MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lmodvsdir Structured version   Visualization version   GIF version
Theorem lmodvsdir 20501
Description: Distributive law for scalar product (right-distributivity). (ax-hvdistr1 30299 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
lmodvsdir.v 𝑉 = (Baseβ€˜π‘Š)
lmodvsdir.a + = (+gβ€˜π‘Š)
lmodvsdir.f 𝐹 = (Scalarβ€˜π‘Š)
lmodvsdir.s Β· = ( ·𝑠 β€˜π‘Š)
lmodvsdir.k 𝐾 = (Baseβ€˜πΉ)
lmodvsdir.p ⨣ = (+gβ€˜πΉ)
Assertion
Ref Expression
lmodvsdir ((π‘Š ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) β†’ ((𝑄 ⨣ 𝑅) Β· 𝑋) = ((𝑄 Β· 𝑋) + (𝑅 Β· 𝑋)))
Proof of Theorem lmodvsdir
StepHypRef Expression
1 lmodvsdir.v . . . . . . . 8 𝑉 = (Baseβ€˜π‘Š)
2 lmodvsdir.a . . . . . . . 8 + = (+gβ€˜π‘Š)
3 lmodvsdir.s . . . . . . . 8 Β· = ( ·𝑠 β€˜π‘Š)
4 lmodvsdir.f . . . . . . . 8 𝐹 = (Scalarβ€˜π‘Š)
5 lmodvsdir.k . . . . . . . 8 𝐾 = (Baseβ€˜πΉ)
6 lmodvsdir.p . . . . . . . 8 ⨣ = (+gβ€˜πΉ)
7 eqid 2732 . . . . . . . 8 (.rβ€˜πΉ) = (.rβ€˜πΉ)
8 eqid 2732 . . . . . . . 8 (1rβ€˜πΉ) = (1rβ€˜πΉ)
91, 2, 3, 4, 5, 6, 7, 8lmodlema 20480 . . . . . . 7 ((π‘Š ∈ LMod ∧ (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)))
109simpld 495 . . . . . 6 ((π‘Š ∈ LMod ∧ (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))))
1110simp3d 1144 . . . . 5 ((π‘Š ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) β†’ ((𝑄 ⨣ 𝑅) Β· 𝑋) = ((𝑄 Β· 𝑋) + (𝑅 Β· 𝑋)))
12113expa 1118 . . . 4 (((π‘Š ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) β†’ ((𝑄 ⨣ 𝑅) Β· 𝑋) = ((𝑄 Β· 𝑋) + (𝑅 Β· 𝑋)))
1312anabsan2 672 . . 3 (((π‘Š ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ 𝑋 ∈ 𝑉) β†’ ((𝑄 ⨣ 𝑅) Β· 𝑋) = ((𝑄 Β· 𝑋) + (𝑅 Β· 𝑋)))
1413exp42 436 . 2 (π‘Š ∈ LMod β†’ (𝑄 ∈ 𝐾 β†’ (𝑅 ∈ 𝐾 β†’ (𝑋 ∈ 𝑉 β†’ ((𝑄 ⨣ 𝑅) Β· 𝑋) = ((𝑄 Β· 𝑋) + (𝑅 Β· 𝑋))))))
15143imp2 1349 1 ((π‘Š ∈ LMod ∧ (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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  β€˜cfv 6543  (class class class)co 7411  Basecbs 17146  +gcplusg 17199  .rcmulr 17200  Scalarcsca 17202   ·𝑠 cvsca 17203  1rcur 20006  LModclmod 20475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  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 7414  df-lmod 20477
This theorem is referenced by:  lmod0vs  20510  lmodvsmmulgdi  20512  lmodvneg1  20520  lmodcom  20523  lmodsubdir  20535  islss3  20575  lss1d  20579  prdslmodd  20585  lspsolvlem  20761  frlmup1  21359  asclghm  21443  scmataddcl  22025  scmatghm  22042  pm2mpghm  22325  clmvsdir  24614  cvsi  24653  lmodvslmhm  32243  imaslmod  32509  lshpkrlem4  38069  baerlem3lem1  40664  baerlem5blem1  40666  hgmapadd  40851  mendlmod  42017  lmodvsmdi  47137  lincsum  47188  ldepsprlem  47231
  Copyright terms: Public domain W3C validator