Theorem lflvsdi1 39702

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 6881	. . 3 ⊢ 𝑉 ∈ V
3	2	a1i 11	. 2 ⊢ (𝜑 → 𝑉 ∈ V)
4		lfldi.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐾)
5		fconst6g 6753	. . 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 39687	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾)
13	7, 8, 12	syl2anc 593	. 2 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾)
14		lfldi1.h	. . 3 ⊢ (𝜑 → 𝐻 ∈ 𝐹)
15	9, 10, 1, 11	lflf 39687	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹) → 𝐻:𝑉⟶𝐾)
16	7, 14, 15	syl2anc 593	. 2 ⊢ (𝜑 → 𝐻:𝑉⟶𝐾)
17	9	lmodring 20935	. . . 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 20312	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
22	18, 21	sylan 589	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
23	3, 6, 13, 16, 22	caofdir 7703	1 ⊢ (𝜑 → ((𝐺 ∘f + 𝐻) ∘f · (𝑉 × {𝑋})) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐻 ∘f · (𝑉 × {𝑋}))))

Syntax hints:   → wi 4   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  Vcvv 3454  {csn 4582   × cxp 5645  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396   ∘_f cof 7658  Basecbs 17245  +_gcplusg 17286  ._rcmulr 17287  Scalarcsca 17289  Ringcrg 20283  LModclmod 20927  LFnlclfn 39681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7660  df-map 8810  df-ring 20285  df-lmod 20929  df-lfl 39682

This theorem is referenced by:  ldualvsdi1  39767