Theorem lflvsdi2 39542

Description: Reverse 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)
lfldi2.y	⊢ (𝜑 → 𝑌 ∈ 𝐾)
lfldi2.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)

Assertion

Ref	Expression
lflvsdi2	⊢ (𝜑 → (𝐺 ∘f · ((𝑉 × {𝑋}) ∘f + (𝑉 × {𝑌}))) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐺 ∘f · (𝑉 × {𝑌}))))

Proof of Theorem lflvsdi2

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 6849	. . 3 ⊢ 𝑉 ∈ V
3	2	a1i 11	. 2 ⊢ (𝜑 → 𝑉 ∈ V)
4		lfldi.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ LMod)
5		lfldi2.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
6		lfldi.r	. . . 4 ⊢ 𝑅 = (Scalar‘𝑊)
7		lfldi.k	. . . 4 ⊢ 𝐾 = (Base‘𝑅)
8		lfldi.f	. . . 4 ⊢ 𝐹 = (LFnl‘𝑊)
9	6, 7, 1, 8	lflf 39526	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾)
10	4, 5, 9	syl2anc 585	. 2 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾)
11		lfldi.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐾)
12		fconst6g 6724	. . 3 ⊢ (𝑋 ∈ 𝐾 → (𝑉 × {𝑋}):𝑉⟶𝐾)
13	11, 12	syl 17	. 2 ⊢ (𝜑 → (𝑉 × {𝑋}):𝑉⟶𝐾)
14		lfldi2.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐾)
15		fconst6g 6724	. . 3 ⊢ (𝑌 ∈ 𝐾 → (𝑉 × {𝑌}):𝑉⟶𝐾)
16	14, 15	syl 17	. 2 ⊢ (𝜑 → (𝑉 × {𝑌}):𝑉⟶𝐾)
17	6	lmodring 20857	. . . 4 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring)
18	4, 17	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
19		lfldi.p	. . . 4 ⊢ + = (+g‘𝑅)
20		lfldi.t	. . . 4 ⊢ · = (.r‘𝑅)
21	7, 19, 20	ringdi 20236	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
22	18, 21	sylan 581	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
23	3, 10, 13, 16, 22	caofdi 7667	1 ⊢ (𝜑 → (𝐺 ∘f · ((𝑉 × {𝑋}) ∘f + (𝑉 × {𝑌}))) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐺 ∘f · (𝑉 × {𝑌}))))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3430  {csn 4568   × cxp 5623  ⟶wf 6489  ‘cfv 6493  (class class class)co 7361   ∘_f cof 7623  Basecbs 17173  +_gcplusg 17214  ._rcmulr 17215  Scalarcsca 17217  Ringcrg 20208  LModclmod 20849  LFnlclfn 39520

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-of 7625  df-map 8769  df-ring 20210  df-lmod 20851  df-lfl 39521

This theorem is referenced by:  lflvsdi2a  39543