Theorem lflvsdi1 36374

Description: Distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-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)
lfldi1.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)
lfldi1.h	⊢ (𝜑 → 𝐻 ∈ 𝐹)

Assertion

Ref	Expression
lflvsdi1	⊢ (𝜑 → ((𝐺 ∘f + 𝐻) ∘f · (𝑉 × {𝑋})) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐻 ∘f · (𝑉 × {𝑋}))))

Proof of Theorem lflvsdi1

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lfldi.v	. . . 4 ⊢ 𝑉 = (Base‘𝑊)
2	1	fvexi 6659	. . 3 ⊢ 𝑉 ∈ V
3	2	a1i 11	. 2 ⊢ (𝜑 → 𝑉 ∈ V)
4		lfldi.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐾)
5		fconst6g 6542	. . 3 ⊢ (𝑋 ∈ 𝐾 → (𝑉 × {𝑋}):𝑉⟶𝐾)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → (𝑉 × {𝑋}):𝑉⟶𝐾)
7		lfldi.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ LMod)
8		lfldi1.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
9		lfldi.r	. . . 4 ⊢ 𝑅 = (Scalar‘𝑊)
10		lfldi.k	. . . 4 ⊢ 𝐾 = (Base‘𝑅)
11		lfldi.f	. . . 4 ⊢ 𝐹 = (LFnl‘𝑊)
12	9, 10, 1, 11	lflf 36359	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾)
13	7, 8, 12	syl2anc 587	. 2 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾)
14		lfldi1.h	. . 3 ⊢ (𝜑 → 𝐻 ∈ 𝐹)
15	9, 10, 1, 11	lflf 36359	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹) → 𝐻:𝑉⟶𝐾)
16	7, 14, 15	syl2anc 587	. 2 ⊢ (𝜑 → 𝐻:𝑉⟶𝐾)
17	9	lmodring 19635	. . . 4 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring)
18	7, 17	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
19		lfldi.p	. . . 4 ⊢ + = (+g‘𝑅)
20		lfldi.t	. . . 4 ⊢ · = (.r‘𝑅)
21	10, 19, 20	ringdir 19313	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
22	18, 21	sylan 583	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
23	3, 6, 13, 16, 22	caofdir 7426	1 ⊢ (𝜑 → ((𝐺 ∘f + 𝐻) ∘f · (𝑉 × {𝑋})) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐻 ∘f · (𝑉 × {𝑋}))))

Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3441  {csn 4525   × cxp 5517  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∘_f cof 7387  Basecbs 16475  +_gcplusg 16557  ._rcmulr 16558  Scalarcsca 16560  Ringcrg 19290  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:  ldualvsdi1  36439