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Theorem lflvsdi1 35216
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 6460 . . 3 𝑉 ∈ V
32a1i 11 . 2 (𝜑𝑉 ∈ V)
4 lfldi.x . . 3 (𝜑𝑋𝐾)
5 fconst6g 6344 . . 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 35201 . . 3 ((𝑊 ∈ LMod ∧ 𝐺𝐹) → 𝐺:𝑉𝐾)
137, 8, 12syl2anc 579 . 2 (𝜑𝐺:𝑉𝐾)
14 lfldi1.h . . 3 (𝜑𝐻𝐹)
159, 10, 1, 11lflf 35201 . . 3 ((𝑊 ∈ LMod ∧ 𝐻𝐹) → 𝐻:𝑉𝐾)
167, 14, 15syl2anc 579 . 2 (𝜑𝐻:𝑉𝐾)
179lmodring 19263 . . . 4 (𝑊 ∈ LMod → 𝑅 ∈ Ring)
187, 17syl 17 . . 3 (𝜑𝑅 ∈ Ring)
19 lfldi.p . . . 4 + = (+g𝑅)
20 lfldi.t . . . 4 · = (.r𝑅)
2110, 19, 20ringdir 18954 . . 3 ((𝑅 ∈ Ring ∧ (𝑥𝐾𝑦𝐾𝑧𝐾)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
2218, 21sylan 575 . 2 ((𝜑 ∧ (𝑥𝐾𝑦𝐾𝑧𝐾)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
233, 6, 13, 16, 22caofdir 7211 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 1071   = wceq 1601  wcel 2106  Vcvv 3397  {csn 4397   × cxp 5353  wf 6131  cfv 6135  (class class class)co 6922  𝑓 cof 7172  Basecbs 16255  +gcplusg 16338  .rcmulr 16339  Scalarcsca 16341  Ringcrg 18934  LModclmod 19255  LFnlclfn 35195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-of 7174  df-map 8142  df-ring 18936  df-lmod 19257  df-lfl 35196
This theorem is referenced by:  ldualvsdi1  35281
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