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Theorem clmvsdi 24608
Description: Distributive law for scalar product (left-distributivity). (lmodvsdi 20495 analog.) (Contributed by NM, 3-Nov-2006.) (Revised by AV, 28-Sep-2021.)
Hypotheses
Ref Expression
clmvscl.v 𝑉 = (Baseβ€˜π‘Š)
clmvscl.f 𝐹 = (Scalarβ€˜π‘Š)
clmvscl.s Β· = ( ·𝑠 β€˜π‘Š)
clmvscl.k 𝐾 = (Baseβ€˜πΉ)
clmvsdir.a + = (+gβ€˜π‘Š)
Assertion
Ref Expression
clmvsdi ((π‘Š ∈ β„‚Mod ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ (𝐴 Β· (𝑋 + π‘Œ)) = ((𝐴 Β· 𝑋) + (𝐴 Β· π‘Œ)))

Proof of Theorem clmvsdi
StepHypRef Expression
1 clmlmod 24583 . 2 (π‘Š ∈ β„‚Mod β†’ π‘Š ∈ LMod)
2 clmvscl.v . . 3 𝑉 = (Baseβ€˜π‘Š)
3 clmvsdir.a . . 3 + = (+gβ€˜π‘Š)
4 clmvscl.f . . 3 𝐹 = (Scalarβ€˜π‘Š)
5 clmvscl.s . . 3 Β· = ( ·𝑠 β€˜π‘Š)
6 clmvscl.k . . 3 𝐾 = (Baseβ€˜πΉ)
72, 3, 4, 5, 6lmodvsdi 20495 . 2 ((π‘Š ∈ LMod ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ (𝐴 Β· (𝑋 + π‘Œ)) = ((𝐴 Β· 𝑋) + (𝐴 Β· π‘Œ)))
81, 7sylan 581 1 ((π‘Š ∈ β„‚Mod ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ (𝐴 Β· (𝑋 + π‘Œ)) = ((𝐴 Β· 𝑋) + (𝐴 Β· π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  +gcplusg 17197  Scalarcsca 17200   ·𝑠 cvsca 17201  LModclmod 20471  β„‚Modcclm 24578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-lmod 20473  df-clm 24579
This theorem is referenced by:  clmnegsubdi2  24621  clmsub4  24622  ncvspi  24673
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