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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 27249 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 2610 | . . . . . . . . 9 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
7 | eqid 2610 | . . . . . . . . 9 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
8 | eqid 2610 | . . . . . . . . 9 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | lmodlema 18691 | . . . . . . . 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 474 | . . . . . . 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 1067 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌))) |
12 | 11 | 3expia 1259 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) → ((𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))) |
13 | 12 | anabsan2 859 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) → ((𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))) |
14 | 13 | exp4b 630 | . . 3 ⊢ (𝑊 ∈ LMod → (𝑅 ∈ 𝐾 → (𝑌 ∈ 𝑉 → (𝑋 ∈ 𝑉 → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))))) |
15 | 14 | com34 89 | . 2 ⊢ (𝑊 ∈ LMod → (𝑅 ∈ 𝐾 → (𝑋 ∈ 𝑉 → (𝑌 ∈ 𝑉 → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))))) |
16 | 15 | 3imp2 1274 | 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 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 .rcmulr 15769 Scalarcsca 15771 ·𝑠 cvsca 15772 1rcur 18324 LModclmod 18686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-lmod 18688 |
This theorem is referenced by: lmodcom 18732 lmodsubdi 18743 lmodvsghm 18747 islss3 18780 prdslmodd 18790 lmodvsinv2 18858 lmhmplusg 18865 lsmcl 18904 pj1lmhm 18921 lspfixed 18949 lspsolvlem 18963 clmvsdi 22700 cvsi 22738 lshpkrlem4 33418 baerlem5alem1 36015 baerlem5blem1 36016 hdmap14lem8 36185 mendlmod 36782 lmodvsmdi 41957 |
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