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Mirrors > Home > MPE Home > Th. List > Mathboxes > slmdvsdir | Structured version Visualization version GIF version |
Description: Distributive law for scalar product. (ax-hvdistr1 31037 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 2735 | . . . . . . . 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 2735 | . . . . . . . 8 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
9 | eqid 2735 | . . . . . . . 8 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
10 | eqid 2735 | . . . . . . . 8 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | slmdlema 33192 | . . . . . . 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 494 | . . . . . 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 1143 | . . . . 5 ⊢ ((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) |
14 | 13 | 3expa 1117 | . . . 4 ⊢ (((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) |
15 | 14 | anabsan2 674 | . . 3 ⊢ (((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ 𝑋 ∈ 𝑉) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) |
16 | 15 | exp42 435 | . 2 ⊢ (𝑊 ∈ SLMod → (𝑄 ∈ 𝐾 → (𝑅 ∈ 𝐾 → (𝑋 ∈ 𝑉 → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))))) |
17 | 16 | 3imp2 1348 | 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 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 .rcmulr 17299 Scalarcsca 17301 ·𝑠 cvsca 17302 0gc0g 17486 1rcur 20199 SLModcslmd 33189 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-slmd 33190 |
This theorem is referenced by: gsumvsca2 33216 |
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