qusvsval - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > qusvsval		Structured version Visualization version GIF version

Theorem qusvsval 33320

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 2734	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺)) = (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))
8		ovex 7446	. . . . . 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 17559	. . . 4 ⊢ (𝜑 → 𝑁 = ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺)) “_s 𝑀))
12	4, 6, 7, 9, 10	quslem 17560	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺)):𝐵–onto→(𝐵 / (𝑀 ~_QG 𝐺)))
13		eqid 2734	. . . 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 1194	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑘 ∈ 𝑆)
22		simpr2 1195	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑢 ∈ 𝐵)
23		simpr3 1196	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑣 ∈ 𝐵)
24	5, 17, 14, 15, 18, 20, 21, 3, 16, 7, 22, 23	qusvscpbl 33319	. . . 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 17555	. . 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 1464	. 2 ⊢ (𝜑 → (𝐾 ∙ ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))‘𝑋)) = ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))‘(𝐾 · 𝑋)))
27		eceq1 8766	. . . . 5 ⊢ (𝑥 = 𝑋 → [𝑥](𝑀 ~_QG 𝐺) = [𝑋](𝑀 ~_QG 𝐺))
28		ecexg 8731	. . . . . 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 6996	. . . 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 7429	. 2 ⊢ (𝜑 → (𝐾 ∙ ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))‘𝑋)) = (𝐾 ∙ [𝑋](𝑀 ~_QG 𝐺)))
33	5, 13, 15, 14	lmodvscl 20845	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → (𝐾 · 𝑋) ∈ 𝐵)
34	10, 1, 2, 33	syl3anc 1372	. . 3 ⊢ (𝜑 → (𝐾 · 𝑋) ∈ 𝐵)
35		eceq1 8766	. . . 4 ⊢ (𝑥 = (𝐾 · 𝑋) → [𝑥](𝑀 ~_QG 𝐺) = [(𝐾 · 𝑋)](𝑀 ~_QG 𝐺))
36		ecexg 8731	. . . . 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 6996	. . 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 2777	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 1539 ∈ wcel 2107 Vcvv 3463 ↦ cmpt 5205 ‘cfv 6541 (class class class)co 7413 [cec 8725 / cqs 8726 Basecbs 17230 Scalarcsca 17277 ·_𝑠 cvsca 17278 /_s cqus 17522 ~_QG cqg 19110 LModclmod 20827 LSubSpclss 20898

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-cnex 11193 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 ax-pre-mulgt0 11214

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4888 df-iun 4973 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7870 df-1st 7996 df-2nd 7997 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-er 8727 df-ec 8729 df-qs 8733 df-en 8968 df-dom 8969 df-sdom 8970 df-fin 8971 df-sup 9464 df-inf 9465 df-pnf 11279 df-mnf 11280 df-xr 11281 df-ltxr 11282 df-le 11283 df-sub 11476 df-neg 11477 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12510 df-z 12597 df-dec 12717 df-uz 12861 df-fz 13530 df-struct 17167 df-sets 17184 df-slot 17202 df-ndx 17214 df-base 17231 df-ress 17254 df-plusg 17287 df-mulr 17288 df-sca 17290 df-vsca 17291 df-ip 17292 df-tset 17293 df-ple 17294 df-ds 17296 df-0g 17458 df-imas 17525 df-qus 17526 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-grp 18924 df-minusg 18925 df-sbg 18926 df-subg 19111 df-eqg 19113 df-cmn 19769 df-abl 19770 df-mgp 20107 df-rng 20119 df-ur 20148 df-ring 20201 df-lmod 20829 df-lss 20899

This theorem is referenced by: lmhmqusker 33385

W3C validator