Theorem lflvsdi1 38451

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 6896	. . 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 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 38436	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾)
13	7, 8, 12	syl2anc 583	. 2 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾)
14		lfldi1.h	. . 3 ⊢ (𝜑 → 𝐻 ∈ 𝐹)
15	9, 10, 1, 11	lflf 38436	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹) → 𝐻:𝑉⟶𝐾)
16	7, 14, 15	syl2anc 583	. 2 ⊢ (𝜑 → 𝐻:𝑉⟶𝐾)
17	9	lmodring 20710	. . . 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 20160	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
22	18, 21	sylan 579	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
23	3, 6, 13, 16, 22	caofdir 7704	1 ⊢ (𝜑 → ((𝐺 ∘f + 𝐻) ∘f · (𝑉 × {𝑋})) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐻 ∘f · (𝑉 × {𝑋}))))

Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3466  {csn 4621   × cxp 5665  ⟶wf 6530  ‘cfv 6534  (class class class)co 7402   ∘_f cof 7662  Basecbs 17149  +_gcplusg 17202  ._rcmulr 17203  Scalarcsca 17205  Ringcrg 20134  LModclmod 20702  LFnlclfn 38430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-of 7664  df-map 8819  df-ring 20136  df-lmod 20704  df-lfl 38431

This theorem is referenced by:  ldualvsdi1  38516