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Theorem slmdvsdi 32401
Description: Distributive law for scalar product. (ax-hvdistr1 30299 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
slmdvsdi.v 𝑉 = (Baseβ€˜π‘Š)
slmdvsdi.a + = (+gβ€˜π‘Š)
slmdvsdi.f 𝐹 = (Scalarβ€˜π‘Š)
slmdvsdi.s Β· = ( ·𝑠 β€˜π‘Š)
slmdvsdi.k 𝐾 = (Baseβ€˜πΉ)
Assertion
Ref Expression
slmdvsdi ((π‘Š ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ (𝑅 Β· (𝑋 + π‘Œ)) = ((𝑅 Β· 𝑋) + (𝑅 Β· π‘Œ)))
Proof of Theorem slmdvsdi
StepHypRef Expression
1 slmdvsdi.v . . . . . . . . 9 𝑉 = (Baseβ€˜π‘Š)
2 slmdvsdi.a . . . . . . . . 9 + = (+gβ€˜π‘Š)
3 slmdvsdi.s . . . . . . . . 9 Β· = ( ·𝑠 β€˜π‘Š)
4 eqid 2732 . . . . . . . . 9 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
5 slmdvsdi.f . . . . . . . . 9 𝐹 = (Scalarβ€˜π‘Š)
6 slmdvsdi.k . . . . . . . . 9 𝐾 = (Baseβ€˜πΉ)
7 eqid 2732 . . . . . . . . 9 (+gβ€˜πΉ) = (+gβ€˜πΉ)
8 eqid 2732 . . . . . . . . 9 (.rβ€˜πΉ) = (.rβ€˜πΉ)
9 eqid 2732 . . . . . . . . 9 (1rβ€˜πΉ) = (1rβ€˜πΉ)
10 eqid 2732 . . . . . . . . 9 (0gβ€˜πΉ) = (0gβ€˜πΉ)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10slmdlema 32389 . . . . . . . 8 ((π‘Š ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (π‘Œ ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) β†’ (((𝑅 Β· 𝑋) ∈ 𝑉 ∧ (𝑅 Β· (𝑋 + π‘Œ)) = ((𝑅 Β· 𝑋) + (𝑅 Β· π‘Œ)) ∧ ((𝑅(+gβ€˜πΉ)𝑅) Β· 𝑋) = ((𝑅 Β· 𝑋) + (𝑅 Β· 𝑋))) ∧ (((𝑅(.rβ€˜πΉ)𝑅) Β· 𝑋) = (𝑅 Β· (𝑅 Β· 𝑋)) ∧ ((1rβ€˜πΉ) Β· 𝑋) = 𝑋 ∧ ((0gβ€˜πΉ) Β· 𝑋) = (0gβ€˜π‘Š))))
1211simpld 495 . . . . . . 7 ((π‘Š ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (π‘Œ ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) β†’ ((𝑅 Β· 𝑋) ∈ 𝑉 ∧ (𝑅 Β· (𝑋 + π‘Œ)) = ((𝑅 Β· 𝑋) + (𝑅 Β· π‘Œ)) ∧ ((𝑅(+gβ€˜πΉ)𝑅) Β· 𝑋) = ((𝑅 Β· 𝑋) + (𝑅 Β· 𝑋))))
1312simp2d 1143 . . . . . 6 ((π‘Š ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (π‘Œ ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) β†’ (𝑅 Β· (𝑋 + π‘Œ)) = ((𝑅 Β· 𝑋) + (𝑅 Β· π‘Œ)))
14133expia 1121 . . . . 5 ((π‘Š ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) β†’ ((π‘Œ ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) β†’ (𝑅 Β· (𝑋 + π‘Œ)) = ((𝑅 Β· 𝑋) + (𝑅 Β· π‘Œ))))
1514anabsan2 672 . . . 4 ((π‘Š ∈ SLMod ∧ 𝑅 ∈ 𝐾) β†’ ((π‘Œ ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) β†’ (𝑅 Β· (𝑋 + π‘Œ)) = ((𝑅 Β· 𝑋) + (𝑅 Β· π‘Œ))))
1615exp4b 431 . . 3 (π‘Š ∈ SLMod β†’ (𝑅 ∈ 𝐾 β†’ (π‘Œ ∈ 𝑉 β†’ (𝑋 ∈ 𝑉 β†’ (𝑅 Β· (𝑋 + π‘Œ)) = ((𝑅 Β· 𝑋) + (𝑅 Β· π‘Œ))))))
1716com34 91 . 2 (π‘Š ∈ SLMod β†’ (𝑅 ∈ 𝐾 β†’ (𝑋 ∈ 𝑉 β†’ (π‘Œ ∈ 𝑉 β†’ (𝑅 Β· (𝑋 + π‘Œ)) = ((𝑅 Β· 𝑋) + (𝑅 Β· π‘Œ))))))
18173imp2 1349 1 ((π‘Š ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ (𝑅 Β· (𝑋 + π‘Œ)) = ((𝑅 Β· 𝑋) + (𝑅 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  .rcmulr 17200  Scalarcsca 17202   ·𝑠 cvsca 17203  0gc0g 17387  1rcur 20006  SLModcslmd 32386
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-slmd 32387
This theorem is referenced by:  gsumvsca1  32412
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