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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 20884 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 25101 | . 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 20884 | . 2 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐴 · (𝑋 + 𝑌)) = ((𝐴 · 𝑋) + (𝐴 · 𝑌))) | 
| 8 | 1, 7 | sylan 580 | 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 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 +gcplusg 17298 Scalarcsca 17301 ·𝑠 cvsca 17302 LModclmod 20859 ℂModcclm 25096 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-nul 5305 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 df-lmod 20861 df-clm 25097 | 
| This theorem is referenced by: clmnegsubdi2 25139 clmsub4 25140 ncvspi 25191 | 
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