qusvsval - Mathbox for Thierry Arnoux

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Theorem qusvsval 33323

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 2729	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺)) = (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))
8		ovex 7420	. . . . . 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 17505	. . . 4 ⊢ (𝜑 → 𝑁 = ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺)) “_s 𝑀))
12	4, 6, 7, 9, 10	quslem 17506	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺)):𝐵–onto→(𝐵 / (𝑀 ~_QG 𝐺)))
13		eqid 2729	. . . 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 33322	. . . 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 17501	. . 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 8710	. . . . 5 ⊢ (𝑥 = 𝑋 → [𝑥](𝑀 ~_QG 𝐺) = [𝑋](𝑀 ~_QG 𝐺))
28		ecexg 8675	. . . . . 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 6968	. . . 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 7403	. 2 ⊢ (𝜑 → (𝐾 ∙ ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))‘𝑋)) = (𝐾 ∙ [𝑋](𝑀 ~_QG 𝐺)))
33	5, 13, 15, 14	lmodvscl 20784	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → (𝐾 · 𝑋) ∈ 𝐵)
34	10, 1, 2, 33	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝐾 · 𝑋) ∈ 𝐵)
35		eceq1 8710	. . . 4 ⊢ (𝑥 = (𝐾 · 𝑋) → [𝑥](𝑀 ~_QG 𝐺) = [(𝐾 · 𝑋)](𝑀 ~_QG 𝐺))
36		ecexg 8675	. . . . 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 6968	. . 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 2772	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 3447 ↦ cmpt 5188 ‘cfv 6511 (class class class)co 7387 [cec 8669 / cqs 8670 Basecbs 17179 Scalarcsca 17223 ·_𝑠 cvsca 17224 /_s cqus 17468 ~_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-rep 5234 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-tp 4594 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-1o 8434 df-er 8671 df-ec 8673 df-qs 8677 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 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-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-fz 13469 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-0g 17404 df-imas 17471 df-qus 17472 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: lmhmqusker 33388

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