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Theorem lflvsdi1 39060
Description: 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)
lfldi1.g (𝜑𝐺𝐹)
lfldi1.h (𝜑𝐻𝐹)
Assertion
Ref Expression
lflvsdi1 (𝜑 → ((𝐺f + 𝐻) ∘f · (𝑉 × {𝑋})) = ((𝐺f · (𝑉 × {𝑋})) ∘f + (𝐻f · (𝑉 × {𝑋}))))

Proof of Theorem lflvsdi1
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lfldi.v . . . 4 𝑉 = (Base‘𝑊)
21fvexi 6921 . . 3 𝑉 ∈ V
32a1i 11 . 2 (𝜑𝑉 ∈ V)
4 lfldi.x . . 3 (𝜑𝑋𝐾)
5 fconst6g 6798 . . 3 (𝑋𝐾 → (𝑉 × {𝑋}):𝑉𝐾)
64, 5syl 17 . 2 (𝜑 → (𝑉 × {𝑋}):𝑉𝐾)
7 lfldi.w . . 3 (𝜑𝑊 ∈ LMod)
8 lfldi1.g . . 3 (𝜑𝐺𝐹)
9 lfldi.r . . . 4 𝑅 = (Scalar‘𝑊)
10 lfldi.k . . . 4 𝐾 = (Base‘𝑅)
11 lfldi.f . . . 4 𝐹 = (LFnl‘𝑊)
129, 10, 1, 11lflf 39045 . . 3 ((𝑊 ∈ LMod ∧ 𝐺𝐹) → 𝐺:𝑉𝐾)
137, 8, 12syl2anc 584 . 2 (𝜑𝐺:𝑉𝐾)
14 lfldi1.h . . 3 (𝜑𝐻𝐹)
159, 10, 1, 11lflf 39045 . . 3 ((𝑊 ∈ LMod ∧ 𝐻𝐹) → 𝐻:𝑉𝐾)
167, 14, 15syl2anc 584 . 2 (𝜑𝐻:𝑉𝐾)
179lmodring 20883 . . . 4 (𝑊 ∈ LMod → 𝑅 ∈ Ring)
187, 17syl 17 . . 3 (𝜑𝑅 ∈ Ring)
19 lfldi.p . . . 4 + = (+g𝑅)
20 lfldi.t . . . 4 · = (.r𝑅)
2110, 19, 20ringdir 20279 . . 3 ((𝑅 ∈ Ring ∧ (𝑥𝐾𝑦𝐾𝑧𝐾)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
2218, 21sylan 580 . 2 ((𝜑 ∧ (𝑥𝐾𝑦𝐾𝑧𝐾)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
233, 6, 13, 16, 22caofdir 7739 1 (𝜑 → ((𝐺f + 𝐻) ∘f · (𝑉 × {𝑋})) = ((𝐺f · (𝑉 × {𝑋})) ∘f + (𝐻f · (𝑉 × {𝑋}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  wcel 2106  Vcvv 3478  {csn 4631   × cxp 5687  wf 6559  cfv 6563  (class class class)co 7431  f cof 7695  Basecbs 17245  +gcplusg 17298  .rcmulr 17299  Scalarcsca 17301  Ringcrg 20251  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:  ldualvsdi1  39125
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