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Mirrors > Home > MPE Home > Th. List > lmodvsdi | Structured version Visualization version GIF version |
Description: Distributive law for scalar product (left-distributivity). (ax-hvdistr1 28779 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) |
Ref | Expression |
---|---|
lmodvsdi.v | ⊢ 𝑉 = (Base‘𝑊) |
lmodvsdi.a | ⊢ + = (+g‘𝑊) |
lmodvsdi.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lmodvsdi.s | ⊢ · = ( ·𝑠 ‘𝑊) |
lmodvsdi.k | ⊢ 𝐾 = (Base‘𝐹) |
Ref | Expression |
---|---|
lmodvsdi | ⊢ ((𝑊 ∈ LMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodvsdi.v | . . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑊) | |
2 | lmodvsdi.a | . . . . . . . . 9 ⊢ + = (+g‘𝑊) | |
3 | lmodvsdi.s | . . . . . . . . 9 ⊢ · = ( ·𝑠 ‘𝑊) | |
4 | lmodvsdi.f | . . . . . . . . 9 ⊢ 𝐹 = (Scalar‘𝑊) | |
5 | lmodvsdi.k | . . . . . . . . 9 ⊢ 𝐾 = (Base‘𝐹) | |
6 | eqid 2821 | . . . . . . . . 9 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
7 | eqid 2821 | . . . . . . . . 9 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
8 | eqid 2821 | . . . . . . . . 9 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | lmodlema 19633 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → (((𝑅 · 𝑋) ∈ 𝑉 ∧ (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)) ∧ ((𝑅(+g‘𝐹)𝑅) · 𝑋) = ((𝑅 · 𝑋) + (𝑅 · 𝑋))) ∧ (((𝑅(.r‘𝐹)𝑅) · 𝑋) = (𝑅 · (𝑅 · 𝑋)) ∧ ((1r‘𝐹) · 𝑋) = 𝑋))) |
10 | 9 | simpld 497 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑅 · 𝑋) ∈ 𝑉 ∧ (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)) ∧ ((𝑅(+g‘𝐹)𝑅) · 𝑋) = ((𝑅 · 𝑋) + (𝑅 · 𝑋)))) |
11 | 10 | simp2d 1139 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌))) |
12 | 11 | 3expia 1117 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) → ((𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))) |
13 | 12 | anabsan2 672 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) → ((𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))) |
14 | 13 | exp4b 433 | . . 3 ⊢ (𝑊 ∈ LMod → (𝑅 ∈ 𝐾 → (𝑌 ∈ 𝑉 → (𝑋 ∈ 𝑉 → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))))) |
15 | 14 | com34 91 | . 2 ⊢ (𝑊 ∈ LMod → (𝑅 ∈ 𝐾 → (𝑋 ∈ 𝑉 → (𝑌 ∈ 𝑉 → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))))) |
16 | 15 | 3imp2 1345 | 1 ⊢ ((𝑊 ∈ LMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 .rcmulr 16560 Scalarcsca 16562 ·𝑠 cvsca 16563 1rcur 19245 LModclmod 19628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-nul 5202 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-iota 6308 df-fv 6357 df-ov 7153 df-lmod 19630 |
This theorem is referenced by: lmodcom 19674 lmodsubdi 19685 lmodvsghm 19689 islss3 19725 prdslmodd 19735 lmodvsinv2 19803 lmhmplusg 19810 lsmcl 19849 pj1lmhm 19866 lspfixed 19894 lspsolvlem 19908 clmvsdi 23690 cvsi 23728 eqgvscpbl 30914 imaslmod 30917 lshpkrlem4 36243 baerlem5alem1 38838 baerlem5blem1 38839 hdmap14lem8 39005 mendlmod 39786 lmodvsmdi 44424 |
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