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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > qusscaval | 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) |
qusscaval.n | ⊢ 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺)) |
qusscaval.m | ⊢ ∙ = ( ·𝑠 ‘𝑁) |
Ref | Expression |
---|---|
qusscaval | ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → (𝐾 ∙ [𝑋](𝑀 ~QG 𝐺)) = [(𝐾 · 𝑋)](𝑀 ~QG 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qusscaval.n | . . . . 5 ⊢ 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺)) | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺))) |
3 | eqgvscpbl.v | . . . . 5 ⊢ 𝐵 = (Base‘𝑀) | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝑀)) |
5 | eqid 2798 | . . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺)) = (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺)) | |
6 | ovex 7168 | . . . . 5 ⊢ (𝑀 ~QG 𝐺) ∈ V | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑀 ~QG 𝐺) ∈ V) |
8 | eqgvscpbl.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ LMod) | |
9 | 2, 4, 5, 7, 8 | qusval 16807 | . . 3 ⊢ (𝜑 → 𝑁 = ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺)) “s 𝑀)) |
10 | 2, 4, 5, 7, 8 | quslem 16808 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺)):𝐵–onto→(𝐵 / (𝑀 ~QG 𝐺))) |
11 | eqid 2798 | . . 3 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀) | |
12 | eqgvscpbl.s | . . 3 ⊢ 𝑆 = (Base‘(Scalar‘𝑀)) | |
13 | eqgvscpbl.p | . . 3 ⊢ · = ( ·𝑠 ‘𝑀) | |
14 | qusscaval.m | . . 3 ⊢ ∙ = ( ·𝑠 ‘𝑁) | |
15 | eqgvscpbl.e | . . . 4 ⊢ ∼ = (𝑀 ~QG 𝐺) | |
16 | 8 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑀 ∈ LMod) |
17 | eqgvscpbl.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝑀)) | |
18 | 17 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝐺 ∈ (LSubSp‘𝑀)) |
19 | simpr1 1191 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑘 ∈ 𝑆) | |
20 | simpr2 1192 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑢 ∈ 𝐵) | |
21 | simpr3 1193 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑣 ∈ 𝐵) | |
22 | 3, 15, 12, 13, 16, 18, 19, 1, 14, 5, 20, 21 | qusvscpbl 30971 | . . 3 ⊢ ((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)))) |
23 | 9, 4, 10, 8, 11, 12, 13, 14, 22 | imasvscaval 16803 | . 2 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → (𝐾 ∙ ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘𝑋)) = ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘(𝐾 · 𝑋))) |
24 | eceq1 8310 | . . . . 5 ⊢ (𝑥 = 𝑋 → [𝑥](𝑀 ~QG 𝐺) = [𝑋](𝑀 ~QG 𝐺)) | |
25 | ecexg 8276 | . . . . . 6 ⊢ ((𝑀 ~QG 𝐺) ∈ V → [𝑋](𝑀 ~QG 𝐺) ∈ V) | |
26 | 6, 25 | ax-mp 5 | . . . . 5 ⊢ [𝑋](𝑀 ~QG 𝐺) ∈ V |
27 | 24, 5, 26 | fvmpt 6745 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘𝑋) = [𝑋](𝑀 ~QG 𝐺)) |
28 | 27 | 3ad2ant3 1132 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘𝑋) = [𝑋](𝑀 ~QG 𝐺)) |
29 | 28 | oveq2d 7151 | . 2 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → (𝐾 ∙ ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘𝑋)) = (𝐾 ∙ [𝑋](𝑀 ~QG 𝐺))) |
30 | 3, 11, 13, 12 | lmodvscl 19644 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → (𝐾 · 𝑋) ∈ 𝐵) |
31 | 8, 30 | syl3an1 1160 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → (𝐾 · 𝑋) ∈ 𝐵) |
32 | eceq1 8310 | . . . 4 ⊢ (𝑥 = (𝐾 · 𝑋) → [𝑥](𝑀 ~QG 𝐺) = [(𝐾 · 𝑋)](𝑀 ~QG 𝐺)) | |
33 | ecexg 8276 | . . . . 5 ⊢ ((𝑀 ~QG 𝐺) ∈ V → [(𝐾 · 𝑋)](𝑀 ~QG 𝐺) ∈ V) | |
34 | 6, 33 | ax-mp 5 | . . . 4 ⊢ [(𝐾 · 𝑋)](𝑀 ~QG 𝐺) ∈ V |
35 | 32, 5, 34 | fvmpt 6745 | . . 3 ⊢ ((𝐾 · 𝑋) ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘(𝐾 · 𝑋)) = [(𝐾 · 𝑋)](𝑀 ~QG 𝐺)) |
36 | 31, 35 | syl 17 | . 2 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → ((𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))‘(𝐾 · 𝑋)) = [(𝐾 · 𝑋)](𝑀 ~QG 𝐺)) |
37 | 23, 29, 36 | 3eqtr3d 2841 | 1 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → (𝐾 ∙ [𝑋](𝑀 ~QG 𝐺)) = [(𝐾 · 𝑋)](𝑀 ~QG 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 [cec 8270 / cqs 8271 Basecbs 16475 Scalarcsca 16560 ·𝑠 cvsca 16561 /s cqus 16770 ~QG cqg 18267 LModclmod 19627 LSubSpclss 19696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-ec 8274 df-qs 8278 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-0g 16707 df-imas 16773 df-qus 16774 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-eqg 18270 df-mgp 19233 df-ur 19245 df-ring 19292 df-lmod 19629 df-lss 19697 |
This theorem is referenced by: (None) |
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