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Theorem lflvsdi2 35154
 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)
21fvexi 6447 . . 3 𝑉 ∈ V
32a1i 11 . 2 (𝜑𝑉 ∈ V)
4 lfldi.w . . 3 (𝜑𝑊 ∈ LMod)
5 lfldi2.g . . 3 (𝜑𝐺𝐹)
6 lfldi.r . . . 4 𝑅 = (Scalar‘𝑊)
7 lfldi.k . . . 4 𝐾 = (Base‘𝑅)
8 lfldi.f . . . 4 𝐹 = (LFnl‘𝑊)
96, 7, 1, 8lflf 35138 . . 3 ((𝑊 ∈ LMod ∧ 𝐺𝐹) → 𝐺:𝑉𝐾)
104, 5, 9syl2anc 581 . 2 (𝜑𝐺:𝑉𝐾)
11 lfldi.x . . 3 (𝜑𝑋𝐾)
12 fconst6g 6331 . . 3 (𝑋𝐾 → (𝑉 × {𝑋}):𝑉𝐾)
1311, 12syl 17 . 2 (𝜑 → (𝑉 × {𝑋}):𝑉𝐾)
14 lfldi2.y . . 3 (𝜑𝑌𝐾)
15 fconst6g 6331 . . 3 (𝑌𝐾 → (𝑉 × {𝑌}):𝑉𝐾)
1614, 15syl 17 . 2 (𝜑 → (𝑉 × {𝑌}):𝑉𝐾)
176lmodring 19227 . . . 4 (𝑊 ∈ LMod → 𝑅 ∈ Ring)
184, 17syl 17 . . 3 (𝜑𝑅 ∈ Ring)
19 lfldi.p . . . 4 + = (+g𝑅)
20 lfldi.t . . . 4 · = (.r𝑅)
217, 19, 20ringdi 18920 . . 3 ((𝑅 ∈ Ring ∧ (𝑥𝐾𝑦𝐾𝑧𝐾)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
2218, 21sylan 577 . 2 ((𝜑 ∧ (𝑥𝐾𝑦𝐾𝑧𝐾)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
233, 10, 13, 16, 22caofdi 7193 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 1113   = wceq 1658   ∈ wcel 2166  Vcvv 3414  {csn 4397   × cxp 5340  ⟶wf 6119  ‘cfv 6123  (class class class)co 6905   ∘𝑓 cof 7155  Basecbs 16222  +gcplusg 16305  .rcmulr 16306  Scalarcsca 16308  Ringcrg 18901  LModclmod 19219  LFnlclfn 35132 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-of 7157  df-map 8124  df-ring 18903  df-lmod 19221  df-lfl 35133 This theorem is referenced by:  lflvsdi2a  35155
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