Theorem lflvsdi2a 39572

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 6841	. . . . 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 7650	. . 3 ⊢ (𝜑 → ((𝑉 × {𝑋}) ∘f + (𝑉 × {𝑌})) = (𝑉 × {(𝑋 + 𝑌)}))
7	6	oveq2d 7372	. 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 39571	. 2 ⊢ (𝜑 → (𝐺 ∘f · ((𝑉 × {𝑋}) ∘f + (𝑉 × {𝑌}))) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐺 ∘f · (𝑉 × {𝑌}))))
16	7, 15	eqtr3d 2776	1 ⊢ (𝜑 → (𝐺 ∘f · (𝑉 × {(𝑋 + 𝑌)})) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐺 ∘f · (𝑉 × {𝑌}))))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  Vcvv 3431  {csn 4555   × cxp 5616  ‘cfv 6485  (class class class)co 7356   ∘_f cof 7618  Basecbs 17170  +_gcplusg 17211  ._rcmulr 17212  Scalarcsca 17214  LModclmod 20850  LFnlclfn 39549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-of 7620  df-map 8765  df-ring 20207  df-lmod 20852  df-lfl 39550

This theorem is referenced by:  ldualvsdi2  39636