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Theorem lmodvsdi 20731
Description: Distributive law for scalar product (left-distributivity). (ax-hvdistr1 30770 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 2726 . . . . . . . . 9 (+gβ€˜πΉ) = (+gβ€˜πΉ)
7 eqid 2726 . . . . . . . . 9 (.rβ€˜πΉ) = (.rβ€˜πΉ)
8 eqid 2726 . . . . . . . . 9 (1rβ€˜πΉ) = (1rβ€˜πΉ)
91, 2, 3, 4, 5, 6, 7, 8lmodlema 20711 . . . . . . . 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 494 . . . . . . 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 671 . . . 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 430 . . 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  +gcplusg 17206  .rcmulr 17207  Scalarcsca 17209   ·𝑠 cvsca 17210  1rcur 20086  LModclmod 20706
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 2697  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6489  df-fv 6545  df-ov 7408  df-lmod 20708
This theorem is referenced by:  lmodcom  20754  lmodsubdi  20765  lmodvsghm  20769  islss3  20806  prdslmodd  20816  lmodvsinv2  20885  lmhmplusg  20892  lsmcl  20931  pj1lmhm  20948  lspfixed  20979  lspsolvlem  20993  clmvsdi  24974  cvsi  25012  eqgvscpbl  32968  imaslmod  32971  lshpkrlem4  38496  baerlem5alem1  41092  baerlem5blem1  41093  hdmap14lem8  41259  mendlmod  42513  lmodvsmdi  47334
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