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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > qusvsval | Structured version Visualization version GIF version |
Description: Value of the scalar multiplication operation on the quotient structure. (Contributed by Thierry Arnoux, 18-May-2023.) |
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) |
Ref | Expression |
---|---|
qusvsval | ⊢ (𝜑 → (𝐾 ∙ [𝑋](𝑀 ~QG 𝐺)) = [(𝐾 · 𝑋)](𝑀 ~QG 𝐺)) |
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 2724 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺)) = (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺)) | |
8 | ovex 7434 | . . . . . 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 17487 | . . . 4 ⊢ (𝜑 → 𝑁 = ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺)) “s 𝑀)) |
12 | 4, 6, 7, 9, 10 | quslem 17488 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺)):𝐵–onto→(𝐵 / (𝑀 ~QG 𝐺))) |
13 | eqid 2724 | . . . 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 1191 | . . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑘 ∈ 𝑆) | |
22 | simpr2 1192 | . . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑢 ∈ 𝐵) | |
23 | simpr3 1193 | . . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑣 ∈ 𝐵) | |
24 | 5, 17, 14, 15, 18, 20, 21, 3, 16, 7, 22, 23 | qusvscpbl 32932 | . . . 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 17483 | . . 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 1459 | . 2 ⊢ (𝜑 → (𝐾 ∙ ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘𝑋)) = ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘(𝐾 · 𝑋))) |
27 | eceq1 8737 | . . . . 5 ⊢ (𝑥 = 𝑋 → [𝑥](𝑀 ~QG 𝐺) = [𝑋](𝑀 ~QG 𝐺)) | |
28 | ecexg 8703 | . . . . . 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 6988 | . . . 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 7417 | . 2 ⊢ (𝜑 → (𝐾 ∙ ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘𝑋)) = (𝐾 ∙ [𝑋](𝑀 ~QG 𝐺))) |
33 | 5, 13, 15, 14 | lmodvscl 20714 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → (𝐾 · 𝑋) ∈ 𝐵) |
34 | 10, 1, 2, 33 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝐾 · 𝑋) ∈ 𝐵) |
35 | eceq1 8737 | . . . 4 ⊢ (𝑥 = (𝐾 · 𝑋) → [𝑥](𝑀 ~QG 𝐺) = [(𝐾 · 𝑋)](𝑀 ~QG 𝐺)) | |
36 | ecexg 8703 | . . . . 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 6988 | . . 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 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ↦ cmpt 5221 ‘cfv 6533 (class class class)co 7401 [cec 8697 / cqs 8698 Basecbs 17143 Scalarcsca 17199 ·𝑠 cvsca 17200 /s cqus 17450 ~QG cqg 19039 LModclmod 20696 LSubSpclss 20768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-ec 8701 df-qs 8705 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-0g 17386 df-imas 17453 df-qus 17454 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-grp 18856 df-minusg 18857 df-sbg 18858 df-subg 19040 df-eqg 19042 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-lmod 20698 df-lss 20769 |
This theorem is referenced by: lmhmqusker 33003 |
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