Theorem lflvsdi1 39910

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 6899	. . 3 ⊢ 𝑉 ∈ V
3	2	a1i 11	. 2 ⊢ (𝜑 → 𝑉 ∈ V)
4		lfldi.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐾)
5		fconst6g 6771	. . 3 ⊢ (𝑋 ∈ 𝐾 → (𝑉 × {𝑋}):𝑉⟶𝐾)
6	4, 5	syl 18	. 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 39895	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾)
13	7, 8, 12	syl2anc 596	. 2 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾)
14		lfldi1.h	. . 3 ⊢ (𝜑 → 𝐻 ∈ 𝐹)
15	9, 10, 1, 11	lflf 39895	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹) → 𝐻:𝑉⟶𝐾)
16	7, 14, 15	syl2anc 596	. 2 ⊢ (𝜑 → 𝐻:𝑉⟶𝐾)
17	9	lmodring 21039	. . . 4 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring)
18	7, 17	syl 18	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
19		lfldi.p	. . . 4 ⊢ + = (+g‘𝑅)
20		lfldi.t	. . . 4 ⊢ · = (.r‘𝑅)
21	10, 19, 20	ringdir 20389	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
22	18, 21	sylan 592	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
23	3, 6, 13, 16, 22	caofdir 7727	1 ⊢ (𝜑 → ((𝐺 ∘f + 𝐻) ∘f · (𝑉 × {𝑋})) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐻 ∘f · (𝑉 × {𝑋}))))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  Vcvv 3457  {csn 4591   × cxp 5661  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419   ∘_f cof 7682  Basecbs 17291  +_gcplusg 17332  ._rcmulr 17333  Scalarcsca 17335  Ringcrg 20359  LModclmod 21031  LFnlclfn 39889

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7684  df-map 8832  df-ring 20361  df-lmod 21033  df-lfl 39890

This theorem is used by:  ldualvsdi1  39975