qusvscpbl - Mathbox for Thierry Arnoux

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Theorem qusvscpbl 33614

Description: The quotient map distributes over the scalar multiplication. (Contributed by Thierry Arnoux, 18-May-2023.)

**Hypotheses**
Ref	Expression
eqgvscpbl.v	⊢ 𝐵 = (Base‘𝑀)
eqgvscpbl.e	⊢ ∼ = (𝑀 ~_QG 𝐺)
eqgvscpbl.s	⊢ 𝑆 = (Base‘(Scalar‘𝑀))
eqgvscpbl.p	⊢ · = ( ·_𝑠 ‘𝑀)
eqgvscpbl.m	⊢ (𝜑 → 𝑀 ∈ LMod)
eqgvscpbl.g	⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝑀))
eqgvscpbl.k	⊢ (𝜑 → 𝐾 ∈ 𝑆)
qusvsval.n	⊢ 𝑁 = (𝑀 /_s (𝑀 ~_QG 𝐺))
qusvsval.m	⊢ ∙ = ( ·_𝑠 ‘𝑁)
qusvscpbl.f	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))
qusvscpbl.u	⊢ (𝜑 → 𝑈 ∈ 𝐵)
qusvscpbl.v	⊢ (𝜑 → 𝑉 ∈ 𝐵)

**Assertion**
Ref	Expression
qusvscpbl	⊢ (𝜑 → ((𝐹‘𝑈) = (𝐹‘𝑉) → (𝐹‘(𝐾 · 𝑈)) = (𝐹‘(𝐾 · 𝑉))))

Distinct variable groups: 𝑥,𝐵 𝑥,𝐺 𝑥,𝐾 𝑥,𝑀 𝑥,𝑈 𝑥,𝑉 𝜑,𝑥 𝑥, · Allowed substitution hints: ∼ (𝑥) 𝑆(𝑥) ∙ (𝑥) 𝐹(𝑥) 𝑁(𝑥)

**Proof of Theorem qusvscpbl**
Step	Hyp	Ref	Expression
1		eqgvscpbl.v	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
2		eqid 2769	. . . 4 ⊢ (𝑀 ~_QG 𝐺) = (𝑀 ~_QG 𝐺)
3		eqgvscpbl.s	. . . 4 ⊢ 𝑆 = (Base‘(Scalar‘𝑀))
4		eqgvscpbl.p	. . . 4 ⊢ · = ( ·_𝑠 ‘𝑀)
5		eqgvscpbl.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ LMod)
6		eqgvscpbl.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝑀))
7		eqgvscpbl.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑆)
8	1, 2, 3, 4, 5, 6, 7	eqgvscpbl 33613	. . 3 ⊢ (𝜑 → (𝑈(𝑀 ~_QG 𝐺)𝑉 → (𝐾 · 𝑈)(𝑀 ~_QG 𝐺)(𝐾 · 𝑉)))
9		eqid 2769	. . . . . . 7 ⊢ (LSubSp‘𝑀) = (LSubSp‘𝑀)
10	9	lsssubg 21056	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝐺 ∈ (LSubSp‘𝑀)) → 𝐺 ∈ (SubGrp‘𝑀))
11	5, 6, 10	syl2anc 595	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (SubGrp‘𝑀))
12	1, 2	eqger 19246	. . . . 5 ⊢ (𝐺 ∈ (SubGrp‘𝑀) → (𝑀 ~_QG 𝐺) Er 𝐵)
13	11, 12	syl 18	. . . 4 ⊢ (𝜑 → (𝑀 ~_QG 𝐺) Er 𝐵)
14		qusvscpbl.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐵)
15	13, 14	erth 8749	. . 3 ⊢ (𝜑 → (𝑈(𝑀 ~_QG 𝐺)𝑉 ↔ [𝑈](𝑀 ~_QG 𝐺) = [𝑉](𝑀 ~_QG 𝐺)))
16		eqid 2769	. . . . . 6 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
17	1, 16, 4, 3	lmodvscl 20977	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝐾 ∈ 𝑆 ∧ 𝑈 ∈ 𝐵) → (𝐾 · 𝑈) ∈ 𝐵)
18	5, 7, 14, 17	syl3anc 1396	. . . 4 ⊢ (𝜑 → (𝐾 · 𝑈) ∈ 𝐵)
19	13, 18	erth 8749	. . 3 ⊢ (𝜑 → ((𝐾 · 𝑈)(𝑀 ~_QG 𝐺)(𝐾 · 𝑉) ↔ [(𝐾 · 𝑈)](𝑀 ~_QG 𝐺) = [(𝐾 · 𝑉)](𝑀 ~_QG 𝐺)))
20	8, 15, 19	3imtr3d 296	. 2 ⊢ (𝜑 → ([𝑈](𝑀 ~_QG 𝐺) = [𝑉](𝑀 ~_QG 𝐺) → [(𝐾 · 𝑈)](𝑀 ~_QG 𝐺) = [(𝐾 · 𝑉)](𝑀 ~_QG 𝐺)))
21		eceq1 8734	. . . . 5 ⊢ (𝑥 = 𝑈 → [𝑥](𝑀 ~_QG 𝐺) = [𝑈](𝑀 ~_QG 𝐺))
22		qusvscpbl.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))
23		ovex 7444	. . . . . 6 ⊢ (𝑀 ~_QG 𝐺) ∈ V
24		ecexg 8698	. . . . . 6 ⊢ ((𝑀 ~_QG 𝐺) ∈ V → [𝑈](𝑀 ~_QG 𝐺) ∈ V)
25	23, 24	ax-mp 5	. . . . 5 ⊢ [𝑈](𝑀 ~_QG 𝐺) ∈ V
26	21, 22, 25	fvmpt 6990	. . . 4 ⊢ (𝑈 ∈ 𝐵 → (𝐹‘𝑈) = [𝑈](𝑀 ~_QG 𝐺))
27	14, 26	syl 18	. . 3 ⊢ (𝜑 → (𝐹‘𝑈) = [𝑈](𝑀 ~_QG 𝐺))
28		qusvscpbl.v	. . . 4 ⊢ (𝜑 → 𝑉 ∈ 𝐵)
29		eceq1 8734	. . . . 5 ⊢ (𝑥 = 𝑉 → [𝑥](𝑀 ~_QG 𝐺) = [𝑉](𝑀 ~_QG 𝐺))
30		ecexg 8698	. . . . . 6 ⊢ ((𝑀 ~_QG 𝐺) ∈ V → [𝑉](𝑀 ~_QG 𝐺) ∈ V)
31	23, 30	ax-mp 5	. . . . 5 ⊢ [𝑉](𝑀 ~_QG 𝐺) ∈ V
32	29, 22, 31	fvmpt 6990	. . . 4 ⊢ (𝑉 ∈ 𝐵 → (𝐹‘𝑉) = [𝑉](𝑀 ~_QG 𝐺))
33	28, 32	syl 18	. . 3 ⊢ (𝜑 → (𝐹‘𝑉) = [𝑉](𝑀 ~_QG 𝐺))
34	27, 33	eqeq12d 2785	. 2 ⊢ (𝜑 → ((𝐹‘𝑈) = (𝐹‘𝑉) ↔ [𝑈](𝑀 ~_QG 𝐺) = [𝑉](𝑀 ~_QG 𝐺)))
35		eceq1 8734	. . . . 5 ⊢ (𝑥 = (𝐾 · 𝑈) → [𝑥](𝑀 ~_QG 𝐺) = [(𝐾 · 𝑈)](𝑀 ~_QG 𝐺))
36		ecexg 8698	. . . . . 6 ⊢ ((𝑀 ~_QG 𝐺) ∈ V → [(𝐾 · 𝑈)](𝑀 ~_QG 𝐺) ∈ V)
37	23, 36	ax-mp 5	. . . . 5 ⊢ [(𝐾 · 𝑈)](𝑀 ~_QG 𝐺) ∈ V
38	35, 22, 37	fvmpt 6990	. . . 4 ⊢ ((𝐾 · 𝑈) ∈ 𝐵 → (𝐹‘(𝐾 · 𝑈)) = [(𝐾 · 𝑈)](𝑀 ~_QG 𝐺))
39	18, 38	syl 18	. . 3 ⊢ (𝜑 → (𝐹‘(𝐾 · 𝑈)) = [(𝐾 · 𝑈)](𝑀 ~_QG 𝐺))
40	1, 16, 4, 3	lmodvscl 20977	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝐾 ∈ 𝑆 ∧ 𝑉 ∈ 𝐵) → (𝐾 · 𝑉) ∈ 𝐵)
41	5, 7, 28, 40	syl3anc 1396	. . . 4 ⊢ (𝜑 → (𝐾 · 𝑉) ∈ 𝐵)
42		eceq1 8734	. . . . 5 ⊢ (𝑥 = (𝐾 · 𝑉) → [𝑥](𝑀 ~_QG 𝐺) = [(𝐾 · 𝑉)](𝑀 ~_QG 𝐺))
43		ecexg 8698	. . . . . 6 ⊢ ((𝑀 ~_QG 𝐺) ∈ V → [(𝐾 · 𝑉)](𝑀 ~_QG 𝐺) ∈ V)
44	23, 43	ax-mp 5	. . . . 5 ⊢ [(𝐾 · 𝑉)](𝑀 ~_QG 𝐺) ∈ V
45	42, 22, 44	fvmpt 6990	. . . 4 ⊢ ((𝐾 · 𝑉) ∈ 𝐵 → (𝐹‘(𝐾 · 𝑉)) = [(𝐾 · 𝑉)](𝑀 ~_QG 𝐺))
46	41, 45	syl 18	. . 3 ⊢ (𝜑 → (𝐹‘(𝐾 · 𝑉)) = [(𝐾 · 𝑉)](𝑀 ~_QG 𝐺))
47	39, 46	eqeq12d 2785	. 2 ⊢ (𝜑 → ((𝐹‘(𝐾 · 𝑈)) = (𝐹‘(𝐾 · 𝑉)) ↔ [(𝐾 · 𝑈)](𝑀 ~_QG 𝐺) = [(𝐾 · 𝑉)](𝑀 ~_QG 𝐺)))
48	20, 34, 47	3imtr4d 297	1 ⊢ (𝜑 → ((𝐹‘𝑈) = (𝐹‘𝑉) → (𝐹‘(𝐾 · 𝑈)) = (𝐹‘(𝐾 · 𝑉))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1567 ∈ wcel 2149 Vcvv 3463 class class class wbr 5113 ↦ cmpt 5196 ‘cfv 6537 (class class class)co 7411 Er wer 8691 [cec 8692 Basecbs 17269 Scalarcsca 17313 ·_𝑠 cvsca 17314 /_s cqus 17559 SubGrpcsubg 19186 ~_QG cqg 19188 LModclmod 20959 LSubSpclss 21030

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-er 8694 df-ec 8696 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-nn 12234 df-2 12303 df-sets 17224 df-slot 17242 df-ndx 17254 df-base 17270 df-ress 17291 df-plusg 17323 df-0g 17494 df-mgm 18698 df-sgrp 18777 df-mnd 18793 df-grp 19003 df-minusg 19004 df-sbg 19005 df-subg 19189 df-eqg 19191 df-cmn 19852 df-abl 19853 df-mgp 20217 df-rng 20231 df-ur 20264 df-ring 20317 df-lmod 20961 df-lss 21031

This theorem is referenced by: qusvsval 33615 quslmod 33621 quslmhm 33622

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