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Theorem lflvsdi2 36324
Description: Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-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
lflvsdi2 (𝜑 → (𝐺f · ((𝑉 × {𝑋}) ∘f + (𝑉 × {𝑌}))) = ((𝐺f · (𝑉 × {𝑋})) ∘f + (𝐺f · (𝑉 × {𝑌}))))

Proof of Theorem lflvsdi2
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lfldi.v . . . 4 𝑉 = (Base‘𝑊)
21fvexi 6675 . . 3 𝑉 ∈ V
32a1i 11 . 2 (𝜑𝑉 ∈ V)
4 lfldi.w . . 3 (𝜑𝑊 ∈ LMod)
5 lfldi2.g . . 3 (𝜑𝐺𝐹)
6 lfldi.r . . . 4 𝑅 = (Scalar‘𝑊)
7 lfldi.k . . . 4 𝐾 = (Base‘𝑅)
8 lfldi.f . . . 4 𝐹 = (LFnl‘𝑊)
96, 7, 1, 8lflf 36308 . . 3 ((𝑊 ∈ LMod ∧ 𝐺𝐹) → 𝐺:𝑉𝐾)
104, 5, 9syl2anc 587 . 2 (𝜑𝐺:𝑉𝐾)
11 lfldi.x . . 3 (𝜑𝑋𝐾)
12 fconst6g 6558 . . 3 (𝑋𝐾 → (𝑉 × {𝑋}):𝑉𝐾)
1311, 12syl 17 . 2 (𝜑 → (𝑉 × {𝑋}):𝑉𝐾)
14 lfldi2.y . . 3 (𝜑𝑌𝐾)
15 fconst6g 6558 . . 3 (𝑌𝐾 → (𝑉 × {𝑌}):𝑉𝐾)
1614, 15syl 17 . 2 (𝜑 → (𝑉 × {𝑌}):𝑉𝐾)
176lmodring 19642 . . . 4 (𝑊 ∈ LMod → 𝑅 ∈ Ring)
184, 17syl 17 . . 3 (𝜑𝑅 ∈ Ring)
19 lfldi.p . . . 4 + = (+g𝑅)
20 lfldi.t . . . 4 · = (.r𝑅)
217, 19, 20ringdi 19319 . . 3 ((𝑅 ∈ Ring ∧ (𝑥𝐾𝑦𝐾𝑧𝐾)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
2218, 21sylan 583 . 2 ((𝜑 ∧ (𝑥𝐾𝑦𝐾𝑧𝐾)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
233, 10, 13, 16, 22caofdi 7439 1 (𝜑 → (𝐺f · ((𝑉 × {𝑋}) ∘f + (𝑉 × {𝑌}))) = ((𝐺f · (𝑉 × {𝑋})) ∘f + (𝐺f · (𝑉 × {𝑌}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2115  Vcvv 3480  {csn 4550   × cxp 5540  wf 6339  cfv 6343  (class class class)co 7149  f cof 7401  Basecbs 16483  +gcplusg 16565  .rcmulr 16566  Scalarcsca 16568  Ringcrg 19297  LModclmod 19634  LFnlclfn 36302
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 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-of 7403  df-map 8404  df-ring 19299  df-lmod 19636  df-lfl 36303
This theorem is referenced by:  lflvsdi2a  36325
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