Theorem clmvsdi 25139

Description: Distributive law for scalar product (left-distributivity). (lmodvsdi 20900 analog.) (Contributed by NM, 3-Nov-2006.) (Revised by AV, 28-Sep-2021.)

Hypotheses

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)

Assertion

Ref	Expression
clmvsdi	⊢ ((𝑊 ∈ ℂMod ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐴 · (𝑋 + 𝑌)) = ((𝐴 · 𝑋) + (𝐴 · 𝑌)))

Proof of Theorem clmvsdi

Step	Hyp	Ref	Expression
1		clmlmod 25114	. 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 20900	. 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)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  +_gcplusg 17298  Scalarcsca 17301   ·_𝑠 cvsca 17302  LModclmod 20875  ℂModcclm 25109

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-lmod 20877  df-clm 25110

This theorem is referenced by:  clmnegsubdi2  25152  clmsub4  25153  ncvspi  25204