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Theorem slmdvsass 30100
Description: Associative law for scalar product. (ax-hvmulass 28194 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 2760 . . . . . . . 8 (+g𝑊) = (+g𝑊)
3 slmdvsass.s . . . . . . . 8 · = ( ·𝑠𝑊)
4 eqid 2760 . . . . . . . 8 (0g𝑊) = (0g𝑊)
5 slmdvsass.f . . . . . . . 8 𝐹 = (Scalar‘𝑊)
6 slmdvsass.k . . . . . . . 8 𝐾 = (Base‘𝐹)
7 eqid 2760 . . . . . . . 8 (+g𝐹) = (+g𝐹)
8 slmdvsass.t . . . . . . . 8 × = (.r𝐹)
9 eqid 2760 . . . . . . . 8 (1r𝐹) = (1r𝐹)
10 eqid 2760 . . . . . . . 8 (0g𝐹) = (0g𝐹)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10slmdlema 30086 . . . . . . 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 482 . . . . . 6 ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾) ∧ (𝑋𝑉𝑋𝑉)) → (((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)) ∧ ((1r𝐹) · 𝑋) = 𝑋 ∧ ((0g𝐹) · 𝑋) = (0g𝑊)))
1312simp1d 1137 . . . . 5 ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾) ∧ (𝑋𝑉𝑋𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
14133expa 1112 . . . 4 (((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾)) ∧ (𝑋𝑉𝑋𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
1514anabsan2 898 . . 3 (((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾)) ∧ 𝑋𝑉) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
1615exp42 640 . 2 (𝑊 ∈ SLMod → (𝑄𝐾 → (𝑅𝐾 → (𝑋𝑉 → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋))))))
17163imp2 1443 1 ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾𝑋𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  cfv 6049  (class class class)co 6814  Basecbs 16079  +gcplusg 16163  .rcmulr 16164  Scalarcsca 16166   ·𝑠 cvsca 16167  0gc0g 16322  1rcur 18721  SLModcslmd 30083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-nul 4941
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-ov 6817  df-slmd 30084
This theorem is referenced by:  slmdvs0  30108
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