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Theorem lflvsdi1 36332
 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 6666 . . 3 𝑉 ∈ V
32a1i 11 . 2 (𝜑𝑉 ∈ V)
4 lfldi.x . . 3 (𝜑𝑋𝐾)
5 fconst6g 6549 . . 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 36317 . . 3 ((𝑊 ∈ LMod ∧ 𝐺𝐹) → 𝐺:𝑉𝐾)
137, 8, 12syl2anc 587 . 2 (𝜑𝐺:𝑉𝐾)
14 lfldi1.h . . 3 (𝜑𝐻𝐹)
159, 10, 1, 11lflf 36317 . . 3 ((𝑊 ∈ LMod ∧ 𝐻𝐹) → 𝐻:𝑉𝐾)
167, 14, 15syl2anc 587 . 2 (𝜑𝐻:𝑉𝐾)
179lmodring 19633 . . . 4 (𝑊 ∈ LMod → 𝑅 ∈ Ring)
187, 17syl 17 . . 3 (𝜑𝑅 ∈ Ring)
19 lfldi.p . . . 4 + = (+g𝑅)
20 lfldi.t . . . 4 · = (.r𝑅)
2110, 19, 20ringdir 19311 . . 3 ((𝑅 ∈ Ring ∧ (𝑥𝐾𝑦𝐾𝑧𝐾)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
2218, 21sylan 583 . 2 ((𝜑 ∧ (𝑥𝐾𝑦𝐾𝑧𝐾)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
233, 6, 13, 16, 22caofdir 7431 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 1084   = wceq 1538   ∈ wcel 2114  Vcvv 3469  {csn 4539   × cxp 5530  ⟶wf 6330  ‘cfv 6334  (class class class)co 7140   ∘f cof 7392  Basecbs 16474  +gcplusg 16556  .rcmulr 16557  Scalarcsca 16559  Ringcrg 19288  LModclmod 19625  LFnlclfn 36311 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446 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 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-of 7394  df-map 8395  df-ring 19290  df-lmod 19627  df-lfl 36312 This theorem is referenced by:  ldualvsdi1  36397
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