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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldualvsdi1 | Structured version Visualization version GIF version |
Description: Distributive law for scalar product operation, using operations from the dual space. (Contributed by NM, 21-Oct-2014.) |
Ref | Expression |
---|---|
ldualvsdi1.f | ⊢ 𝐹 = (LFnl‘𝑊) |
ldualvsdi1.r | ⊢ 𝑅 = (Scalar‘𝑊) |
ldualvsdi1.k | ⊢ 𝐾 = (Base‘𝑅) |
ldualvsdi1.d | ⊢ 𝐷 = (LDual‘𝑊) |
ldualvsdi1.p | ⊢ + = (+g‘𝐷) |
ldualvsdi1.s | ⊢ · = ( ·𝑠 ‘𝐷) |
ldualvsdi1.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
ldualvsdi1.x | ⊢ (𝜑 → 𝑋 ∈ 𝐾) |
ldualvsdi1.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
ldualvsdi1.h | ⊢ (𝜑 → 𝐻 ∈ 𝐹) |
Ref | Expression |
---|---|
ldualvsdi1 | ⊢ (𝜑 → (𝑋 · (𝐺 + 𝐻)) = ((𝑋 · 𝐺) + (𝑋 · 𝐻))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ldualvsdi1.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
2 | eqid 2733 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
3 | ldualvsdi1.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑊) | |
4 | ldualvsdi1.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
5 | eqid 2733 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
6 | ldualvsdi1.d | . . . 4 ⊢ 𝐷 = (LDual‘𝑊) | |
7 | ldualvsdi1.s | . . . 4 ⊢ · = ( ·𝑠 ‘𝐷) | |
8 | ldualvsdi1.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
9 | ldualvsdi1.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
10 | ldualvsdi1.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | ldualvs 37945 | . . 3 ⊢ (𝜑 → (𝑋 · 𝐺) = (𝐺 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑋}))) |
12 | ldualvsdi1.h | . . . 4 ⊢ (𝜑 → 𝐻 ∈ 𝐹) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 12 | ldualvs 37945 | . . 3 ⊢ (𝜑 → (𝑋 · 𝐻) = (𝐻 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑋}))) |
14 | 11, 13 | oveq12d 7422 | . 2 ⊢ (𝜑 → ((𝑋 · 𝐺) ∘f (+g‘𝑅)(𝑋 · 𝐻)) = ((𝐺 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑋})) ∘f (+g‘𝑅)(𝐻 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑋})))) |
15 | eqid 2733 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
16 | ldualvsdi1.p | . . 3 ⊢ + = (+g‘𝐷) | |
17 | 1, 3, 4, 6, 7, 8, 9, 10 | ldualvscl 37947 | . . 3 ⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝐹) |
18 | 1, 3, 4, 6, 7, 8, 9, 12 | ldualvscl 37947 | . . 3 ⊢ (𝜑 → (𝑋 · 𝐻) ∈ 𝐹) |
19 | 1, 3, 15, 6, 16, 8, 17, 18 | ldualvadd 37937 | . 2 ⊢ (𝜑 → ((𝑋 · 𝐺) + (𝑋 · 𝐻)) = ((𝑋 · 𝐺) ∘f (+g‘𝑅)(𝑋 · 𝐻))) |
20 | 1, 6, 16, 8, 10, 12 | ldualvaddcl 37938 | . . . 4 ⊢ (𝜑 → (𝐺 + 𝐻) ∈ 𝐹) |
21 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 20 | ldualvs 37945 | . . 3 ⊢ (𝜑 → (𝑋 · (𝐺 + 𝐻)) = ((𝐺 + 𝐻) ∘f (.r‘𝑅)((Base‘𝑊) × {𝑋}))) |
22 | 1, 3, 15, 6, 16, 8, 10, 12 | ldualvadd 37937 | . . . 4 ⊢ (𝜑 → (𝐺 + 𝐻) = (𝐺 ∘f (+g‘𝑅)𝐻)) |
23 | 22 | oveq1d 7419 | . . 3 ⊢ (𝜑 → ((𝐺 + 𝐻) ∘f (.r‘𝑅)((Base‘𝑊) × {𝑋})) = ((𝐺 ∘f (+g‘𝑅)𝐻) ∘f (.r‘𝑅)((Base‘𝑊) × {𝑋}))) |
24 | 2, 3, 4, 15, 5, 1, 8, 9, 10, 12 | lflvsdi1 37886 | . . 3 ⊢ (𝜑 → ((𝐺 ∘f (+g‘𝑅)𝐻) ∘f (.r‘𝑅)((Base‘𝑊) × {𝑋})) = ((𝐺 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑋})) ∘f (+g‘𝑅)(𝐻 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑋})))) |
25 | 21, 23, 24 | 3eqtrd 2777 | . 2 ⊢ (𝜑 → (𝑋 · (𝐺 + 𝐻)) = ((𝐺 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑋})) ∘f (+g‘𝑅)(𝐻 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑋})))) |
26 | 14, 19, 25 | 3eqtr4rd 2784 | 1 ⊢ (𝜑 → (𝑋 · (𝐺 + 𝐻)) = ((𝑋 · 𝐺) + (𝑋 · 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {csn 4627 × cxp 5673 ‘cfv 6540 (class class class)co 7404 ∘f cof 7663 Basecbs 17140 +gcplusg 17193 .rcmulr 17194 Scalarcsca 17196 ·𝑠 cvsca 17197 LModclmod 20459 LFnlclfn 37865 LDualcld 37931 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 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 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7665 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-sca 17209 df-vsca 17210 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-sbg 18820 df-cmn 19643 df-abl 19644 df-mgp 19980 df-ur 19997 df-ring 20049 df-lmod 20461 df-lfl 37866 df-ldual 37932 |
This theorem is referenced by: lduallmodlem 37960 |
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