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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 27993 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 2651 | . . . . . . . 8 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
8 | eqid 2651 | . . . . . . . 8 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | lmodlema 18916 | . . . . . . 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 474 | . . . . . 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 1095 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) |
12 | 11 | 3expa 1284 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) |
13 | 12 | anabsan2 880 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ 𝑋 ∈ 𝑉) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) |
14 | 13 | exp42 638 | . 2 ⊢ (𝑊 ∈ LMod → (𝑄 ∈ 𝐾 → (𝑅 ∈ 𝐾 → (𝑋 ∈ 𝑉 → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))))) |
15 | 14 | 3imp2 1304 | 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 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 +gcplusg 15988 .rcmulr 15989 Scalarcsca 15991 ·𝑠 cvsca 15992 1rcur 18547 LModclmod 18911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-nul 4822 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-iota 5889 df-fv 5934 df-ov 6693 df-lmod 18913 |
This theorem is referenced by: lmod0vs 18944 lmodvsmmulgdi 18946 lmodvneg1 18954 lmodcom 18957 lmodsubdir 18969 islss3 19007 lss1d 19011 prdslmodd 19017 lspsolvlem 19190 asclghm 19386 frlmup1 20185 scmataddcl 20370 scmatghm 20387 pm2mpghm 20669 clmvsdir 22937 cvsi 22976 lshpkrlem4 34718 baerlem3lem1 37313 baerlem5blem1 37315 hgmapadd 37503 mendlmod 38080 lmodvsmdi 42488 lincsum 42543 ldepsprlem 42586 |
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