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Theorem lflvsdi2a 36322
 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 6676 . . . . 5 𝑉 ∈ V
32a1i 11 . . . 4 (𝜑𝑉 ∈ V)
4 lfldi.x . . . 4 (𝜑𝑋𝐾)
5 lfldi2.y . . . 4 (𝜑𝑌𝐾)
63, 4, 5ofc12 7429 . . 3 (𝜑 → ((𝑉 × {𝑋}) ∘f + (𝑉 × {𝑌})) = (𝑉 × {(𝑋 + 𝑌)}))
76oveq2d 7166 . 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 36321 . 2 (𝜑 → (𝐺f · ((𝑉 × {𝑋}) ∘f + (𝑉 × {𝑌}))) = ((𝐺f · (𝑉 × {𝑋})) ∘f + (𝐺f · (𝑉 × {𝑌}))))
167, 15eqtr3d 2861 1 (𝜑 → (𝐺f · (𝑉 × {(𝑋 + 𝑌)})) = ((𝐺f · (𝑉 × {𝑋})) ∘f + (𝐺f · (𝑉 × {𝑌}))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115  Vcvv 3481  {csn 4551   × cxp 5541  ‘cfv 6344  (class class class)co 7150   ∘f cof 7402  Basecbs 16486  +gcplusg 16568  .rcmulr 16569  Scalarcsca 16571  LModclmod 19637  LFnlclfn 36299 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7404  df-map 8405  df-ring 19302  df-lmod 19639  df-lfl 36300 This theorem is referenced by:  ldualvsdi2  36386
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