Theorem lflvsdi2a 36376

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 6659	. . . . 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 7414	. . 3 ⊢ (𝜑 → ((𝑉 × {𝑋}) ∘f + (𝑉 × {𝑌})) = (𝑉 × {(𝑋 + 𝑌)}))
7	6	oveq2d 7151	. 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 36375	. 2 ⊢ (𝜑 → (𝐺 ∘f · ((𝑉 × {𝑋}) ∘f + (𝑉 × {𝑌}))) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐺 ∘f · (𝑉 × {𝑌}))))
16	7, 15	eqtr3d 2835	1 ⊢ (𝜑 → (𝐺 ∘f · (𝑉 × {(𝑋 + 𝑌)})) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐺 ∘f · (𝑉 × {𝑌}))))

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3441  {csn 4525   × cxp 5517  ‘cfv 6324  (class class class)co 7135   ∘_f cof 7387  Basecbs 16475  +_gcplusg 16557  ._rcmulr 16558  Scalarcsca 16560  LModclmod 19627  LFnlclfn 36353

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389  df-map 8391  df-ring 19292  df-lmod 19629  df-lfl 36354

This theorem is referenced by:  ldualvsdi2  36440