Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  slmdvsdir Structured version   Visualization version   GIF version
Theorem slmdvsdir 32632
Description: Distributive law for scalar product. (ax-hvdistr1 30529 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
slmdvsdir.v 𝑉 = (Baseβ€˜π‘Š)
slmdvsdir.a + = (+gβ€˜π‘Š)
slmdvsdir.f 𝐹 = (Scalarβ€˜π‘Š)
slmdvsdir.s Β· = ( ·𝑠 β€˜π‘Š)
slmdvsdir.k 𝐾 = (Baseβ€˜πΉ)
slmdvsdir.p ⨣ = (+gβ€˜πΉ)
Assertion
Ref Expression
slmdvsdir ((π‘Š ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) β†’ ((𝑄 ⨣ 𝑅) Β· 𝑋) = ((𝑄 Β· 𝑋) + (𝑅 Β· 𝑋)))
Proof of Theorem slmdvsdir
StepHypRef Expression
1 slmdvsdir.v . . . . . . . 8 𝑉 = (Baseβ€˜π‘Š)
2 slmdvsdir.a . . . . . . . 8 + = (+gβ€˜π‘Š)
3 slmdvsdir.s . . . . . . . 8 Β· = ( ·𝑠 β€˜π‘Š)
4 eqid 2731 . . . . . . . 8 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
5 slmdvsdir.f . . . . . . . 8 𝐹 = (Scalarβ€˜π‘Š)
6 slmdvsdir.k . . . . . . . 8 𝐾 = (Baseβ€˜πΉ)
7 slmdvsdir.p . . . . . . . 8 ⨣ = (+gβ€˜πΉ)
8 eqid 2731 . . . . . . . 8 (.rβ€˜πΉ) = (.rβ€˜πΉ)
9 eqid 2731 . . . . . . . 8 (1rβ€˜πΉ) = (1rβ€˜πΉ)
10 eqid 2731 . . . . . . . 8 (0gβ€˜πΉ) = (0gβ€˜πΉ)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10slmdlema 32619 . . . . . . 7 ((π‘Š ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) β†’ (((𝑅 Β· 𝑋) ∈ 𝑉 ∧ (𝑅 Β· (𝑋 + 𝑋)) = ((𝑅 Β· 𝑋) + (𝑅 Β· 𝑋)) ∧ ((𝑄 ⨣ 𝑅) Β· 𝑋) = ((𝑄 Β· 𝑋) + (𝑅 Β· 𝑋))) ∧ (((𝑄(.rβ€˜πΉ)𝑅) Β· 𝑋) = (𝑄 Β· (𝑅 Β· 𝑋)) ∧ ((1rβ€˜πΉ) Β· 𝑋) = 𝑋 ∧ ((0gβ€˜πΉ) Β· 𝑋) = (0gβ€˜π‘Š))))
1211simpld 494 . . . . . 6 ((π‘Š ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) β†’ ((𝑅 Β· 𝑋) ∈ 𝑉 ∧ (𝑅 Β· (𝑋 + 𝑋)) = ((𝑅 Β· 𝑋) + (𝑅 Β· 𝑋)) ∧ ((𝑄 ⨣ 𝑅) Β· 𝑋) = ((𝑄 Β· 𝑋) + (𝑅 Β· 𝑋))))
1312simp3d 1143 . . . . 5 ((π‘Š ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) β†’ ((𝑄 ⨣ 𝑅) Β· 𝑋) = ((𝑄 Β· 𝑋) + (𝑅 Β· 𝑋)))
14133expa 1117 . . . 4 (((π‘Š ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) β†’ ((𝑄 ⨣ 𝑅) Β· 𝑋) = ((𝑄 Β· 𝑋) + (𝑅 Β· 𝑋)))
1514anabsan2 671 . . 3 (((π‘Š ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ 𝑋 ∈ 𝑉) β†’ ((𝑄 ⨣ 𝑅) Β· 𝑋) = ((𝑄 Β· 𝑋) + (𝑅 Β· 𝑋)))
1615exp42 435 . 2 (π‘Š ∈ SLMod β†’ (𝑄 ∈ 𝐾 β†’ (𝑅 ∈ 𝐾 β†’ (𝑋 ∈ 𝑉 β†’ ((𝑄 ⨣ 𝑅) Β· 𝑋) = ((𝑄 Β· 𝑋) + (𝑅 Β· 𝑋))))))
17163imp2 1348 1 ((π‘Š ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) β†’ ((𝑄 ⨣ 𝑅) Β· 𝑋) = ((𝑄 Β· 𝑋) + (𝑅 Β· 𝑋)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  β€˜cfv 6543  (class class class)co 7412  Basecbs 17149  +gcplusg 17202  .rcmulr 17203  Scalarcsca 17205   ·𝑠 cvsca 17206  0gc0g 17390  1rcur 20076  SLModcslmd 32616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7415  df-slmd 32617
This theorem is referenced by:  gsumvsca2  32643
  Copyright terms: Public domain W3C validator