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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 19660 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 23674 | . 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 19660 | . 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 1536 ∈ wcel 2113 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 +gcplusg 16568 Scalarcsca 16571 ·𝑠 cvsca 16572 LModclmod 19637 ℂModcclm 23669 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-nul 5213 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-iota 6317 df-fv 6366 df-ov 7162 df-lmod 19639 df-clm 23670 |
This theorem is referenced by: clmnegsubdi2 23712 clmsub4 23713 ncvspi 23763 |
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