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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 2734 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
3 | ldualvsdi1.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑊) | |
4 | ldualvsdi1.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
5 | eqid 2734 | . . . 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 36845 | . . 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 36845 | . . 3 ⊢ (𝜑 → (𝑋 · 𝐻) = (𝐻 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑋}))) |
14 | 11, 13 | oveq12d 7220 | . 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 2734 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
16 | ldualvsdi1.p | . . 3 ⊢ + = (+g‘𝐷) | |
17 | 1, 3, 4, 6, 7, 8, 9, 10 | ldualvscl 36847 | . . 3 ⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝐹) |
18 | 1, 3, 4, 6, 7, 8, 9, 12 | ldualvscl 36847 | . . 3 ⊢ (𝜑 → (𝑋 · 𝐻) ∈ 𝐹) |
19 | 1, 3, 15, 6, 16, 8, 17, 18 | ldualvadd 36837 | . 2 ⊢ (𝜑 → ((𝑋 · 𝐺) + (𝑋 · 𝐻)) = ((𝑋 · 𝐺) ∘f (+g‘𝑅)(𝑋 · 𝐻))) |
20 | 1, 6, 16, 8, 10, 12 | ldualvaddcl 36838 | . . . 4 ⊢ (𝜑 → (𝐺 + 𝐻) ∈ 𝐹) |
21 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 20 | ldualvs 36845 | . . 3 ⊢ (𝜑 → (𝑋 · (𝐺 + 𝐻)) = ((𝐺 + 𝐻) ∘f (.r‘𝑅)((Base‘𝑊) × {𝑋}))) |
22 | 1, 3, 15, 6, 16, 8, 10, 12 | ldualvadd 36837 | . . . 4 ⊢ (𝜑 → (𝐺 + 𝐻) = (𝐺 ∘f (+g‘𝑅)𝐻)) |
23 | 22 | oveq1d 7217 | . . 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 36786 | . . 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 2778 | . 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 2785 | 1 ⊢ (𝜑 → (𝑋 · (𝐺 + 𝐻)) = ((𝑋 · 𝐺) + (𝑋 · 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 {csn 4531 × cxp 5538 ‘cfv 6369 (class class class)co 7202 ∘f cof 7456 Basecbs 16684 +gcplusg 16767 .rcmulr 16768 Scalarcsca 16770 ·𝑠 cvsca 16771 LModclmod 19871 LFnlclfn 36765 LDualcld 36831 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-n0 12074 df-z 12160 df-uz 12422 df-fz 13079 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-plusg 16780 df-sca 16783 df-vsca 16784 df-0g 16918 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-grp 18340 df-minusg 18341 df-sbg 18342 df-cmn 19144 df-abl 19145 df-mgp 19477 df-ur 19489 df-ring 19536 df-lmod 19873 df-lfl 36766 df-ldual 36832 |
This theorem is referenced by: lduallmodlem 36860 |
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