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Theorem slmdvsass 30336
Description: Associative law for scalar product. (ax-hvmulass 28440 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 2778 . . . . . . . 8 (+g𝑊) = (+g𝑊)
3 slmdvsass.s . . . . . . . 8 · = ( ·𝑠𝑊)
4 eqid 2778 . . . . . . . 8 (0g𝑊) = (0g𝑊)
5 slmdvsass.f . . . . . . . 8 𝐹 = (Scalar‘𝑊)
6 slmdvsass.k . . . . . . . 8 𝐾 = (Base‘𝐹)
7 eqid 2778 . . . . . . . 8 (+g𝐹) = (+g𝐹)
8 slmdvsass.t . . . . . . . 8 × = (.r𝐹)
9 eqid 2778 . . . . . . . 8 (1r𝐹) = (1r𝐹)
10 eqid 2778 . . . . . . . 8 (0g𝐹) = (0g𝐹)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10slmdlema 30322 . . . . . . 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 491 . . . . . 6 ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾) ∧ (𝑋𝑉𝑋𝑉)) → (((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)) ∧ ((1r𝐹) · 𝑋) = 𝑋 ∧ ((0g𝐹) · 𝑋) = (0g𝑊)))
1312simp1d 1133 . . . . 5 ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾) ∧ (𝑋𝑉𝑋𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
14133expa 1108 . . . 4 (((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾)) ∧ (𝑋𝑉𝑋𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
1514anabsan2 664 . . 3 (((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾)) ∧ 𝑋𝑉) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
1615exp42 428 . 2 (𝑊 ∈ SLMod → (𝑄𝐾 → (𝑅𝐾 → (𝑋𝑉 → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋))))))
17163imp2 1411 1 ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾𝑋𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2107  cfv 6137  (class class class)co 6924  Basecbs 16259  +gcplusg 16342  .rcmulr 16343  Scalarcsca 16345   ·𝑠 cvsca 16346  0gc0g 16490  1rcur 18892  SLModcslmd 30319
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 5027
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 4674  df-br 4889  df-iota 6101  df-fv 6145  df-ov 6927  df-slmd 30320
This theorem is referenced by:  slmdvs0  30344
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