Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > slmdvsdir | Structured version Visualization version GIF version |
Description: Distributive law for scalar product. (ax-hvdistr1 27249 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
Ref | Expression |
---|---|
slmdvsdir.v | ⊢ 𝑉 = (Base‘𝑊) |
slmdvsdir.a | ⊢ + = (+g‘𝑊) |
slmdvsdir.f | ⊢ 𝐹 = (Scalar‘𝑊) |
slmdvsdir.s | ⊢ · = ( ·𝑠 ‘𝑊) |
slmdvsdir.k | ⊢ 𝐾 = (Base‘𝐹) |
slmdvsdir.p | ⊢ ⨣ = (+g‘𝐹) |
Ref | Expression |
---|---|
slmdvsdir | ⊢ ((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | slmdvsdir.v | . . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊) | |
2 | slmdvsdir.a | . . . . . . . 8 ⊢ + = (+g‘𝑊) | |
3 | slmdvsdir.s | . . . . . . . 8 ⊢ · = ( ·𝑠 ‘𝑊) | |
4 | eqid 2610 | . . . . . . . 8 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
5 | slmdvsdir.f | . . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊) | |
6 | slmdvsdir.k | . . . . . . . 8 ⊢ 𝐾 = (Base‘𝐹) | |
7 | slmdvsdir.p | . . . . . . . 8 ⊢ ⨣ = (+g‘𝐹) | |
8 | eqid 2610 | . . . . . . . 8 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
9 | eqid 2610 | . . . . . . . 8 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
10 | eqid 2610 | . . . . . . . 8 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | slmdlema 29087 | . . . . . . 7 ⊢ ((𝑊 ∈ SLMod ∧ (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 ∧ ((0g‘𝐹) · 𝑋) = (0g‘𝑊)))) |
12 | 11 | simpld 474 | . . . . . 6 ⊢ ((𝑊 ∈ SLMod ∧ (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)))) |
13 | 12 | simp3d 1068 | . . . . 5 ⊢ ((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) |
14 | 13 | 3expa 1257 | . . . 4 ⊢ (((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) |
15 | 14 | anabsan2 859 | . . 3 ⊢ (((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ 𝑋 ∈ 𝑉) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) |
16 | 15 | exp42 637 | . 2 ⊢ (𝑊 ∈ SLMod → (𝑄 ∈ 𝐾 → (𝑅 ∈ 𝐾 → (𝑋 ∈ 𝑉 → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))))) |
17 | 16 | 3imp2 1274 | 1 ⊢ ((𝑊 ∈ SLMod ∧ (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 1031 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 .rcmulr 15769 Scalarcsca 15771 ·𝑠 cvsca 15772 0gc0g 15923 1rcur 18324 SLModcslmd 29084 |
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-slmd 29085 |
This theorem is referenced by: gsumvsca2 29114 |
Copyright terms: Public domain | W3C validator |