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Theorem lflvsdi1 33383
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)
2 fvex 6113 . . . 4 (Base‘𝑊) ∈ V
31, 2eqeltri 2684 . . 3 𝑉 ∈ V
43a1i 11 . 2 (𝜑𝑉 ∈ V)
5 lfldi.x . . 3 (𝜑𝑋𝐾)
6 fconst6g 6007 . . 3 (𝑋𝐾 → (𝑉 × {𝑋}):𝑉𝐾)
75, 6syl 17 . 2 (𝜑 → (𝑉 × {𝑋}):𝑉𝐾)
8 lfldi.w . . 3 (𝜑𝑊 ∈ LMod)
9 lfldi1.g . . 3 (𝜑𝐺𝐹)
10 lfldi.r . . . 4 𝑅 = (Scalar‘𝑊)
11 lfldi.k . . . 4 𝐾 = (Base‘𝑅)
12 lfldi.f . . . 4 𝐹 = (LFnl‘𝑊)
1310, 11, 1, 12lflf 33368 . . 3 ((𝑊 ∈ LMod ∧ 𝐺𝐹) → 𝐺:𝑉𝐾)
148, 9, 13syl2anc 691 . 2 (𝜑𝐺:𝑉𝐾)
15 lfldi1.h . . 3 (𝜑𝐻𝐹)
1610, 11, 1, 12lflf 33368 . . 3 ((𝑊 ∈ LMod ∧ 𝐻𝐹) → 𝐻:𝑉𝐾)
178, 15, 16syl2anc 691 . 2 (𝜑𝐻:𝑉𝐾)
1810lmodring 18694 . . . 4 (𝑊 ∈ LMod → 𝑅 ∈ Ring)
198, 18syl 17 . . 3 (𝜑𝑅 ∈ Ring)
20 lfldi.p . . . 4 + = (+g𝑅)
21 lfldi.t . . . 4 · = (.r𝑅)
2211, 20, 21ringdir 18390 . . 3 ((𝑅 ∈ Ring ∧ (𝑥𝐾𝑦𝐾𝑧𝐾)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
2319, 22sylan 487 . 2 ((𝜑 ∧ (𝑥𝐾𝑦𝐾𝑧𝐾)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
244, 7, 14, 17, 23caofdir 6832 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 1031   = wceq 1475  wcel 1977  Vcvv 3173  {csn 4125   × cxp 5036  wf 5800  cfv 5804  (class class class)co 6549  𝑓 cof 6793  Basecbs 15695  +gcplusg 15768  .rcmulr 15769  Scalarcsca 15771  Ringcrg 18370  LModclmod 18686  LFnlclfn 33362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-map 7746  df-ring 18372  df-lmod 18688  df-lfl 33363
This theorem is referenced by:  ldualvsdi1  33448
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