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Theorem slmdvsdi 30330
Description: Distributive law for scalar product. (ax-hvdistr1 28437 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 2778 . . . . . . . . 9 (0g𝑊) = (0g𝑊)
5 slmdvsdi.f . . . . . . . . 9 𝐹 = (Scalar‘𝑊)
6 slmdvsdi.k . . . . . . . . 9 𝐾 = (Base‘𝐹)
7 eqid 2778 . . . . . . . . 9 (+g𝐹) = (+g𝐹)
8 eqid 2778 . . . . . . . . 9 (.r𝐹) = (.r𝐹)
9 eqid 2778 . . . . . . . . 9 (1r𝐹) = (1r𝐹)
10 eqid 2778 . . . . . . . . 9 (0g𝐹) = (0g𝐹)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10slmdlema 30318 . . . . . . . 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 490 . . . . . . 7 ((𝑊 ∈ SLMod ∧ (𝑅𝐾𝑅𝐾) ∧ (𝑌𝑉𝑋𝑉)) → ((𝑅 · 𝑋) ∈ 𝑉 ∧ (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)) ∧ ((𝑅(+g𝐹)𝑅) · 𝑋) = ((𝑅 · 𝑋) + (𝑅 · 𝑋))))
1312simp2d 1134 . . . . . 6 ((𝑊 ∈ SLMod ∧ (𝑅𝐾𝑅𝐾) ∧ (𝑌𝑉𝑋𝑉)) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))
14133expia 1111 . . . . 5 ((𝑊 ∈ SLMod ∧ (𝑅𝐾𝑅𝐾)) → ((𝑌𝑉𝑋𝑉) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌))))
1514anabsan2 664 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑅𝐾) → ((𝑌𝑉𝑋𝑉) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌))))
1615exp4b 423 . . 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 1411 1 ((𝑊 ∈ SLMod ∧ (𝑅𝐾𝑋𝑉𝑌𝑉)) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2107  cfv 6135  (class class class)co 6922  Basecbs 16255  +gcplusg 16338  .rcmulr 16339  Scalarcsca 16341   ·𝑠 cvsca 16342  0gc0g 16486  1rcur 18888  SLModcslmd 30315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-nul 5025
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-iota 6099  df-fv 6143  df-ov 6925  df-slmd 30316
This theorem is referenced by:  gsumvsca1  30344
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