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Theorem lmodvsdi 20782
Description: Distributive law for scalar product (left-distributivity). (ax-hvdistr1 30846 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
lmodvsdi.v 𝑉 = (Baseβ€˜π‘Š)
lmodvsdi.a + = (+gβ€˜π‘Š)
lmodvsdi.f 𝐹 = (Scalarβ€˜π‘Š)
lmodvsdi.s Β· = ( ·𝑠 β€˜π‘Š)
lmodvsdi.k 𝐾 = (Baseβ€˜πΉ)
Assertion
Ref Expression
lmodvsdi ((π‘Š ∈ LMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ (𝑅 Β· (𝑋 + π‘Œ)) = ((𝑅 Β· 𝑋) + (𝑅 Β· π‘Œ)))
Proof of Theorem lmodvsdi
StepHypRef Expression
1 lmodvsdi.v . . . . . . . . 9 𝑉 = (Baseβ€˜π‘Š)
2 lmodvsdi.a . . . . . . . . 9 + = (+gβ€˜π‘Š)
3 lmodvsdi.s . . . . . . . . 9 Β· = ( ·𝑠 β€˜π‘Š)
4 lmodvsdi.f . . . . . . . . 9 𝐹 = (Scalarβ€˜π‘Š)
5 lmodvsdi.k . . . . . . . . 9 𝐾 = (Baseβ€˜πΉ)
6 eqid 2728 . . . . . . . . 9 (+gβ€˜πΉ) = (+gβ€˜πΉ)
7 eqid 2728 . . . . . . . . 9 (.rβ€˜πΉ) = (.rβ€˜πΉ)
8 eqid 2728 . . . . . . . . 9 (1rβ€˜πΉ) = (1rβ€˜πΉ)
91, 2, 3, 4, 5, 6, 7, 8lmodlema 20762 . . . . . . . 8 ((π‘Š ∈ LMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (π‘Œ ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) β†’ (((𝑅 Β· 𝑋) ∈ 𝑉 ∧ (𝑅 Β· (𝑋 + π‘Œ)) = ((𝑅 Β· 𝑋) + (𝑅 Β· π‘Œ)) ∧ ((𝑅(+gβ€˜πΉ)𝑅) Β· 𝑋) = ((𝑅 Β· 𝑋) + (𝑅 Β· 𝑋))) ∧ (((𝑅(.rβ€˜πΉ)𝑅) Β· 𝑋) = (𝑅 Β· (𝑅 Β· 𝑋)) ∧ ((1rβ€˜πΉ) Β· 𝑋) = 𝑋)))
109simpld 493 . . . . . . 7 ((π‘Š ∈ LMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (π‘Œ ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) β†’ ((𝑅 Β· 𝑋) ∈ 𝑉 ∧ (𝑅 Β· (𝑋 + π‘Œ)) = ((𝑅 Β· 𝑋) + (𝑅 Β· π‘Œ)) ∧ ((𝑅(+gβ€˜πΉ)𝑅) Β· 𝑋) = ((𝑅 Β· 𝑋) + (𝑅 Β· 𝑋))))
1110simp2d 1140 . . . . . 6 ((π‘Š ∈ LMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (π‘Œ ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) β†’ (𝑅 Β· (𝑋 + π‘Œ)) = ((𝑅 Β· 𝑋) + (𝑅 Β· π‘Œ)))
12113expia 1118 . . . . 5 ((π‘Š ∈ LMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) β†’ ((π‘Œ ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) β†’ (𝑅 Β· (𝑋 + π‘Œ)) = ((𝑅 Β· 𝑋) + (𝑅 Β· π‘Œ))))
1312anabsan2 672 . . . 4 ((π‘Š ∈ LMod ∧ 𝑅 ∈ 𝐾) β†’ ((π‘Œ ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) β†’ (𝑅 Β· (𝑋 + π‘Œ)) = ((𝑅 Β· 𝑋) + (𝑅 Β· π‘Œ))))
1413exp4b 429 . . 3 (π‘Š ∈ LMod β†’ (𝑅 ∈ 𝐾 β†’ (π‘Œ ∈ 𝑉 β†’ (𝑋 ∈ 𝑉 β†’ (𝑅 Β· (𝑋 + π‘Œ)) = ((𝑅 Β· 𝑋) + (𝑅 Β· π‘Œ))))))
1514com34 91 . 2 (π‘Š ∈ LMod β†’ (𝑅 ∈ 𝐾 β†’ (𝑋 ∈ 𝑉 β†’ (π‘Œ ∈ 𝑉 β†’ (𝑅 Β· (𝑋 + π‘Œ)) = ((𝑅 Β· 𝑋) + (𝑅 Β· π‘Œ))))))
16153imp2 1346 1 ((π‘Š ∈ LMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ (𝑅 Β· (𝑋 + π‘Œ)) = ((𝑅 Β· 𝑋) + (𝑅 Β· π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  β€˜cfv 6553  (class class class)co 7426  Basecbs 17189  +gcplusg 17242  .rcmulr 17243  Scalarcsca 17245   ·𝑠 cvsca 17246  1rcur 20135  LModclmod 20757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-nul 5310
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7429  df-lmod 20759
This theorem is referenced by:  lmodcom  20805  lmodsubdi  20816  lmodvsghm  20820  islss3  20857  prdslmodd  20867  lmodvsinv2  20936  lmhmplusg  20943  lsmcl  20982  pj1lmhm  20999  lspfixed  21030  lspsolvlem  21044  clmvsdi  25047  cvsi  25085  eqgvscpbl  33094  imaslmod  33097  lshpkrlem4  38625  baerlem5alem1  41221  baerlem5blem1  41222  hdmap14lem8  41388  mendlmod  42666  lmodvsmdi  47542
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