qusvscpbl - Mathbox for Thierry Arnoux

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Theorem qusvscpbl 33322

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 2729	. . . 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 33321	. . 3 ⊢ (𝜑 → (𝑈(𝑀 ~_QG 𝐺)𝑉 → (𝐾 · 𝑈)(𝑀 ~_QG 𝐺)(𝐾 · 𝑉)))
9		eqid 2729	. . . . . . 7 ⊢ (LSubSp‘𝑀) = (LSubSp‘𝑀)
10	9	lsssubg 20863	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝐺 ∈ (LSubSp‘𝑀)) → 𝐺 ∈ (SubGrp‘𝑀))
11	5, 6, 10	syl2anc 584	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (SubGrp‘𝑀))
12	1, 2	eqger 19110	. . . . 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 8725	. . 3 ⊢ (𝜑 → (𝑈(𝑀 ~_QG 𝐺)𝑉 ↔ [𝑈](𝑀 ~_QG 𝐺) = [𝑉](𝑀 ~_QG 𝐺)))
16		eqid 2729	. . . . . 6 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
17	1, 16, 4, 3	lmodvscl 20784	. . . . 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 8725	. . 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 8710	. . . . 5 ⊢ (𝑥 = 𝑈 → [𝑥](𝑀 ~_QG 𝐺) = [𝑈](𝑀 ~_QG 𝐺))
22		qusvscpbl.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))
23		ovex 7420	. . . . . 6 ⊢ (𝑀 ~_QG 𝐺) ∈ V
24		ecexg 8675	. . . . . 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 6968	. . . 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 8710	. . . . 5 ⊢ (𝑥 = 𝑉 → [𝑥](𝑀 ~_QG 𝐺) = [𝑉](𝑀 ~_QG 𝐺))
30		ecexg 8675	. . . . . 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 6968	. . . 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 2745	. 2 ⊢ (𝜑 → ((𝐹‘𝑈) = (𝐹‘𝑉) ↔ [𝑈](𝑀 ~_QG 𝐺) = [𝑉](𝑀 ~_QG 𝐺)))
35		eceq1 8710	. . . . 5 ⊢ (𝑥 = (𝐾 · 𝑈) → [𝑥](𝑀 ~_QG 𝐺) = [(𝐾 · 𝑈)](𝑀 ~_QG 𝐺))
36		ecexg 8675	. . . . . 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 6968	. . . 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 20784	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝐾 ∈ 𝑆 ∧ 𝑉 ∈ 𝐵) → (𝐾 · 𝑉) ∈ 𝐵)
41	5, 7, 28, 40	syl3anc 1373	. . . 4 ⊢ (𝜑 → (𝐾 · 𝑉) ∈ 𝐵)
42		eceq1 8710	. . . . 5 ⊢ (𝑥 = (𝐾 · 𝑉) → [𝑥](𝑀 ~_QG 𝐺) = [(𝐾 · 𝑉)](𝑀 ~_QG 𝐺))
43		ecexg 8675	. . . . . 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 6968	. . . 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 2745	. 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 1540 ∈ wcel 2109 Vcvv 3447 class class class wbr 5107 ↦ cmpt 5188 ‘cfv 6511 (class class class)co 7387 Er wer 8668 [cec 8669 Basecbs 17179 Scalarcsca 17223 ·_𝑠 cvsca 17224 /_s cqus 17468 SubGrpcsubg 19052 ~_QG cqg 19054 LModclmod 20766 LSubSpclss 20837

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-er 8671 df-ec 8673 df-en 8919 df-dom 8920 df-sdom 8921 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-0g 17404 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-grp 18868 df-minusg 18869 df-sbg 18870 df-subg 19055 df-eqg 19057 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-lmod 20768 df-lss 20838

This theorem is referenced by: qusvsval 33323 quslmod 33329 quslmhm 33330

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