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Theorem lflvsdi1 33845
 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 6158 . . . 4 (Base‘𝑊) ∈ V
31, 2eqeltri 2694 . . 3 𝑉 ∈ V
43a1i 11 . 2 (𝜑𝑉 ∈ V)
5 lfldi.x . . 3 (𝜑𝑋𝐾)
6 fconst6g 6051 . . 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 33830 . . 3 ((𝑊 ∈ LMod ∧ 𝐺𝐹) → 𝐺:𝑉𝐾)
148, 9, 13syl2anc 692 . 2 (𝜑𝐺:𝑉𝐾)
15 lfldi1.h . . 3 (𝜑𝐻𝐹)
1610, 11, 1, 12lflf 33830 . . 3 ((𝑊 ∈ LMod ∧ 𝐻𝐹) → 𝐻:𝑉𝐾)
178, 15, 16syl2anc 692 . 2 (𝜑𝐻:𝑉𝐾)
1810lmodring 18792 . . . 4 (𝑊 ∈ LMod → 𝑅 ∈ Ring)
198, 18syl 17 . . 3 (𝜑𝑅 ∈ Ring)
20 lfldi.p . . . 4 + = (+g𝑅)
21 lfldi.t . . . 4 · = (.r𝑅)
2211, 20, 21ringdir 18488 . . 3 ((𝑅 ∈ Ring ∧ (𝑥𝐾𝑦𝐾𝑧𝐾)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
2319, 22sylan 488 . 2 ((𝜑 ∧ (𝑥𝐾𝑦𝐾𝑧𝐾)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
244, 7, 14, 17, 23caofdir 6887 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 1036   = wceq 1480   ∈ wcel 1987  Vcvv 3186  {csn 4148   × cxp 5072  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604   ∘𝑓 cof 6848  Basecbs 15781  +gcplusg 15862  .rcmulr 15863  Scalarcsca 15865  Ringcrg 18468  LModclmod 18784  LFnlclfn 33824 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-map 7804  df-ring 18470  df-lmod 18786  df-lfl 33825 This theorem is referenced by:  ldualvsdi1  33910
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