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Theorem slmdvsass 33081
Description: Associative law for scalar product. (ax-hvmulass 30940 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 2726 . . . . . . . 8 (+g𝑊) = (+g𝑊)
3 slmdvsass.s . . . . . . . 8 · = ( ·𝑠𝑊)
4 eqid 2726 . . . . . . . 8 (0g𝑊) = (0g𝑊)
5 slmdvsass.f . . . . . . . 8 𝐹 = (Scalar‘𝑊)
6 slmdvsass.k . . . . . . . 8 𝐾 = (Base‘𝐹)
7 eqid 2726 . . . . . . . 8 (+g𝐹) = (+g𝐹)
8 slmdvsass.t . . . . . . . 8 × = (.r𝐹)
9 eqid 2726 . . . . . . . 8 (1r𝐹) = (1r𝐹)
10 eqid 2726 . . . . . . . 8 (0g𝐹) = (0g𝐹)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10slmdlema 33067 . . . . . . 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 494 . . . . . 6 ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾) ∧ (𝑋𝑉𝑋𝑉)) → (((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)) ∧ ((1r𝐹) · 𝑋) = 𝑋 ∧ ((0g𝐹) · 𝑋) = (0g𝑊)))
1312simp1d 1139 . . . . 5 ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾) ∧ (𝑋𝑉𝑋𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
14133expa 1115 . . . 4 (((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾)) ∧ (𝑋𝑉𝑋𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
1514anabsan2 672 . . 3 (((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾)) ∧ 𝑋𝑉) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
1615exp42 434 . 2 (𝑊 ∈ SLMod → (𝑄𝐾 → (𝑅𝐾 → (𝑋𝑉 → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋))))))
17163imp2 1346 1 ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾𝑋𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1534  wcel 2099  cfv 6554  (class class class)co 7424  Basecbs 17213  +gcplusg 17266  .rcmulr 17267  Scalarcsca 17269   ·𝑠 cvsca 17270  0gc0g 17454  1rcur 20164  SLModcslmd 33064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-nul 5311
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-iota 6506  df-fv 6562  df-ov 7427  df-slmd 33065
This theorem is referenced by:  slmdvs0  33089
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