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Theorem slmdvsdi 30878
Description: Distributive law for scalar product. (ax-hvdistr1 28797 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 2824 . . . . . . . . 9 (0g𝑊) = (0g𝑊)
5 slmdvsdi.f . . . . . . . . 9 𝐹 = (Scalar‘𝑊)
6 slmdvsdi.k . . . . . . . . 9 𝐾 = (Base‘𝐹)
7 eqid 2824 . . . . . . . . 9 (+g𝐹) = (+g𝐹)
8 eqid 2824 . . . . . . . . 9 (.r𝐹) = (.r𝐹)
9 eqid 2824 . . . . . . . . 9 (1r𝐹) = (1r𝐹)
10 eqid 2824 . . . . . . . . 9 (0g𝐹) = (0g𝐹)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10slmdlema 30866 . . . . . . . 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 498 . . . . . . 7 ((𝑊 ∈ SLMod ∧ (𝑅𝐾𝑅𝐾) ∧ (𝑌𝑉𝑋𝑉)) → ((𝑅 · 𝑋) ∈ 𝑉 ∧ (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)) ∧ ((𝑅(+g𝐹)𝑅) · 𝑋) = ((𝑅 · 𝑋) + (𝑅 · 𝑋))))
1312simp2d 1140 . . . . . 6 ((𝑊 ∈ SLMod ∧ (𝑅𝐾𝑅𝐾) ∧ (𝑌𝑉𝑋𝑉)) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))
14133expia 1118 . . . . 5 ((𝑊 ∈ SLMod ∧ (𝑅𝐾𝑅𝐾)) → ((𝑌𝑉𝑋𝑉) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌))))
1514anabsan2 673 . . . 4 ((𝑊 ∈ SLMod ∧ 𝑅𝐾) → ((𝑌𝑉𝑋𝑉) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌))))
1615exp4b 434 . . 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 1346 1 ((𝑊 ∈ SLMod ∧ (𝑅𝐾𝑋𝑉𝑌𝑉)) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  cfv 6343  (class class class)co 7149  Basecbs 16483  +gcplusg 16565  .rcmulr 16566  Scalarcsca 16568   ·𝑠 cvsca 16569  0gc0g 16713  1rcur 19251  SLModcslmd 30863
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5196
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-ov 7152  df-slmd 30864
This theorem is referenced by:  gsumvsca1  30889
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