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 33442

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 2740	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺)) = (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))
8		ovex 7396	. . . . . 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 17504	. . . 4 ⊢ (𝜑 → 𝑁 = ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺)) “_s 𝑀))
12	4, 6, 7, 9, 10	quslem 17505	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺)):𝐵–onto→(𝐵 / (𝑀 ~_QG 𝐺)))
13		eqid 2740	. . . 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 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑀 ∈ LMod)
19		eqgvscpbl.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝑀))
20	19	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝐺 ∈ (LSubSp‘𝑀))
21		simpr1 1201	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑘 ∈ 𝑆)
22		simpr2 1202	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑢 ∈ 𝐵)
23		simpr3 1203	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑣 ∈ 𝐵)
24	5, 17, 14, 15, 18, 20, 21, 3, 16, 7, 22, 23	qusvscpbl 33441	. . . 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 17500	. . 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 1471	. 2 ⊢ (𝜑 → (𝐾 ∙ ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))‘𝑋)) = ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))‘(𝐾 · 𝑋)))
27		eceq1 8680	. . . . 5 ⊢ (𝑥 = 𝑋 → [𝑥](𝑀 ~_QG 𝐺) = [𝑋](𝑀 ~_QG 𝐺))
28		ecexg 8644	. . . . . 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 6942	. . . 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 7379	. 2 ⊢ (𝜑 → (𝐾 ∙ ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~_QG 𝐺))‘𝑋)) = (𝐾 ∙ [𝑋](𝑀 ~_QG 𝐺)))
33	5, 13, 15, 14	lmodvscl 20875	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → (𝐾 · 𝑋) ∈ 𝐵)
34	10, 1, 2, 33	syl3anc 1379	. . 3 ⊢ (𝜑 → (𝐾 · 𝑋) ∈ 𝐵)
35		eceq1 8680	. . . 4 ⊢ (𝑥 = (𝐾 · 𝑋) → [𝑥](𝑀 ~_QG 𝐺) = [(𝐾 · 𝑋)](𝑀 ~_QG 𝐺))
36		ecexg 8644	. . . . 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 6942	. . 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 2783	1 ⊢ (𝜑 → (𝐾 ∙ [𝑋](𝑀 ~_QG 𝐺)) = [(𝐾 · 𝑋)](𝑀 ~_QG 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 Vcvv 3432 ↦ cmpt 5160 ‘cfv 6492 (class class class)co 7363 [cec 8638 / cqs 8639 Basecbs 17177 Scalarcsca 17221 ·_𝑠 cvsca 17222 /_s cqus 17467 ~_QG cqg 19096 LModclmod 20857 LSubSpclss 20928

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-er 8640 df-ec 8642 df-qs 8646 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9352 df-inf 9353 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-fz 13460 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-0g 17402 df-imas 17470 df-qus 17471 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-grp 18910 df-minusg 18911 df-sbg 18912 df-subg 19097 df-eqg 19099 df-cmn 19755 df-abl 19756 df-mgp 20120 df-rng 20132 df-ur 20161 df-ring 20214 df-lmod 20859 df-lss 20929

This theorem is referenced by: lmhmqusker 33507

W3C validator