Theorem lflvsdi2a 39119

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	⊢ + = (+g‘𝑅)
lfldi.t	⊢ · = (.r‘𝑅)
lfldi.f	⊢ 𝐹 = (LFnl‘𝑊)
lfldi.w	⊢ (𝜑 → 𝑊 ∈ LMod)
lfldi.x	⊢ (𝜑 → 𝑋 ∈ 𝐾)
lfldi2.y	⊢ (𝜑 → 𝑌 ∈ 𝐾)
lfldi2.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)

Assertion

Ref	Expression
lflvsdi2a	⊢ (𝜑 → (𝐺 ∘f · (𝑉 × {(𝑋 + 𝑌)})) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐺 ∘f · (𝑉 × {𝑌}))))

Proof of Theorem lflvsdi2a

Step	Hyp	Ref	Expression
1		lfldi.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑊)
2	1	fvexi 6831	. . . . 5 ⊢ 𝑉 ∈ V
3	2	a1i 11	. . . 4 ⊢ (𝜑 → 𝑉 ∈ V)
4		lfldi.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐾)
5		lfldi2.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐾)
6	3, 4, 5	ofc12 7635	. . 3 ⊢ (𝜑 → ((𝑉 × {𝑋}) ∘f + (𝑉 × {𝑌})) = (𝑉 × {(𝑋 + 𝑌)}))
7	6	oveq2d 7357	. 2 ⊢ (𝜑 → (𝐺 ∘f · ((𝑉 × {𝑋}) ∘f + (𝑉 × {𝑌}))) = (𝐺 ∘f · (𝑉 × {(𝑋 + 𝑌)})))
8		lfldi.r	. . 3 ⊢ 𝑅 = (Scalar‘𝑊)
9		lfldi.k	. . 3 ⊢ 𝐾 = (Base‘𝑅)
10		lfldi.p	. . 3 ⊢ + = (+g‘𝑅)
11		lfldi.t	. . 3 ⊢ · = (.r‘𝑅)
12		lfldi.f	. . 3 ⊢ 𝐹 = (LFnl‘𝑊)
13		lfldi.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ LMod)
14		lfldi2.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
15	1, 8, 9, 10, 11, 12, 13, 4, 5, 14	lflvsdi2 39118	. 2 ⊢ (𝜑 → (𝐺 ∘f · ((𝑉 × {𝑋}) ∘f + (𝑉 × {𝑌}))) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐺 ∘f · (𝑉 × {𝑌}))))
16	7, 15	eqtr3d 2768	1 ⊢ (𝜑 → (𝐺 ∘f · (𝑉 × {(𝑋 + 𝑌)})) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐺 ∘f · (𝑉 × {𝑌}))))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  Vcvv 3436  {csn 4571   × cxp 5609  ‘cfv 6476  (class class class)co 7341   ∘_f cof 7603  Basecbs 17115  +_gcplusg 17156  ._rcmulr 17157  Scalarcsca 17159  LModclmod 20788  LFnlclfn 39096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-of 7605  df-map 8747  df-ring 20148  df-lmod 20790  df-lfl 39097

This theorem is referenced by:  ldualvsdi2  39183