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Mirrors > Home > MPE Home > Th. List > Mathboxes > slmdvsdi | Structured version Visualization version GIF version |
Description: Distributive law for scalar product. (ax-hvdistr1 28785 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
Ref | Expression |
---|---|
slmdvsdi.v | ⊢ 𝑉 = (Base‘𝑊) |
slmdvsdi.a | ⊢ + = (+g‘𝑊) |
slmdvsdi.f | ⊢ 𝐹 = (Scalar‘𝑊) |
slmdvsdi.s | ⊢ · = ( ·𝑠 ‘𝑊) |
slmdvsdi.k | ⊢ 𝐾 = (Base‘𝐹) |
Ref | Expression |
---|---|
slmdvsdi | ⊢ ((𝑊 ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | slmdvsdi.v | . . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑊) | |
2 | slmdvsdi.a | . . . . . . . . 9 ⊢ + = (+g‘𝑊) | |
3 | slmdvsdi.s | . . . . . . . . 9 ⊢ · = ( ·𝑠 ‘𝑊) | |
4 | eqid 2821 | . . . . . . . . 9 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
5 | slmdvsdi.f | . . . . . . . . 9 ⊢ 𝐹 = (Scalar‘𝑊) | |
6 | slmdvsdi.k | . . . . . . . . 9 ⊢ 𝐾 = (Base‘𝐹) | |
7 | eqid 2821 | . . . . . . . . 9 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
8 | eqid 2821 | . . . . . . . . 9 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
9 | eqid 2821 | . . . . . . . . 9 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
10 | eqid 2821 | . . . . . . . . 9 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | slmdlema 30831 | . . . . . . . 8 ⊢ ((𝑊 ∈ SLMod ∧ (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 ∧ ((0g‘𝐹) · 𝑋) = (0g‘𝑊)))) |
12 | 11 | simpld 497 | . . . . . . 7 ⊢ ((𝑊 ∈ SLMod ∧ (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)))) |
13 | 12 | simp2d 1139 | . . . . . 6 ⊢ ((𝑊 ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌))) |
14 | 13 | 3expia 1117 | . . . . 5 ⊢ ((𝑊 ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) → ((𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))) |
15 | 14 | anabsan2 672 | . . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑅 ∈ 𝐾) → ((𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))) |
16 | 15 | exp4b 433 | . . 3 ⊢ (𝑊 ∈ SLMod → (𝑅 ∈ 𝐾 → (𝑌 ∈ 𝑉 → (𝑋 ∈ 𝑉 → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))))) |
17 | 16 | com34 91 | . 2 ⊢ (𝑊 ∈ SLMod → (𝑅 ∈ 𝐾 → (𝑋 ∈ 𝑉 → (𝑌 ∈ 𝑉 → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))))) |
18 | 17 | 3imp2 1345 | 1 ⊢ ((𝑊 ∈ SLMod ∧ (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 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 .rcmulr 16566 Scalarcsca 16568 ·𝑠 cvsca 16569 0gc0g 16713 1rcur 19251 SLModcslmd 30828 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-nul 5210 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 df-slmd 30829 |
This theorem is referenced by: gsumvsca1 30854 |
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