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Theorem lflvsdi2a 34186
 Description: Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 21-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
lflvsdi2a (𝜑 → (𝐺𝑓 · (𝑉 × {(𝑋 + 𝑌)})) = ((𝐺𝑓 · (𝑉 × {𝑋})) ∘𝑓 + (𝐺𝑓 · (𝑉 × {𝑌}))))

Proof of Theorem lflvsdi2a
StepHypRef Expression
1 lfldi.v . . . . . 6 𝑉 = (Base‘𝑊)
2 fvex 6188 . . . . . 6 (Base‘𝑊) ∈ V
31, 2eqeltri 2695 . . . . 5 𝑉 ∈ V
43a1i 11 . . . 4 (𝜑𝑉 ∈ V)
5 lfldi.x . . . 4 (𝜑𝑋𝐾)
6 lfldi2.y . . . 4 (𝜑𝑌𝐾)
74, 5, 6ofc12 6907 . . 3 (𝜑 → ((𝑉 × {𝑋}) ∘𝑓 + (𝑉 × {𝑌})) = (𝑉 × {(𝑋 + 𝑌)}))
87oveq2d 6651 . 2 (𝜑 → (𝐺𝑓 · ((𝑉 × {𝑋}) ∘𝑓 + (𝑉 × {𝑌}))) = (𝐺𝑓 · (𝑉 × {(𝑋 + 𝑌)})))
9 lfldi.r . . 3 𝑅 = (Scalar‘𝑊)
10 lfldi.k . . 3 𝐾 = (Base‘𝑅)
11 lfldi.p . . 3 + = (+g𝑅)
12 lfldi.t . . 3 · = (.r𝑅)
13 lfldi.f . . 3 𝐹 = (LFnl‘𝑊)
14 lfldi.w . . 3 (𝜑𝑊 ∈ LMod)
15 lfldi2.g . . 3 (𝜑𝐺𝐹)
161, 9, 10, 11, 12, 13, 14, 5, 6, 15lflvsdi2 34185 . 2 (𝜑 → (𝐺𝑓 · ((𝑉 × {𝑋}) ∘𝑓 + (𝑉 × {𝑌}))) = ((𝐺𝑓 · (𝑉 × {𝑋})) ∘𝑓 + (𝐺𝑓 · (𝑉 × {𝑌}))))
178, 16eqtr3d 2656 1 (𝜑 → (𝐺𝑓 · (𝑉 × {(𝑋 + 𝑌)})) = ((𝐺𝑓 · (𝑉 × {𝑋})) ∘𝑓 + (𝐺𝑓 · (𝑉 × {𝑌}))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1481   ∈ wcel 1988  Vcvv 3195  {csn 4168   × cxp 5102  ‘cfv 5876  (class class class)co 6635   ∘𝑓 cof 6880  Basecbs 15838  +gcplusg 15922  .rcmulr 15923  Scalarcsca 15925  LModclmod 18844  LFnlclfn 34163 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-of 6882  df-map 7844  df-ring 18530  df-lmod 18846  df-lfl 34164 This theorem is referenced by:  ldualvsdi2  34250
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