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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 2622 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
3 | ldualvsdi1.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑊) | |
4 | ldualvsdi1.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
5 | eqid 2622 | . . . 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 34424 | . . 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 34424 | . . 3 ⊢ (𝜑 → (𝑋 · 𝐻) = (𝐻 ∘𝑓 (.r‘𝑅)((Base‘𝑊) × {𝑋}))) |
14 | 11, 13 | oveq12d 6668 | . 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 2622 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
16 | ldualvsdi1.p | . . 3 ⊢ + = (+g‘𝐷) | |
17 | 1, 3, 4, 6, 7, 8, 9, 10 | ldualvscl 34426 | . . 3 ⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝐹) |
18 | 1, 3, 4, 6, 7, 8, 9, 12 | ldualvscl 34426 | . . 3 ⊢ (𝜑 → (𝑋 · 𝐻) ∈ 𝐹) |
19 | 1, 3, 15, 6, 16, 8, 17, 18 | ldualvadd 34416 | . 2 ⊢ (𝜑 → ((𝑋 · 𝐺) + (𝑋 · 𝐻)) = ((𝑋 · 𝐺) ∘𝑓 (+g‘𝑅)(𝑋 · 𝐻))) |
20 | 1, 6, 16, 8, 10, 12 | ldualvaddcl 34417 | . . . 4 ⊢ (𝜑 → (𝐺 + 𝐻) ∈ 𝐹) |
21 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 20 | ldualvs 34424 | . . 3 ⊢ (𝜑 → (𝑋 · (𝐺 + 𝐻)) = ((𝐺 + 𝐻) ∘𝑓 (.r‘𝑅)((Base‘𝑊) × {𝑋}))) |
22 | 1, 3, 15, 6, 16, 8, 10, 12 | ldualvadd 34416 | . . . 4 ⊢ (𝜑 → (𝐺 + 𝐻) = (𝐺 ∘𝑓 (+g‘𝑅)𝐻)) |
23 | 22 | oveq1d 6665 | . . 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 34365 | . . 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 2660 | . 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 2667 | 1 ⊢ (𝜑 → (𝑋 · (𝐺 + 𝐻)) = ((𝑋 · 𝐺) + (𝑋 · 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 {csn 4177 × cxp 5112 ‘cfv 5888 (class class class)co 6650 ∘𝑓 cof 6895 Basecbs 15857 +gcplusg 15941 .rcmulr 15942 Scalarcsca 15944 ·𝑠 cvsca 15945 LModclmod 18863 LFnlclfn 34344 LDualcld 34410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-sca 15957 df-vsca 15958 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-lmod 18865 df-lfl 34345 df-ldual 34411 |
This theorem is referenced by: lduallmodlem 34439 |
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