qusvscpbl - Mathbox for Thierry Arnoux

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Theorem qusvscpbl 33344

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 2740	. . . 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 33343	. . 3 ⊢ (𝜑 → (𝑈(𝑀 ~_QG 𝐺)𝑉 → (𝐾 · 𝑈)(𝑀 ~_QG 𝐺)(𝐾 · 𝑉)))
9		eqid 2740	. . . . . . 7 ⊢ (LSubSp‘𝑀) = (LSubSp‘𝑀)
10	9	lsssubg 20978	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝐺 ∈ (LSubSp‘𝑀)) → 𝐺 ∈ (SubGrp‘𝑀))
11	5, 6, 10	syl2anc 583	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (SubGrp‘𝑀))
12	1, 2	eqger 19218	. . . . 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 8814	. . 3 ⊢ (𝜑 → (𝑈(𝑀 ~_QG 𝐺)𝑉 ↔ [𝑈](𝑀 ~_QG 𝐺) = [𝑉](𝑀 ~_QG 𝐺)))
16		eqid 2740	. . . . . 6 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
17	1, 16, 4, 3	lmodvscl 20898	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝐾 ∈ 𝑆 ∧ 𝑈 ∈ 𝐵) → (𝐾 · 𝑈) ∈ 𝐵)
18	5, 7, 14, 17	syl3anc 1371	. . . 4 ⊢ (𝜑 → (𝐾 · 𝑈) ∈ 𝐵)
19	13, 18	erth 8814	. . 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 8802	. . . . 5 ⊢ (𝑥 = 𝑈 → [𝑥](𝑀 ~_QG 𝐺) = [𝑈](𝑀 ~_QG 𝐺))
22		qusvscpbl.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))
23		ovex 7481	. . . . . 6 ⊢ (𝑀 ~_QG 𝐺) ∈ V
24		ecexg 8767	. . . . . 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 7029	. . . 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 8802	. . . . 5 ⊢ (𝑥 = 𝑉 → [𝑥](𝑀 ~_QG 𝐺) = [𝑉](𝑀 ~_QG 𝐺))
30		ecexg 8767	. . . . . 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 7029	. . . 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 2756	. 2 ⊢ (𝜑 → ((𝐹‘𝑈) = (𝐹‘𝑉) ↔ [𝑈](𝑀 ~_QG 𝐺) = [𝑉](𝑀 ~_QG 𝐺)))
35		eceq1 8802	. . . . 5 ⊢ (𝑥 = (𝐾 · 𝑈) → [𝑥](𝑀 ~_QG 𝐺) = [(𝐾 · 𝑈)](𝑀 ~_QG 𝐺))
36		ecexg 8767	. . . . . 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 7029	. . . 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 20898	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝐾 ∈ 𝑆 ∧ 𝑉 ∈ 𝐵) → (𝐾 · 𝑉) ∈ 𝐵)
41	5, 7, 28, 40	syl3anc 1371	. . . 4 ⊢ (𝜑 → (𝐾 · 𝑉) ∈ 𝐵)
42		eceq1 8802	. . . . 5 ⊢ (𝑥 = (𝐾 · 𝑉) → [𝑥](𝑀 ~_QG 𝐺) = [(𝐾 · 𝑉)](𝑀 ~_QG 𝐺))
43		ecexg 8767	. . . . . 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 7029	. . . 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 2756	. 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 1537 ∈ wcel 2108 Vcvv 3488 class class class wbr 5166 ↦ cmpt 5249 ‘cfv 6573 (class class class)co 7448 Er wer 8760 [cec 8761 Basecbs 17258 Scalarcsca 17314 ·_𝑠 cvsca 17315 /_s cqus 17565 SubGrpcsubg 19160 ~_QG cqg 19162 LModclmod 20880 LSubSpclss 20952

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-ec 8765 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-0g 17501 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-grp 18976 df-minusg 18977 df-sbg 18978 df-subg 19163 df-eqg 19165 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-lmod 20882 df-lss 20953

This theorem is referenced by: qusvsval 33345 quslmod 33351 quslmhm 33352

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