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Theorem lmodvsdi 20639
Description: Distributive law for scalar product (left-distributivity). (ax-hvdistr1 30528 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 2730 . . . . . . . . 9 (+gβ€˜πΉ) = (+gβ€˜πΉ)
7 eqid 2730 . . . . . . . . 9 (.rβ€˜πΉ) = (.rβ€˜πΉ)
8 eqid 2730 . . . . . . . . 9 (1rβ€˜πΉ) = (1rβ€˜πΉ)
91, 2, 3, 4, 5, 6, 7, 8lmodlema 20619 . . . . . . . 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 1141 . . . . . 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 1119 . . . . 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 670 . . . 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 1347 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 1085   = wceq 1539   ∈ wcel 2104  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148  +gcplusg 17201  .rcmulr 17202  Scalarcsca 17204   ·𝑠 cvsca 17205  1rcur 20075  LModclmod 20614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-nul 5305
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6494  df-fv 6550  df-ov 7414  df-lmod 20616
This theorem is referenced by:  lmodcom  20662  lmodsubdi  20673  lmodvsghm  20677  islss3  20714  prdslmodd  20724  lmodvsinv2  20792  lmhmplusg  20799  lsmcl  20838  pj1lmhm  20855  lspfixed  20886  lspsolvlem  20900  clmvsdi  24839  cvsi  24877  eqgvscpbl  32735  imaslmod  32738  lshpkrlem4  38286  baerlem5alem1  40882  baerlem5blem1  40883  hdmap14lem8  41049  mendlmod  42237  lmodvsmdi  47146
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