 8710	. . . . . . 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 7396	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ~_QG 𝐾) ∈ V)
21		reldmghm 19183	. . . . . . . . . . 11 ⊢ Rel dom GrpHom
22	21	ovrcl 7402	. . . . . . . . . 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 17503	. . . . . . 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 2839	. . . . 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 6016	. . . . . 6 ⊢ (𝑞 = [𝑥](𝐺 ~_QG 𝐾) → (𝐹 “ 𝑞) = (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)))
33	32	unieqd 4864	. . . . 5 ⊢ (𝑞 = [𝑥](𝐺 ~_QG 𝐾) → ∪ (𝐹 “ 𝑞) = ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)))
34	27, 29, 31, 33	fmptco 7077	. . . 4 ⊢ (𝜑 → (𝐽 ∘ 𝐿) = (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾))))
35	34	fneq1d 6586	. . 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 8641	. . . . . 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 6636	. . . 4 ⊢ 𝐿 Fn 𝐵
40		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
41		fvco2 6932	. . . 4 ⊢ ((𝐿 Fn 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝐽 ∘ 𝐿)‘𝑥) = (𝐽‘(𝐿‘𝑥)))
42	39, 40, 41	sylancr 588	. . 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 6956	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐿‘𝑥) = [𝑥](𝐺 ~_QG 𝐾))
45	44	fveq2d 6839	. . 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 2847	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (Base‘𝐺))
49	46, 7, 47, 17, 30, 48	ghmquskerlem1 19252	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐽‘[𝑥](𝐺 ~_QG 𝐾)) = (𝐹‘𝑥))
50	42, 45, 49	3eqtrrd 2777	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = ((𝐽 ∘ 𝐿)‘𝑥))
51	6, 36, 50	eqfnfvd 6981	1 ⊢ (𝜑 → 𝐹 = (𝐽 ∘ 𝐿))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 Vcvv 3430 {csn 4568 ∪ cuni 4851 ↦ cmpt 5167 ^◡ccnv 5624 “ cima 5628 ∘ ccom 5629 Fn wfn 6488 ⟶wf 6489 ‘cfv 6493 (class class class)co 7361 [cec 8635 / cqs 8636 Basecbs 17173 0_gc0g 17396 /_s cqus 17463 ~_QG cqg 19092 GrpHom cghm 19181

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-ec 8639 df-qs 8643 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-inf 9350 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-z 12519 df-dec 12639 df-uz 12783 df-fz 13456 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-sca 17230 df-vsca 17231 df-ip 17232 df-tset 17233 df-ple 17234 df-ds 17236 df-0g 17398 df-imas 17466 df-qus 17467 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-submnd 18746 df-grp 18906 df-minusg 18907 df-sbg 18908 df-subg 19093 df-nsg 19094 df-eqg 19095 df-ghm 19182

This theorem is referenced by: algextdeglem4 33883 aks6d1c6lem5 42633

W3C validator