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Theorem lflvsdi1 37590
Description: 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)
lfldi1.g (πœ‘ β†’ 𝐺 ∈ 𝐹)
lfldi1.h (πœ‘ β†’ 𝐻 ∈ 𝐹)
Assertion
Ref Expression
lflvsdi1 (πœ‘ β†’ ((𝐺 ∘f + 𝐻) ∘f Β· (𝑉 Γ— {𝑋})) = ((𝐺 ∘f Β· (𝑉 Γ— {𝑋})) ∘f + (𝐻 ∘f Β· (𝑉 Γ— {𝑋}))))

Proof of Theorem lflvsdi1
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 6860 . . 3 𝑉 ∈ V
32a1i 11 . 2 (πœ‘ β†’ 𝑉 ∈ V)
4 lfldi.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝐾)
5 fconst6g 6735 . . 3 (𝑋 ∈ 𝐾 β†’ (𝑉 Γ— {𝑋}):π‘‰βŸΆπΎ)
64, 5syl 17 . 2 (πœ‘ β†’ (𝑉 Γ— {𝑋}):π‘‰βŸΆπΎ)
7 lfldi.w . . 3 (πœ‘ β†’ π‘Š ∈ LMod)
8 lfldi1.g . . 3 (πœ‘ β†’ 𝐺 ∈ 𝐹)
9 lfldi.r . . . 4 𝑅 = (Scalarβ€˜π‘Š)
10 lfldi.k . . . 4 𝐾 = (Baseβ€˜π‘…)
11 lfldi.f . . . 4 𝐹 = (LFnlβ€˜π‘Š)
129, 10, 1, 11lflf 37575 . . 3 ((π‘Š ∈ LMod ∧ 𝐺 ∈ 𝐹) β†’ 𝐺:π‘‰βŸΆπΎ)
137, 8, 12syl2anc 585 . 2 (πœ‘ β†’ 𝐺:π‘‰βŸΆπΎ)
14 lfldi1.h . . 3 (πœ‘ β†’ 𝐻 ∈ 𝐹)
159, 10, 1, 11lflf 37575 . . 3 ((π‘Š ∈ LMod ∧ 𝐻 ∈ 𝐹) β†’ 𝐻:π‘‰βŸΆπΎ)
167, 14, 15syl2anc 585 . 2 (πœ‘ β†’ 𝐻:π‘‰βŸΆπΎ)
179lmodring 20373 . . . 4 (π‘Š ∈ LMod β†’ 𝑅 ∈ Ring)
187, 17syl 17 . . 3 (πœ‘ β†’ 𝑅 ∈ Ring)
19 lfldi.p . . . 4 + = (+gβ€˜π‘…)
20 lfldi.t . . . 4 Β· = (.rβ€˜π‘…)
2110, 19, 20ringdir 19996 . . 3 ((𝑅 ∈ Ring ∧ (π‘₯ ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) β†’ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧)))
2218, 21sylan 581 . 2 ((πœ‘ ∧ (π‘₯ ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) β†’ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧)))
233, 6, 13, 16, 22caofdir 7661 1 (πœ‘ β†’ ((𝐺 ∘f + 𝐻) ∘f Β· (𝑉 Γ— {𝑋})) = ((𝐺 ∘f Β· (𝑉 Γ— {𝑋})) ∘f + (𝐻 ∘f Β· (𝑉 Γ— {𝑋}))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  Vcvv 3447  {csn 4590   Γ— cxp 5635  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361   ∘f cof 7619  Basecbs 17091  +gcplusg 17141  .rcmulr 17142  Scalarcsca 17144  Ringcrg 19972  LModclmod 20365  LFnlclfn 37569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-of 7621  df-map 8773  df-ring 19974  df-lmod 20367  df-lfl 37570
This theorem is referenced by:  ldualvsdi1  37655
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