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Theorem slmdvsdi 29550
Description: Distributive law for scalar product. (ax-hvdistr1 27711 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 = (Base‘𝑊)
slmdvsdi.a + = (+g𝑊)
slmdvsdi.f 𝐹 = (Scalar‘𝑊)
slmdvsdi.s · = ( ·𝑠𝑊)
slmdvsdi.k 𝐾 = (Base‘𝐹)
Assertion
Ref Expression
slmdvsdi ((𝑊 ∈ SLMod ∧ (𝑅𝐾𝑋𝑉𝑌𝑉)) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))

Proof of Theorem slmdvsdi
StepHypRef Expression
1 slmdvsdi.v . . . . . . . . 9 𝑉 = (Base‘𝑊)
2 slmdvsdi.a . . . . . . . . 9 + = (+g𝑊)
3 slmdvsdi.s . . . . . . . . 9 · = ( ·𝑠𝑊)
4 eqid 2621 . . . . . . . . 9 (0g𝑊) = (0g𝑊)
5 slmdvsdi.f . . . . . . . . 9 𝐹 = (Scalar‘𝑊)
6 slmdvsdi.k . . . . . . . . 9 𝐾 = (Base‘𝐹)
7 eqid 2621 . . . . . . . . 9 (+g𝐹) = (+g𝐹)
8 eqid 2621 . . . . . . . . 9 (.r𝐹) = (.r𝐹)
9 eqid 2621 . . . . . . . . 9 (1r𝐹) = (1r𝐹)
10 eqid 2621 . . . . . . . . 9 (0g𝐹) = (0g𝐹)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10slmdlema 29538 . . . . . . . 8 ((𝑊 ∈ SLMod ∧ (𝑅𝐾𝑅𝐾) ∧ (𝑌𝑉𝑋𝑉)) → (((𝑅 · 𝑋) ∈ 𝑉 ∧ (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)) ∧ ((𝑅(+g𝐹)𝑅) · 𝑋) = ((𝑅 · 𝑋) + (𝑅 · 𝑋))) ∧ (((𝑅(.r𝐹)𝑅) · 𝑋) = (𝑅 · (𝑅 · 𝑋)) ∧ ((1r𝐹) · 𝑋) = 𝑋 ∧ ((0g𝐹) · 𝑋) = (0g𝑊))))
1211simpld 475 . . . . . . 7 ((𝑊 ∈ SLMod ∧ (𝑅𝐾𝑅𝐾) ∧ (𝑌𝑉𝑋𝑉)) → ((𝑅 · 𝑋) ∈ 𝑉 ∧ (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)) ∧ ((𝑅(+g𝐹)𝑅) · 𝑋) = ((𝑅 · 𝑋) + (𝑅 · 𝑋))))
1312simp2d 1072 . . . . . 6 ((𝑊 ∈ SLMod ∧ (𝑅𝐾𝑅𝐾) ∧ (𝑌𝑉𝑋𝑉)) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))
14133expia 1264 . . . . 5 ((𝑊 ∈ SLMod ∧ (𝑅𝐾𝑅𝐾)) → ((𝑌𝑉𝑋𝑉) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌))))
1514anabsan2 862 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑅𝐾) → ((𝑌𝑉𝑋𝑉) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌))))
1615exp4b 631 . . 3 (𝑊 ∈ SLMod → (𝑅𝐾 → (𝑌𝑉 → (𝑋𝑉 → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌))))))
1716com34 91 . 2 (𝑊 ∈ SLMod → (𝑅𝐾 → (𝑋𝑉 → (𝑌𝑉 → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌))))))
18173imp2 1279 1 ((𝑊 ∈ SLMod ∧ (𝑅𝐾𝑋𝑉𝑌𝑉)) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  .rcmulr 15863  Scalarcsca 15865   ·𝑠 cvsca 15866  0gc0g 16021  1rcur 18422  SLModcslmd 29535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-ov 6607  df-slmd 29536
This theorem is referenced by:  gsumvsca1  29564
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