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Theorem lflvsdi2 38452
Description: Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
Hypotheses
Ref Expression
lfldi.v 𝑉 = (Baseβ€˜π‘Š)
lfldi.r 𝑅 = (Scalarβ€˜π‘Š)
lfldi.k 𝐾 = (Baseβ€˜π‘…)
lfldi.p + = (+gβ€˜π‘…)
lfldi.t Β· = (.rβ€˜π‘…)
lfldi.f 𝐹 = (LFnlβ€˜π‘Š)
lfldi.w (πœ‘ β†’ π‘Š ∈ LMod)
lfldi.x (πœ‘ β†’ 𝑋 ∈ 𝐾)
lfldi2.y (πœ‘ β†’ π‘Œ ∈ 𝐾)
lfldi2.g (πœ‘ β†’ 𝐺 ∈ 𝐹)
Assertion
Ref Expression
lflvsdi2 (πœ‘ β†’ (𝐺 ∘f Β· ((𝑉 Γ— {𝑋}) ∘f + (𝑉 Γ— {π‘Œ}))) = ((𝐺 ∘f Β· (𝑉 Γ— {𝑋})) ∘f + (𝐺 ∘f Β· (𝑉 Γ— {π‘Œ}))))

Proof of Theorem lflvsdi2
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lfldi.v . . . 4 𝑉 = (Baseβ€˜π‘Š)
21fvexi 6896 . . 3 𝑉 ∈ V
32a1i 11 . 2 (πœ‘ β†’ 𝑉 ∈ V)
4 lfldi.w . . 3 (πœ‘ β†’ π‘Š ∈ LMod)
5 lfldi2.g . . 3 (πœ‘ β†’ 𝐺 ∈ 𝐹)
6 lfldi.r . . . 4 𝑅 = (Scalarβ€˜π‘Š)
7 lfldi.k . . . 4 𝐾 = (Baseβ€˜π‘…)
8 lfldi.f . . . 4 𝐹 = (LFnlβ€˜π‘Š)
96, 7, 1, 8lflf 38436 . . 3 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) β†’ 𝐺:π‘‰βŸΆπΎ)
104, 5, 9syl2anc 583 . 2 (πœ‘ β†’ 𝐺:π‘‰βŸΆπΎ)
11 lfldi.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝐾)
12 fconst6g 6771 . . 3 (𝑋 ∈ 𝐾 β†’ (𝑉 Γ— {𝑋}):π‘‰βŸΆπΎ)
1311, 12syl 17 . 2 (πœ‘ β†’ (𝑉 Γ— {𝑋}):π‘‰βŸΆπΎ)
14 lfldi2.y . . 3 (πœ‘ β†’ π‘Œ ∈ 𝐾)
15 fconst6g 6771 . . 3 (π‘Œ ∈ 𝐾 β†’ (𝑉 Γ— {π‘Œ}):π‘‰βŸΆπΎ)
1614, 15syl 17 . 2 (πœ‘ β†’ (𝑉 Γ— {π‘Œ}):π‘‰βŸΆπΎ)
176lmodring 20710 . . . 4 (π‘Š ∈ LMod β†’ 𝑅 ∈ Ring)
184, 17syl 17 . . 3 (πœ‘ β†’ 𝑅 ∈ Ring)
19 lfldi.p . . . 4 + = (+gβ€˜π‘…)
20 lfldi.t . . . 4 Β· = (.rβ€˜π‘…)
217, 19, 20ringdi 20159 . . 3 ((𝑅 ∈ Ring ∧ (π‘₯ ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) β†’ (π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)))
2218, 21sylan 579 . 2 ((πœ‘ ∧ (π‘₯ ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) β†’ (π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)))
233, 10, 13, 16, 22caofdi 7703 1 (πœ‘ β†’ (𝐺 ∘f Β· ((𝑉 Γ— {𝑋}) ∘f + (𝑉 Γ— {π‘Œ}))) = ((𝐺 ∘f Β· (𝑉 Γ— {𝑋})) ∘f + (𝐺 ∘f Β· (𝑉 Γ— {π‘Œ}))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3466  {csn 4621   Γ— cxp 5665  βŸΆwf 6530  β€˜cfv 6534  (class class class)co 7402   ∘f cof 7662  Basecbs 17149  +gcplusg 17202  .rcmulr 17203  Scalarcsca 17205  Ringcrg 20134  LModclmod 20702  LFnlclfn 38430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-of 7664  df-map 8819  df-ring 20136  df-lmod 20704  df-lfl 38431
This theorem is referenced by:  lflvsdi2a  38453
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