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Theorem slmdvsass 31189
Description: Associative law for scalar product. (ax-hvmulass 29088 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
slmdvsass.v 𝑉 = (Base‘𝑊)
slmdvsass.f 𝐹 = (Scalar‘𝑊)
slmdvsass.s · = ( ·𝑠𝑊)
slmdvsass.k 𝐾 = (Base‘𝐹)
slmdvsass.t × = (.r𝐹)
Assertion
Ref Expression
slmdvsass ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾𝑋𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))

Proof of Theorem slmdvsass
StepHypRef Expression
1 slmdvsass.v . . . . . . . 8 𝑉 = (Base‘𝑊)
2 eqid 2737 . . . . . . . 8 (+g𝑊) = (+g𝑊)
3 slmdvsass.s . . . . . . . 8 · = ( ·𝑠𝑊)
4 eqid 2737 . . . . . . . 8 (0g𝑊) = (0g𝑊)
5 slmdvsass.f . . . . . . . 8 𝐹 = (Scalar‘𝑊)
6 slmdvsass.k . . . . . . . 8 𝐾 = (Base‘𝐹)
7 eqid 2737 . . . . . . . 8 (+g𝐹) = (+g𝐹)
8 slmdvsass.t . . . . . . . 8 × = (.r𝐹)
9 eqid 2737 . . . . . . . 8 (1r𝐹) = (1r𝐹)
10 eqid 2737 . . . . . . . 8 (0g𝐹) = (0g𝐹)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10slmdlema 31175 . . . . . . 7 ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾) ∧ (𝑋𝑉𝑋𝑉)) → (((𝑅 · 𝑋) ∈ 𝑉 ∧ (𝑅 · (𝑋(+g𝑊)𝑋)) = ((𝑅 · 𝑋)(+g𝑊)(𝑅 · 𝑋)) ∧ ((𝑄(+g𝐹)𝑅) · 𝑋) = ((𝑄 · 𝑋)(+g𝑊)(𝑅 · 𝑋))) ∧ (((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)) ∧ ((1r𝐹) · 𝑋) = 𝑋 ∧ ((0g𝐹) · 𝑋) = (0g𝑊))))
1211simprd 499 . . . . . 6 ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾) ∧ (𝑋𝑉𝑋𝑉)) → (((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)) ∧ ((1r𝐹) · 𝑋) = 𝑋 ∧ ((0g𝐹) · 𝑋) = (0g𝑊)))
1312simp1d 1144 . . . . 5 ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾) ∧ (𝑋𝑉𝑋𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
14133expa 1120 . . . 4 (((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾)) ∧ (𝑋𝑉𝑋𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
1514anabsan2 674 . . 3 (((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾)) ∧ 𝑋𝑉) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
1615exp42 439 . 2 (𝑊 ∈ SLMod → (𝑄𝐾 → (𝑅𝐾 → (𝑋𝑉 → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋))))))
17163imp2 1351 1 ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾𝑋𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110  cfv 6380  (class class class)co 7213  Basecbs 16760  +gcplusg 16802  .rcmulr 16803  Scalarcsca 16805   ·𝑠 cvsca 16806  0gc0g 16944  1rcur 19516  SLModcslmd 31172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-nul 5199
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-iota 6338  df-fv 6388  df-ov 7216  df-slmd 31173
This theorem is referenced by:  slmdvs0  31197
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