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Theorem clmvsdi 24615
Description: Distributive law for scalar product (left-distributivity). (lmodvsdi 20500 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 24590 . 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 20500 . 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 580 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  β€˜cfv 6543  (class class class)co 7411  Basecbs 17146  +gcplusg 17199  Scalarcsca 17202   ·𝑠 cvsca 17203  LModclmod 20475  β„‚Modcclm 24585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7414  df-lmod 20477  df-clm 24586
This theorem is referenced by:  clmnegsubdi2  24628  clmsub4  24629  ncvspi  24680
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