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Theorem lflvsdi2a 39062
Description: Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 21-Oct-2014.)
Hypotheses
Ref Expression
lfldi.v 𝑉 = (Base‘𝑊)
lfldi.r 𝑅 = (Scalar‘𝑊)
lfldi.k 𝐾 = (Base‘𝑅)
lfldi.p + = (+g𝑅)
lfldi.t · = (.r𝑅)
lfldi.f 𝐹 = (LFnl‘𝑊)
lfldi.w (𝜑𝑊 ∈ LMod)
lfldi.x (𝜑𝑋𝐾)
lfldi2.y (𝜑𝑌𝐾)
lfldi2.g (𝜑𝐺𝐹)
Assertion
Ref Expression
lflvsdi2a (𝜑 → (𝐺f · (𝑉 × {(𝑋 + 𝑌)})) = ((𝐺f · (𝑉 × {𝑋})) ∘f + (𝐺f · (𝑉 × {𝑌}))))

Proof of Theorem lflvsdi2a
StepHypRef Expression
1 lfldi.v . . . . . 6 𝑉 = (Base‘𝑊)
21fvexi 6921 . . . . 5 𝑉 ∈ V
32a1i 11 . . . 4 (𝜑𝑉 ∈ V)
4 lfldi.x . . . 4 (𝜑𝑋𝐾)
5 lfldi2.y . . . 4 (𝜑𝑌𝐾)
63, 4, 5ofc12 7727 . . 3 (𝜑 → ((𝑉 × {𝑋}) ∘f + (𝑉 × {𝑌})) = (𝑉 × {(𝑋 + 𝑌)}))
76oveq2d 7447 . 2 (𝜑 → (𝐺f · ((𝑉 × {𝑋}) ∘f + (𝑉 × {𝑌}))) = (𝐺f · (𝑉 × {(𝑋 + 𝑌)})))
8 lfldi.r . . 3 𝑅 = (Scalar‘𝑊)
9 lfldi.k . . 3 𝐾 = (Base‘𝑅)
10 lfldi.p . . 3 + = (+g𝑅)
11 lfldi.t . . 3 · = (.r𝑅)
12 lfldi.f . . 3 𝐹 = (LFnl‘𝑊)
13 lfldi.w . . 3 (𝜑𝑊 ∈ LMod)
14 lfldi2.g . . 3 (𝜑𝐺𝐹)
151, 8, 9, 10, 11, 12, 13, 4, 5, 14lflvsdi2 39061 . 2 (𝜑 → (𝐺f · ((𝑉 × {𝑋}) ∘f + (𝑉 × {𝑌}))) = ((𝐺f · (𝑉 × {𝑋})) ∘f + (𝐺f · (𝑉 × {𝑌}))))
167, 15eqtr3d 2777 1 (𝜑 → (𝐺f · (𝑉 × {(𝑋 + 𝑌)})) = ((𝐺f · (𝑉 × {𝑋})) ∘f + (𝐺f · (𝑉 × {𝑌}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  {csn 4631   × cxp 5687  cfv 6563  (class class class)co 7431  f cof 7695  Basecbs 17245  +gcplusg 17298  .rcmulr 17299  Scalarcsca 17301  LModclmod 20875  LFnlclfn 39039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-map 8867  df-ring 20253  df-lmod 20877  df-lfl 39040
This theorem is referenced by:  ldualvsdi2  39126
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