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Mirrors > Home > MPE Home > Th. List > clmvsdi | Structured version Visualization version GIF version |
Description: Distributive law for scalar product (left-distributivity). (lmodvsdi 19651 analog.) (Contributed by NM, 3-Nov-2006.) (Revised by AV, 28-Sep-2021.) |
Ref | Expression |
---|---|
clmvscl.v | ⊢ 𝑉 = (Base‘𝑊) |
clmvscl.f | ⊢ 𝐹 = (Scalar‘𝑊) |
clmvscl.s | ⊢ · = ( ·𝑠 ‘𝑊) |
clmvscl.k | ⊢ 𝐾 = (Base‘𝐹) |
clmvsdir.a | ⊢ + = (+g‘𝑊) |
Ref | Expression |
---|---|
clmvsdi | ⊢ ((𝑊 ∈ ℂMod ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐴 · (𝑋 + 𝑌)) = ((𝐴 · 𝑋) + (𝐴 · 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clmlmod 23665 | . 2 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod) | |
2 | clmvscl.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | clmvsdir.a | . . 3 ⊢ + = (+g‘𝑊) | |
4 | clmvscl.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
5 | clmvscl.s | . . 3 ⊢ · = ( ·𝑠 ‘𝑊) | |
6 | clmvscl.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
7 | 2, 3, 4, 5, 6 | lmodvsdi 19651 | . 2 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐴 · (𝑋 + 𝑌)) = ((𝐴 · 𝑋) + (𝐴 · 𝑌))) |
8 | 1, 7 | sylan 582 | 1 ⊢ ((𝑊 ∈ ℂMod ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐴 · (𝑋 + 𝑌)) = ((𝐴 · 𝑋) + (𝐴 · 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 Scalarcsca 16562 ·𝑠 cvsca 16563 LModclmod 19628 ℂModcclm 23660 |
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 df-clm 23661 |
This theorem is referenced by: clmnegsubdi2 23703 clmsub4 23704 ncvspi 23754 |
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