Theorem lflvsdi2a 37398

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 6843	. . . . 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 7627	. . 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 37397	. 2 ⊢ (𝜑 → (𝐺 ∘f · ((𝑉 × {𝑋}) ∘f + (𝑉 × {𝑌}))) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐺 ∘f · (𝑉 × {𝑌}))))
16	7, 15	eqtr3d 2779	1 ⊢ (𝜑 → (𝐺 ∘f · (𝑉 × {(𝑋 + 𝑌)})) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐺 ∘f · (𝑉 × {𝑌}))))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  Vcvv 3442  {csn 4577   × cxp 5622  ‘cfv 6483  (class class class)co 7341   ∘_f cof 7597  Basecbs 17009  +_gcplusg 17059  ._rcmulr 17060  Scalarcsca 17062  LModclmod 20228  LFnlclfn 37375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5233  ax-sep 5247  ax-nul 5254  ax-pow 5312  ax-pr 5376  ax-un 7654

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3731  df-csb 3847  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4274  df-if 4478  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4857  df-iun 4947  df-br 5097  df-opab 5159  df-mpt 5180  df-id 5522  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6435  df-fun 6485  df-fn 6486  df-f 6487  df-f1 6488  df-fo 6489  df-f1o 6490  df-fv 6491  df-ov 7344  df-oprab 7345  df-mpo 7346  df-of 7599  df-map 8692  df-ring 19879  df-lmod 20230  df-lfl 37376

This theorem is referenced by:  ldualvsdi2  37462