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Theorem lflvsdi1 38550
Description: Distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
Hypotheses
Ref Expression
lfldi.v 𝑉 = (Baseβ€˜π‘Š)
lfldi.r 𝑅 = (Scalarβ€˜π‘Š)
lfldi.k 𝐾 = (Baseβ€˜π‘…)
lfldi.p + = (+gβ€˜π‘…)
lfldi.t Β· = (.rβ€˜π‘…)
lfldi.f 𝐹 = (LFnlβ€˜π‘Š)
lfldi.w (πœ‘ β†’ π‘Š ∈ LMod)
lfldi.x (πœ‘ β†’ 𝑋 ∈ 𝐾)
lfldi1.g (πœ‘ β†’ 𝐺 ∈ 𝐹)
lfldi1.h (πœ‘ β†’ 𝐻 ∈ 𝐹)
Assertion
Ref Expression
lflvsdi1 (πœ‘ β†’ ((𝐺 ∘f + 𝐻) ∘f Β· (𝑉 Γ— {𝑋})) = ((𝐺 ∘f Β· (𝑉 Γ— {𝑋})) ∘f + (𝐻 ∘f Β· (𝑉 Γ— {𝑋}))))

Proof of Theorem lflvsdi1
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lfldi.v . . . 4 𝑉 = (Baseβ€˜π‘Š)
21fvexi 6911 . . 3 𝑉 ∈ V
32a1i 11 . 2 (πœ‘ β†’ 𝑉 ∈ V)
4 lfldi.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝐾)
5 fconst6g 6786 . . 3 (𝑋 ∈ 𝐾 β†’ (𝑉 Γ— {𝑋}):π‘‰βŸΆπΎ)
64, 5syl 17 . 2 (πœ‘ β†’ (𝑉 Γ— {𝑋}):π‘‰βŸΆπΎ)
7 lfldi.w . . 3 (πœ‘ β†’ π‘Š ∈ LMod)
8 lfldi1.g . . 3 (πœ‘ β†’ 𝐺 ∈ 𝐹)
9 lfldi.r . . . 4 𝑅 = (Scalarβ€˜π‘Š)
10 lfldi.k . . . 4 𝐾 = (Baseβ€˜π‘…)
11 lfldi.f . . . 4 𝐹 = (LFnlβ€˜π‘Š)
129, 10, 1, 11lflf 38535 . . 3 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) β†’ 𝐺:π‘‰βŸΆπΎ)
137, 8, 12syl2anc 583 . 2 (πœ‘ β†’ 𝐺:π‘‰βŸΆπΎ)
14 lfldi1.h . . 3 (πœ‘ β†’ 𝐻 ∈ 𝐹)
159, 10, 1, 11lflf 38535 . . 3 ((π‘Š ∈ LMod ∧ 𝐻 ∈ 𝐹) β†’ 𝐻:π‘‰βŸΆπΎ)
167, 14, 15syl2anc 583 . 2 (πœ‘ β†’ 𝐻:π‘‰βŸΆπΎ)
179lmodring 20750 . . . 4 (π‘Š ∈ LMod β†’ 𝑅 ∈ Ring)
187, 17syl 17 . . 3 (πœ‘ β†’ 𝑅 ∈ Ring)
19 lfldi.p . . . 4 + = (+gβ€˜π‘…)
20 lfldi.t . . . 4 Β· = (.rβ€˜π‘…)
2110, 19, 20ringdir 20200 . . 3 ((𝑅 ∈ Ring ∧ (π‘₯ ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) β†’ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧)))
2218, 21sylan 579 . 2 ((πœ‘ ∧ (π‘₯ ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) β†’ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧)))
233, 6, 13, 16, 22caofdir 7725 1 (πœ‘ β†’ ((𝐺 ∘f + 𝐻) ∘f Β· (𝑉 Γ— {𝑋})) = ((𝐺 ∘f Β· (𝑉 Γ— {𝑋})) ∘f + (𝐻 ∘f Β· (𝑉 Γ— {𝑋}))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  Vcvv 3471  {csn 4629   Γ— cxp 5676  βŸΆwf 6544  β€˜cfv 6548  (class class class)co 7420   ∘f cof 7683  Basecbs 17179  +gcplusg 17232  .rcmulr 17233  Scalarcsca 17235  Ringcrg 20172  LModclmod 20742  LFnlclfn 38529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7685  df-map 8846  df-ring 20174  df-lmod 20744  df-lfl 38530
This theorem is referenced by:  ldualvsdi1  38615
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