Theorem lflvsdi2 34366

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		fvex 6201	. . . 4 ⊢ (Base‘𝑊) ∈ V
3	1, 2	eqeltri 2697	. . 3 ⊢ 𝑉 ∈ V
4	3	a1i 11	. 2 ⊢ (𝜑 → 𝑉 ∈ V)
5		lfldi.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ LMod)
6		lfldi2.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
7		lfldi.r	. . . 4 ⊢ 𝑅 = (Scalar‘𝑊)
8		lfldi.k	. . . 4 ⊢ 𝐾 = (Base‘𝑅)
9		lfldi.f	. . . 4 ⊢ 𝐹 = (LFnl‘𝑊)
10	7, 8, 1, 9	lflf 34350	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾)
11	5, 6, 10	syl2anc 693	. 2 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾)
12		lfldi.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐾)
13		fconst6g 6094	. . 3 ⊢ (𝑋 ∈ 𝐾 → (𝑉 × {𝑋}):𝑉⟶𝐾)
14	12, 13	syl 17	. 2 ⊢ (𝜑 → (𝑉 × {𝑋}):𝑉⟶𝐾)
15		lfldi2.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐾)
16		fconst6g 6094	. . 3 ⊢ (𝑌 ∈ 𝐾 → (𝑉 × {𝑌}):𝑉⟶𝐾)
17	15, 16	syl 17	. 2 ⊢ (𝜑 → (𝑉 × {𝑌}):𝑉⟶𝐾)
18	7	lmodring 18871	. . . 4 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring)
19	5, 18	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
20		lfldi.p	. . . 4 ⊢ + = (+g‘𝑅)
21		lfldi.t	. . . 4 ⊢ · = (.r‘𝑅)
22	8, 20, 21	ringdi 18566	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
23	19, 22	sylan 488	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
24	4, 11, 14, 17, 23	caofdi 6933	1 ⊢ (𝜑 → (𝐺 ∘𝑓 · ((𝑉 × {𝑋}) ∘𝑓 + (𝑉 × {𝑌}))) = ((𝐺 ∘𝑓 · (𝑉 × {𝑋})) ∘𝑓 + (𝐺 ∘𝑓 · (𝑉 × {𝑌}))))

Syntax hints:   → wi 4   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  Vcvv 3200  {csn 4177   × cxp 5112  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895  Basecbs 15857  +_gcplusg 15941  ._rcmulr 15942  Scalarcsca 15944  Ringcrg 18547  LModclmod 18863  LFnlclfn 34344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-map 7859  df-ring 18549  df-lmod 18865  df-lfl 34345

This theorem is referenced by:  lflvsdi2a  34367