qusvscpbl - Mathbox for Thierry Arnoux

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Theorem qusvscpbl 33381

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 2734	. . . 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 33380	. . 3 ⊢ (𝜑 → (𝑈(𝑀 ~_QG 𝐺)𝑉 → (𝐾 · 𝑈)(𝑀 ~_QG 𝐺)(𝐾 · 𝑉)))
9		eqid 2734	. . . . . . 7 ⊢ (LSubSp‘𝑀) = (LSubSp‘𝑀)
10	9	lsssubg 20906	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝐺 ∈ (LSubSp‘𝑀)) → 𝐺 ∈ (SubGrp‘𝑀))
11	5, 6, 10	syl2anc 584	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (SubGrp‘𝑀))
12	1, 2	eqger 19105	. . . . 5 ⊢ (𝐺 ∈ (SubGrp‘𝑀) → (𝑀 ~_QG 𝐺) Er 𝐵)
13	11, 12	syl 17	. . . 4 ⊢ (𝜑 → (𝑀 ~_QG 𝐺) Er 𝐵)
14		qusvscpbl.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐵)
15	13, 14	erth 8687	. . 3 ⊢ (𝜑 → (𝑈(𝑀 ~_QG 𝐺)𝑉 ↔ [𝑈](𝑀 ~_QG 𝐺) = [𝑉](𝑀 ~_QG 𝐺)))
16		eqid 2734	. . . . . 6 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
17	1, 16, 4, 3	lmodvscl 20827	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝐾 ∈ 𝑆 ∧ 𝑈 ∈ 𝐵) → (𝐾 · 𝑈) ∈ 𝐵)
18	5, 7, 14, 17	syl3anc 1373	. . . 4 ⊢ (𝜑 → (𝐾 · 𝑈) ∈ 𝐵)
19	13, 18	erth 8687	. . 3 ⊢ (𝜑 → ((𝐾 · 𝑈)(𝑀 ~_QG 𝐺)(𝐾 · 𝑉) ↔ [(𝐾 · 𝑈)](𝑀 ~_QG 𝐺) = [(𝐾 · 𝑉)](𝑀 ~_QG 𝐺)))
20	8, 15, 19	3imtr3d 293	. 2 ⊢ (𝜑 → ([𝑈](𝑀 ~_QG 𝐺) = [𝑉](𝑀 ~_QG 𝐺) → [(𝐾 · 𝑈)](𝑀 ~_QG 𝐺) = [(𝐾 · 𝑉)](𝑀 ~_QG 𝐺)))
21		eceq1 8672	. . . . 5 ⊢ (𝑥 = 𝑈 → [𝑥](𝑀 ~_QG 𝐺) = [𝑈](𝑀 ~_QG 𝐺))
22		qusvscpbl.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))
23		ovex 7389	. . . . . 6 ⊢ (𝑀 ~_QG 𝐺) ∈ V
24		ecexg 8637	. . . . . 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 6939	. . . 4 ⊢ (𝑈 ∈ 𝐵 → (𝐹‘𝑈) = [𝑈](𝑀 ~_QG 𝐺))
27	14, 26	syl 17	. . 3 ⊢ (𝜑 → (𝐹‘𝑈) = [𝑈](𝑀 ~_QG 𝐺))
28		qusvscpbl.v	. . . 4 ⊢ (𝜑 → 𝑉 ∈ 𝐵)
29		eceq1 8672	. . . . 5 ⊢ (𝑥 = 𝑉 → [𝑥](𝑀 ~_QG 𝐺) = [𝑉](𝑀 ~_QG 𝐺))
30		ecexg 8637	. . . . . 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 6939	. . . 4 ⊢ (𝑉 ∈ 𝐵 → (𝐹‘𝑉) = [𝑉](𝑀 ~_QG 𝐺))
33	28, 32	syl 17	. . 3 ⊢ (𝜑 → (𝐹‘𝑉) = [𝑉](𝑀 ~_QG 𝐺))
34	27, 33	eqeq12d 2750	. 2 ⊢ (𝜑 → ((𝐹‘𝑈) = (𝐹‘𝑉) ↔ [𝑈](𝑀 ~_QG 𝐺) = [𝑉](𝑀 ~_QG 𝐺)))
35		eceq1 8672	. . . . 5 ⊢ (𝑥 = (𝐾 · 𝑈) → [𝑥](𝑀 ~_QG 𝐺) = [(𝐾 · 𝑈)](𝑀 ~_QG 𝐺))
36		ecexg 8637	. . . . . 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 6939	. . . 4 ⊢ ((𝐾 · 𝑈) ∈ 𝐵 → (𝐹‘(𝐾 · 𝑈)) = [(𝐾 · 𝑈)](𝑀 ~_QG 𝐺))
39	18, 38	syl 17	. . 3 ⊢ (𝜑 → (𝐹‘(𝐾 · 𝑈)) = [(𝐾 · 𝑈)](𝑀 ~_QG 𝐺))
40	1, 16, 4, 3	lmodvscl 20827	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝐾 ∈ 𝑆 ∧ 𝑉 ∈ 𝐵) → (𝐾 · 𝑉) ∈ 𝐵)
41	5, 7, 28, 40	syl3anc 1373	. . . 4 ⊢ (𝜑 → (𝐾 · 𝑉) ∈ 𝐵)
42		eceq1 8672	. . . . 5 ⊢ (𝑥 = (𝐾 · 𝑉) → [𝑥](𝑀 ~_QG 𝐺) = [(𝐾 · 𝑉)](𝑀 ~_QG 𝐺))
43		ecexg 8637	. . . . . 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 6939	. . . 4 ⊢ ((𝐾 · 𝑉) ∈ 𝐵 → (𝐹‘(𝐾 · 𝑉)) = [(𝐾 · 𝑉)](𝑀 ~_QG 𝐺))
46	41, 45	syl 17	. . 3 ⊢ (𝜑 → (𝐹‘(𝐾 · 𝑉)) = [(𝐾 · 𝑉)](𝑀 ~_QG 𝐺))
47	39, 46	eqeq12d 2750	. 2 ⊢ (𝜑 → ((𝐹‘(𝐾 · 𝑈)) = (𝐹‘(𝐾 · 𝑉)) ↔ [(𝐾 · 𝑈)](𝑀 ~_QG 𝐺) = [(𝐾 · 𝑉)](𝑀 ~_QG 𝐺)))
48	20, 34, 47	3imtr4d 294	1 ⊢ (𝜑 → ((𝐹‘𝑈) = (𝐹‘𝑉) → (𝐹‘(𝐾 · 𝑈)) = (𝐹‘(𝐾 · 𝑉))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2113 Vcvv 3438 class class class wbr 5096 ↦ cmpt 5177 ‘cfv 6490 (class class class)co 7356 Er wer 8630 [cec 8631 Basecbs 17134 Scalarcsca 17178 ·_𝑠 cvsca 17179 /_s cqus 17424 SubGrpcsubg 19048 ~_QG cqg 19050 LModclmod 20809 LSubSpclss 20880

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8633 df-ec 8635 df-en 8882 df-dom 8883 df-sdom 8884 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-nn 12144 df-2 12206 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-0g 17359 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19051 df-eqg 19053 df-cmn 19709 df-abl 19710 df-mgp 20074 df-rng 20086 df-ur 20115 df-ring 20168 df-lmod 20811 df-lss 20881

This theorem is referenced by: qusvsval 33382 quslmod 33388 quslmhm 33389

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