qusvscpbl - Mathbox for Thierry Arnoux

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Theorem qusvscpbl 30920

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)
qusscaval.n	⊢ 𝑁 = (𝑀 /_s (𝑀 ~_QG 𝐺))
qusscaval.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 2821	. . . 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 30919	. . 3 ⊢ (𝜑 → (𝑈(𝑀 ~_QG 𝐺)𝑉 → (𝐾 · 𝑈)(𝑀 ~_QG 𝐺)(𝐾 · 𝑉)))
9		eqid 2821	. . . . . . 7 ⊢ (LSubSp‘𝑀) = (LSubSp‘𝑀)
10	9	lsssubg 19729	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝐺 ∈ (LSubSp‘𝑀)) → 𝐺 ∈ (SubGrp‘𝑀))
11	5, 6, 10	syl2anc 586	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (SubGrp‘𝑀))
12	1, 2	eqger 18330	. . . . 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 8338	. . 3 ⊢ (𝜑 → (𝑈(𝑀 ~_QG 𝐺)𝑉 ↔ [𝑈](𝑀 ~_QG 𝐺) = [𝑉](𝑀 ~_QG 𝐺)))
16		eqid 2821	. . . . . 6 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
17	1, 16, 4, 3	lmodvscl 19651	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝐾 ∈ 𝑆 ∧ 𝑈 ∈ 𝐵) → (𝐾 · 𝑈) ∈ 𝐵)
18	5, 7, 14, 17	syl3anc 1367	. . . 4 ⊢ (𝜑 → (𝐾 · 𝑈) ∈ 𝐵)
19	13, 18	erth 8338	. . 3 ⊢ (𝜑 → ((𝐾 · 𝑈)(𝑀 ~_QG 𝐺)(𝐾 · 𝑉) ↔ [(𝐾 · 𝑈)](𝑀 ~_QG 𝐺) = [(𝐾 · 𝑉)](𝑀 ~_QG 𝐺)))
20	8, 15, 19	3imtr3d 295	. 2 ⊢ (𝜑 → ([𝑈](𝑀 ~_QG 𝐺) = [𝑉](𝑀 ~_QG 𝐺) → [(𝐾 · 𝑈)](𝑀 ~_QG 𝐺) = [(𝐾 · 𝑉)](𝑀 ~_QG 𝐺)))
21		eceq1 8327	. . . . 5 ⊢ (𝑥 = 𝑈 → [𝑥](𝑀 ~_QG 𝐺) = [𝑈](𝑀 ~_QG 𝐺))
22		qusvscpbl.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))
23		ovex 7189	. . . . . 6 ⊢ (𝑀 ~_QG 𝐺) ∈ V
24		ecexg 8293	. . . . . 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 6768	. . . 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 8327	. . . . 5 ⊢ (𝑥 = 𝑉 → [𝑥](𝑀 ~_QG 𝐺) = [𝑉](𝑀 ~_QG 𝐺))
30		ecexg 8293	. . . . . 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 6768	. . . 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 2837	. 2 ⊢ (𝜑 → ((𝐹‘𝑈) = (𝐹‘𝑉) ↔ [𝑈](𝑀 ~_QG 𝐺) = [𝑉](𝑀 ~_QG 𝐺)))
35		eceq1 8327	. . . . 5 ⊢ (𝑥 = (𝐾 · 𝑈) → [𝑥](𝑀 ~_QG 𝐺) = [(𝐾 · 𝑈)](𝑀 ~_QG 𝐺))
36		ecexg 8293	. . . . . 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 6768	. . . 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 19651	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝐾 ∈ 𝑆 ∧ 𝑉 ∈ 𝐵) → (𝐾 · 𝑉) ∈ 𝐵)
41	5, 7, 28, 40	syl3anc 1367	. . . 4 ⊢ (𝜑 → (𝐾 · 𝑉) ∈ 𝐵)
42		eceq1 8327	. . . . 5 ⊢ (𝑥 = (𝐾 · 𝑉) → [𝑥](𝑀 ~_QG 𝐺) = [(𝐾 · 𝑉)](𝑀 ~_QG 𝐺))
43		ecexg 8293	. . . . . 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 6768	. . . 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 2837	. 2 ⊢ (𝜑 → ((𝐹‘(𝐾 · 𝑈)) = (𝐹‘(𝐾 · 𝑉)) ↔ [(𝐾 · 𝑈)](𝑀 ~_QG 𝐺) = [(𝐾 · 𝑉)](𝑀 ~_QG 𝐺)))
48	20, 34, 47	3imtr4d 296	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 2114 Vcvv 3494 class class class wbr 5066 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 Er wer 8286 [cec 8287 Basecbs 16483 Scalarcsca 16568 ·_𝑠 cvsca 16569 /_s cqus 16778 SubGrpcsubg 18273 ~_QG cqg 18275 LModclmod 19634 LSubSpclss 19703

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-ec 8291 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-eqg 18278 df-mgp 19240 df-ur 19252 df-ring 19299 df-lmod 19636 df-lss 19704

This theorem is referenced by: qusscaval 30921 quslmod 30923 quslmhm 30924

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