Theorem lflvsdi1 35216

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 6460	. . 3 ⊢ 𝑉 ∈ V
3	2	a1i 11	. 2 ⊢ (𝜑 → 𝑉 ∈ V)
4		lfldi.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐾)
5		fconst6g 6344	. . 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 35201	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾)
13	7, 8, 12	syl2anc 579	. 2 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾)
14		lfldi1.h	. . 3 ⊢ (𝜑 → 𝐻 ∈ 𝐹)
15	9, 10, 1, 11	lflf 35201	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹) → 𝐻:𝑉⟶𝐾)
16	7, 14, 15	syl2anc 579	. 2 ⊢ (𝜑 → 𝐻:𝑉⟶𝐾)
17	9	lmodring 19263	. . . 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 18954	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
22	18, 21	sylan 575	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
23	3, 6, 13, 16, 22	caofdir 7211	1 ⊢ (𝜑 → ((𝐺 ∘𝑓 + 𝐻) ∘𝑓 · (𝑉 × {𝑋})) = ((𝐺 ∘𝑓 · (𝑉 × {𝑋})) ∘𝑓 + (𝐻 ∘𝑓 · (𝑉 × {𝑋}))))

Syntax hints:   → wi 4   ∧ w3a 1071   = wceq 1601   ∈ wcel 2106  Vcvv 3397  {csn 4397   × cxp 5353  ⟶wf 6131  ‘cfv 6135  (class class class)co 6922   ∘_𝑓 cof 7172  Basecbs 16255  +_gcplusg 16338  ._rcmulr 16339  Scalarcsca 16341  Ringcrg 18934  LModclmod 19255  LFnlclfn 35195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-of 7174  df-map 8142  df-ring 18936  df-lmod 19257  df-lfl 35196

This theorem is referenced by:  ldualvsdi1  35281