qusvsval - Mathbox for Thierry Arnoux

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Theorem qusvsval 33330

Description: Value of the scalar multiplication operation on the quotient structure. (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)
qusvsval.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

**Assertion**
Ref	Expression
qusvsval	⊢ (𝜑 → (𝐾 ∙ [𝑋](𝑀 ~_QG 𝐺)) = [(𝐾 · 𝑋)](𝑀 ~_QG 𝐺))

Proof of Theorem qusvsval Dummy variables 𝑘 𝑢 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqgvscpbl.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑆)
2		qusvsval.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
3		qusvsval.n	. . . . . 6 ⊢ 𝑁 = (𝑀 /_s (𝑀 ~_QG 𝐺))
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑁 = (𝑀 /_s (𝑀 ~_QG 𝐺)))
5		eqgvscpbl.v	. . . . . 6 ⊢ 𝐵 = (Base‘𝑀)
6	5	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝑀))
7		eqid 2730	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺)) = (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))
8		ovex 7423	. . . . . 6 ⊢ (𝑀 ~_QG 𝐺) ∈ V
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑀 ~_QG 𝐺) ∈ V)
10		eqgvscpbl.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ LMod)
11	4, 6, 7, 9, 10	qusval 17512	. . . 4 ⊢ (𝜑 → 𝑁 = ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺)) “_s 𝑀))
12	4, 6, 7, 9, 10	quslem 17513	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺)):𝐵–onto→(𝐵 / (𝑀 ~_QG 𝐺)))
13		eqid 2730	. . . 4 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
14		eqgvscpbl.s	. . . 4 ⊢ 𝑆 = (Base‘(Scalar‘𝑀))
15		eqgvscpbl.p	. . . 4 ⊢ · = ( ·_𝑠 ‘𝑀)
16		qusvsval.m	. . . 4 ⊢ ∙ = ( ·_𝑠 ‘𝑁)
17		eqgvscpbl.e	. . . . 5 ⊢ ∼ = (𝑀 ~_QG 𝐺)
18	10	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑀 ∈ LMod)
19		eqgvscpbl.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝑀))
20	19	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝐺 ∈ (LSubSp‘𝑀))
21		simpr1 1195	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑘 ∈ 𝑆)
22		simpr2 1196	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑢 ∈ 𝐵)
23		simpr3 1197	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑣 ∈ 𝐵)
24	5, 17, 14, 15, 18, 20, 21, 3, 16, 7, 22, 23	qusvscpbl 33329	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))‘𝑢) = ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))‘𝑣) → ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))‘(𝑘 · 𝑢)) = ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))‘(𝑘 · 𝑣))))
25	11, 6, 12, 10, 13, 14, 15, 16, 24	imasvscaval 17508	. . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → (𝐾 ∙ ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))‘𝑋)) = ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))‘(𝐾 · 𝑋)))
26	1, 2, 25	mpd3an23 1465	. 2 ⊢ (𝜑 → (𝐾 ∙ ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))‘𝑋)) = ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))‘(𝐾 · 𝑋)))
27		eceq1 8713	. . . . 5 ⊢ (𝑥 = 𝑋 → [𝑥](𝑀 ~_QG 𝐺) = [𝑋](𝑀 ~_QG 𝐺))
28		ecexg 8678	. . . . . 6 ⊢ ((𝑀 ~_QG 𝐺) ∈ V → [𝑋](𝑀 ~_QG 𝐺) ∈ V)
29	8, 28	ax-mp 5	. . . . 5 ⊢ [𝑋](𝑀 ~_QG 𝐺) ∈ V
30	27, 7, 29	fvmpt 6971	. . . 4 ⊢ (𝑋 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))‘𝑋) = [𝑋](𝑀 ~_QG 𝐺))
31	2, 30	syl 17	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))‘𝑋) = [𝑋](𝑀 ~_QG 𝐺))
32	31	oveq2d 7406	. 2 ⊢ (𝜑 → (𝐾 ∙ ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))‘𝑋)) = (𝐾 ∙ [𝑋](𝑀 ~_QG 𝐺)))
33	5, 13, 15, 14	lmodvscl 20791	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → (𝐾 · 𝑋) ∈ 𝐵)
34	10, 1, 2, 33	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝐾 · 𝑋) ∈ 𝐵)
35		eceq1 8713	. . . 4 ⊢ (𝑥 = (𝐾 · 𝑋) → [𝑥](𝑀 ~_QG 𝐺) = [(𝐾 · 𝑋)](𝑀 ~_QG 𝐺))
36		ecexg 8678	. . . . 5 ⊢ ((𝑀 ~_QG 𝐺) ∈ V → [(𝐾 · 𝑋)](𝑀 ~_QG 𝐺) ∈ V)
37	8, 36	ax-mp 5	. . . 4 ⊢ [(𝐾 · 𝑋)](𝑀 ~_QG 𝐺) ∈ V
38	35, 7, 37	fvmpt 6971	. . 3 ⊢ ((𝐾 · 𝑋) ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))‘(𝐾 · 𝑋)) = [(𝐾 · 𝑋)](𝑀 ~_QG 𝐺))
39	34, 38	syl 17	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))‘(𝐾 · 𝑋)) = [(𝐾 · 𝑋)](𝑀 ~_QG 𝐺))
40	26, 32, 39	3eqtr3d 2773	1 ⊢ (𝜑 → (𝐾 ∙ [𝑋](𝑀 ~_QG 𝐺)) = [(𝐾 · 𝑋)](𝑀 ~_QG 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 Vcvv 3450 ↦ cmpt 5191 ‘cfv 6514 (class class class)co 7390 [cec 8672 / cqs 8673 Basecbs 17186 Scalarcsca 17230 ·_𝑠 cvsca 17231 /_s cqus 17475 ~_QG cqg 19061 LModclmod 20773 LSubSpclss 20844

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-ec 8676 df-qs 8680 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9400 df-inf 9401 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-fz 13476 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-0g 17411 df-imas 17478 df-qus 17479 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-grp 18875 df-minusg 18876 df-sbg 18877 df-subg 19062 df-eqg 19064 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-lmod 20775 df-lss 20845

This theorem is referenced by: lmhmqusker 33395

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