ghmquskerco - Metamath Proof Explorer

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Theorem ghmquskerco 19259

Description: In the case of theorem ghmqusker 19262, 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 2736	. . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻)
4	2, 3	ghmf 19195	. . . 4 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐹:𝐵⟶(Base‘𝐻))
5	1, 4	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐻))
6	5	ffnd 6669	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐵)
7	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
8	7	imaexd 7867	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)) ∈ V)
9	8	uniexd 7696	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)) ∈ V)
10	9	ralrimiva 3129	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)) ∈ V)
11		eqid 2736	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾))) = (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)))
12	11	fnmpt 6638	. . . 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 7400	. . . . . . . 8 ⊢ (𝐺 ~_QG 𝐾) ∈ V
15	14	ecelqsi 8716	. . . . . . 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 7402	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ~_QG 𝐾) ∈ V)
21		reldmghm 19189	. . . . . . . . . . 11 ⊢ Rel dom GrpHom
22	21	ovrcl 7408	. . . . . . . . . 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 17509	. . . . . . 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 2838	. . . . 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 6021	. . . . . 6 ⊢ (𝑞 = [𝑥](𝐺 ~_QG 𝐾) → (𝐹 “ 𝑞) = (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)))
33	32	unieqd 4863	. . . . 5 ⊢ (𝑞 = [𝑥](𝐺 ~_QG 𝐾) → ∪ (𝐹 “ 𝑞) = ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾)))
34	27, 29, 31, 33	fmptco 7082	. . . 4 ⊢ (𝜑 → (𝐽 ∘ 𝐿) = (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~_QG 𝐾))))
35	34	fneq1d 6591	. . 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 8647	. . . . . 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 6641	. . . 4 ⊢ 𝐿 Fn 𝐵
40		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
41		fvco2 6937	. . . 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 6961	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐿‘𝑥) = [𝑥](𝐺 ~_QG 𝐾))
45	44	fveq2d 6844	. . 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 2846	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (Base‘𝐺))
49	46, 7, 47, 17, 30, 48	ghmquskerlem1 19258	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐽‘[𝑥](𝐺 ~_QG 𝐾)) = (𝐹‘𝑥))
50	42, 45, 49	3eqtrrd 2776	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = ((𝐽 ∘ 𝐿)‘𝑥))
51	6, 36, 50	eqfnfvd 6986	1 ⊢ (𝜑 → 𝐹 = (𝐽 ∘ 𝐿))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3051 Vcvv 3429 {csn 4567 ∪ cuni 4850 ↦ cmpt 5166 ^◡ccnv 5630 “ cima 5634 ∘ ccom 5635 Fn wfn 6493 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 [cec 8641 / cqs 8642 Basecbs 17179 0_gc0g 17402 /_s cqus 17469 ~_QG cqg 19098 GrpHom cghm 19187

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 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-ec 8645 df-qs 8649 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-inf 9356 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-fz 13462 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-0g 17404 df-imas 17472 df-qus 17473 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-grp 18912 df-minusg 18913 df-sbg 18914 df-subg 19099 df-nsg 19100 df-eqg 19101 df-ghm 19188

This theorem is referenced by: algextdeglem4 33864 aks6d1c6lem5 42616

W3C validator