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Theorem lflvsdi2 33881
Description: Reverse 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)
lfldi2.y (𝜑𝑌𝐾)
lfldi2.g (𝜑𝐺𝐹)
Assertion
Ref Expression
lflvsdi2 (𝜑 → (𝐺𝑓 · ((𝑉 × {𝑋}) ∘𝑓 + (𝑉 × {𝑌}))) = ((𝐺𝑓 · (𝑉 × {𝑋})) ∘𝑓 + (𝐺𝑓 · (𝑉 × {𝑌}))))

Proof of Theorem lflvsdi2
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 6163 . . . 4 (Base‘𝑊) ∈ V
31, 2eqeltri 2694 . . 3 𝑉 ∈ V
43a1i 11 . 2 (𝜑𝑉 ∈ V)
5 lfldi.w . . 3 (𝜑𝑊 ∈ LMod)
6 lfldi2.g . . 3 (𝜑𝐺𝐹)
7 lfldi.r . . . 4 𝑅 = (Scalar‘𝑊)
8 lfldi.k . . . 4 𝐾 = (Base‘𝑅)
9 lfldi.f . . . 4 𝐹 = (LFnl‘𝑊)
107, 8, 1, 9lflf 33865 . . 3 ((𝑊 ∈ LMod ∧ 𝐺𝐹) → 𝐺:𝑉𝐾)
115, 6, 10syl2anc 692 . 2 (𝜑𝐺:𝑉𝐾)
12 lfldi.x . . 3 (𝜑𝑋𝐾)
13 fconst6g 6056 . . 3 (𝑋𝐾 → (𝑉 × {𝑋}):𝑉𝐾)
1412, 13syl 17 . 2 (𝜑 → (𝑉 × {𝑋}):𝑉𝐾)
15 lfldi2.y . . 3 (𝜑𝑌𝐾)
16 fconst6g 6056 . . 3 (𝑌𝐾 → (𝑉 × {𝑌}):𝑉𝐾)
1715, 16syl 17 . 2 (𝜑 → (𝑉 × {𝑌}):𝑉𝐾)
187lmodring 18803 . . . 4 (𝑊 ∈ LMod → 𝑅 ∈ Ring)
195, 18syl 17 . . 3 (𝜑𝑅 ∈ Ring)
20 lfldi.p . . . 4 + = (+g𝑅)
21 lfldi.t . . . 4 · = (.r𝑅)
228, 20, 21ringdi 18498 . . 3 ((𝑅 ∈ Ring ∧ (𝑥𝐾𝑦𝐾𝑧𝐾)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
2319, 22sylan 488 . 2 ((𝜑 ∧ (𝑥𝐾𝑦𝐾𝑧𝐾)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
244, 11, 14, 17, 23caofdi 6893 1 (𝜑 → (𝐺𝑓 · ((𝑉 × {𝑋}) ∘𝑓 + (𝑉 × {𝑌}))) = ((𝐺𝑓 · (𝑉 × {𝑋})) ∘𝑓 + (𝐺𝑓 · (𝑉 × {𝑌}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3189  {csn 4153   × cxp 5077  wf 5848  cfv 5852  (class class class)co 6610  𝑓 cof 6855  Basecbs 15792  +gcplusg 15873  .rcmulr 15874  Scalarcsca 15876  Ringcrg 18479  LModclmod 18795  LFnlclfn 33859
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 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
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 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-map 7811  df-ring 18481  df-lmod 18797  df-lfl 33860
This theorem is referenced by:  lflvsdi2a  33882
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