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Theorem clmvsdi 22700
 Description: Distributive law for scalar product (left-distributivity). (lmodvsdi 18709 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 · = ( ·𝑠𝑊)
clmvscl.k 𝐾 = (Base‘𝐹)
clmvsdir.a + = (+g𝑊)
Assertion
Ref Expression
clmvsdi ((𝑊 ∈ ℂMod ∧ (𝐴𝐾𝑋𝑉𝑌𝑉)) → (𝐴 · (𝑋 + 𝑌)) = ((𝐴 · 𝑋) + (𝐴 · 𝑌)))

Proof of Theorem clmvsdi
StepHypRef Expression
1 clmlmod 22675 . 2 (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
2 clmvscl.v . . 3 𝑉 = (Base‘𝑊)
3 clmvsdir.a . . 3 + = (+g𝑊)
4 clmvscl.f . . 3 𝐹 = (Scalar‘𝑊)
5 clmvscl.s . . 3 · = ( ·𝑠𝑊)
6 clmvscl.k . . 3 𝐾 = (Base‘𝐹)
72, 3, 4, 5, 6lmodvsdi 18709 . 2 ((𝑊 ∈ LMod ∧ (𝐴𝐾𝑋𝑉𝑌𝑉)) → (𝐴 · (𝑋 + 𝑌)) = ((𝐴 · 𝑋) + (𝐴 · 𝑌)))
81, 7sylan 487 1 ((𝑊 ∈ ℂMod ∧ (𝐴𝐾𝑋𝑉𝑌𝑉)) → (𝐴 · (𝑋 + 𝑌)) = ((𝐴 · 𝑋) + (𝐴 · 𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  Scalarcsca 15771   ·𝑠 cvsca 15772  LModclmod 18686  ℂModcclm 22670 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-lmod 18688  df-clm 22671 This theorem is referenced by:  clmnegsubdi2  22713  clmsub4  22714  ncvspi  22764
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