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Theorem slmdvsdir 33222
Description: Distributive law for scalar product. (ax-hvdistr1 31027 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
slmdvsdir.v 𝑉 = (Base‘𝑊)
slmdvsdir.a + = (+g𝑊)
slmdvsdir.f 𝐹 = (Scalar‘𝑊)
slmdvsdir.s · = ( ·𝑠𝑊)
slmdvsdir.k 𝐾 = (Base‘𝐹)
slmdvsdir.p = (+g𝐹)
Assertion
Ref Expression
slmdvsdir ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾𝑋𝑉)) → ((𝑄 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))

Proof of Theorem slmdvsdir
StepHypRef Expression
1 slmdvsdir.v . . . . . . . 8 𝑉 = (Base‘𝑊)
2 slmdvsdir.a . . . . . . . 8 + = (+g𝑊)
3 slmdvsdir.s . . . . . . . 8 · = ( ·𝑠𝑊)
4 eqid 2737 . . . . . . . 8 (0g𝑊) = (0g𝑊)
5 slmdvsdir.f . . . . . . . 8 𝐹 = (Scalar‘𝑊)
6 slmdvsdir.k . . . . . . . 8 𝐾 = (Base‘𝐹)
7 slmdvsdir.p . . . . . . . 8 = (+g𝐹)
8 eqid 2737 . . . . . . . 8 (.r𝐹) = (.r𝐹)
9 eqid 2737 . . . . . . . 8 (1r𝐹) = (1r𝐹)
10 eqid 2737 . . . . . . . 8 (0g𝐹) = (0g𝐹)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10slmdlema 33209 . . . . . . 7 ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾) ∧ (𝑋𝑉𝑋𝑉)) → (((𝑅 · 𝑋) ∈ 𝑉 ∧ (𝑅 · (𝑋 + 𝑋)) = ((𝑅 · 𝑋) + (𝑅 · 𝑋)) ∧ ((𝑄 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) ∧ (((𝑄(.r𝐹)𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)) ∧ ((1r𝐹) · 𝑋) = 𝑋 ∧ ((0g𝐹) · 𝑋) = (0g𝑊))))
1211simpld 494 . . . . . 6 ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾) ∧ (𝑋𝑉𝑋𝑉)) → ((𝑅 · 𝑋) ∈ 𝑉 ∧ (𝑅 · (𝑋 + 𝑋)) = ((𝑅 · 𝑋) + (𝑅 · 𝑋)) ∧ ((𝑄 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))))
1312simp3d 1145 . . . . 5 ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾) ∧ (𝑋𝑉𝑋𝑉)) → ((𝑄 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))
14133expa 1119 . . . 4 (((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾)) ∧ (𝑋𝑉𝑋𝑉)) → ((𝑄 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))
1514anabsan2 674 . . 3 (((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾)) ∧ 𝑋𝑉) → ((𝑄 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))
1615exp42 435 . 2 (𝑊 ∈ SLMod → (𝑄𝐾 → (𝑅𝐾 → (𝑋𝑉 → ((𝑄 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))))))
17163imp2 1350 1 ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾𝑋𝑉)) → ((𝑄 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  .rcmulr 17298  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17484  1rcur 20178  SLModcslmd 33206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-slmd 33207
This theorem is referenced by:  gsumvsca2  33233
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