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Mirrors > Home > MPE Home > Th. List > lmodvsdir | Structured version Visualization version GIF version |
Description: Distributive law for scalar product (right-distributivity). (ax-hvdistr1 28791 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) |
Ref | Expression |
---|---|
lmodvsdir.v | ⊢ 𝑉 = (Base‘𝑊) |
lmodvsdir.a | ⊢ + = (+g‘𝑊) |
lmodvsdir.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lmodvsdir.s | ⊢ · = ( ·𝑠 ‘𝑊) |
lmodvsdir.k | ⊢ 𝐾 = (Base‘𝐹) |
lmodvsdir.p | ⊢ ⨣ = (+g‘𝐹) |
Ref | Expression |
---|---|
lmodvsdir | ⊢ ((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodvsdir.v | . . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊) | |
2 | lmodvsdir.a | . . . . . . . 8 ⊢ + = (+g‘𝑊) | |
3 | lmodvsdir.s | . . . . . . . 8 ⊢ · = ( ·𝑠 ‘𝑊) | |
4 | lmodvsdir.f | . . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊) | |
5 | lmodvsdir.k | . . . . . . . 8 ⊢ 𝐾 = (Base‘𝐹) | |
6 | lmodvsdir.p | . . . . . . . 8 ⊢ ⨣ = (+g‘𝐹) | |
7 | eqid 2798 | . . . . . . . 8 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
8 | eqid 2798 | . . . . . . . 8 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | lmodlema 19632 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → (((𝑅 · 𝑋) ∈ 𝑉 ∧ (𝑅 · (𝑋 + 𝑋)) = ((𝑅 · 𝑋) + (𝑅 · 𝑋)) ∧ ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) ∧ (((𝑄(.r‘𝐹)𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)) ∧ ((1r‘𝐹) · 𝑋) = 𝑋))) |
10 | 9 | simpld 498 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑅 · 𝑋) ∈ 𝑉 ∧ (𝑅 · (𝑋 + 𝑋)) = ((𝑅 · 𝑋) + (𝑅 · 𝑋)) ∧ ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))) |
11 | 10 | simp3d 1141 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) |
12 | 11 | 3expa 1115 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) |
13 | 12 | anabsan2 673 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ 𝑋 ∈ 𝑉) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) |
14 | 13 | exp42 439 | . 2 ⊢ (𝑊 ∈ LMod → (𝑄 ∈ 𝐾 → (𝑅 ∈ 𝐾 → (𝑋 ∈ 𝑉 → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))))) |
15 | 14 | 3imp2 1346 | 1 ⊢ ((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 .rcmulr 16558 Scalarcsca 16560 ·𝑠 cvsca 16561 1rcur 19244 LModclmod 19627 |
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-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 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-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 df-lmod 19629 |
This theorem is referenced by: lmod0vs 19660 lmodvsmmulgdi 19662 lmodvneg1 19670 lmodcom 19673 lmodsubdir 19685 islss3 19724 lss1d 19728 prdslmodd 19734 lspsolvlem 19907 frlmup1 20487 asclghm 20569 scmataddcl 21121 scmatghm 21138 pm2mpghm 21421 clmvsdir 23696 cvsi 23735 lmodvslmhm 30735 imaslmod 30973 lshpkrlem4 36409 baerlem3lem1 39003 baerlem5blem1 39005 hgmapadd 39190 mendlmod 40137 lmodvsmdi 44784 lincsum 44838 ldepsprlem 44881 |
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