Theorem lflset 33364

Description: The set of linear functionals in a left module or left vector space. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)

Hypotheses

Ref	Expression
lflset.v	⊢ 𝑉 = (Base‘𝑊)
lflset.a	⊢ + = (+g‘𝑊)
lflset.d	⊢ 𝐷 = (Scalar‘𝑊)
lflset.s	⊢ · = ( ·𝑠 ‘𝑊)
lflset.k	⊢ 𝐾 = (Base‘𝐷)
lflset.p	⊢ ⨣ = (+g‘𝐷)
lflset.t	⊢ × = (.r‘𝐷)
lflset.f	⊢ 𝐹 = (LFnl‘𝑊)

Assertion

Ref	Expression
lflset	⊢ (𝑊 ∈ 𝑋 → 𝐹 = {𝑓 ∈ (𝐾 ↑𝑚 𝑉) ∣ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓‘𝑥)) ⨣ (𝑓‘𝑦))})

Distinct variable groups:   𝑓,𝑟,𝐾   𝑥,𝑓,𝑦,𝑉   𝑓,𝑊   𝑥,𝑟,𝑦,𝑊

Allowed substitution hints:   𝐷(𝑥,𝑦,𝑓,𝑟)   + (𝑥,𝑦,𝑓,𝑟)   ⨣ (𝑥,𝑦,𝑓,𝑟)   · (𝑥,𝑦,𝑓,𝑟)   × (𝑥,𝑦,𝑓,𝑟)   𝐹(𝑥,𝑦,𝑓,𝑟)   𝐾(𝑥,𝑦)   𝑉(𝑟)   𝑋(𝑥,𝑦,𝑓,𝑟)

Proof of Theorem lflset

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3185	. 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
2		lflset.f	. . 3 ⊢ 𝐹 = (LFnl‘𝑊)
3		fveq2 6103	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
4		lflset.d	. . . . . . . . 9 ⊢ 𝐷 = (Scalar‘𝑊)
5	3, 4	syl6eqr 2662	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐷)
6	5	fveq2d 6107	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐷))
7		lflset.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝐷)
8	6, 7	syl6eqr 2662	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
9		fveq2 6103	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
10		lflset.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑊)
11	9, 10	syl6eqr 2662	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
12	8, 11	oveq12d 6567	. . . . 5 ⊢ (𝑤 = 𝑊 → ((Base‘(Scalar‘𝑤)) ↑𝑚 (Base‘𝑤)) = (𝐾 ↑𝑚 𝑉))
13		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (+g‘𝑤) = (+g‘𝑊))
14		lflset.a	. . . . . . . . . . . 12 ⊢ + = (+g‘𝑊)
15	13, 14	syl6eqr 2662	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (+g‘𝑤) = + )
16		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
17		lflset.s	. . . . . . . . . . . . 13 ⊢ · = ( ·𝑠 ‘𝑊)
18	16, 17	syl6eqr 2662	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = · )
19	18	oveqd 6566	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (𝑟( ·𝑠 ‘𝑤)𝑥) = (𝑟 · 𝑥))
20		eqidd 2611	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → 𝑦 = 𝑦)
21	15, 19, 20	oveq123d 6570	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ((𝑟( ·𝑠 ‘𝑤)𝑥)(+g‘𝑤)𝑦) = ((𝑟 · 𝑥) + 𝑦))
22	21	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑓‘((𝑟( ·𝑠 ‘𝑤)𝑥)(+g‘𝑤)𝑦)) = (𝑓‘((𝑟 · 𝑥) + 𝑦)))
23	5	fveq2d 6107	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = (+g‘𝐷))
24		lflset.p	. . . . . . . . . . 11 ⊢ ⨣ = (+g‘𝐷)
25	23, 24	syl6eqr 2662	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = ⨣ )
26	5	fveq2d 6107	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = (.r‘𝐷))
27		lflset.t	. . . . . . . . . . . 12 ⊢ × = (.r‘𝐷)
28	26, 27	syl6eqr 2662	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = × )
29	28	oveqd 6566	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (𝑟(.r‘(Scalar‘𝑤))(𝑓‘𝑥)) = (𝑟 × (𝑓‘𝑥)))
30		eqidd 2611	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (𝑓‘𝑦) = (𝑓‘𝑦))
31	25, 29, 30	oveq123d 6570	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((𝑟(.r‘(Scalar‘𝑤))(𝑓‘𝑥))(+g‘(Scalar‘𝑤))(𝑓‘𝑦)) = ((𝑟 × (𝑓‘𝑥)) ⨣ (𝑓‘𝑦)))
32	22, 31	eqeq12d 2625	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((𝑓‘((𝑟( ·𝑠 ‘𝑤)𝑥)(+g‘𝑤)𝑦)) = ((𝑟(.r‘(Scalar‘𝑤))(𝑓‘𝑥))(+g‘(Scalar‘𝑤))(𝑓‘𝑦)) ↔ (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓‘𝑥)) ⨣ (𝑓‘𝑦))))
33	11, 32	raleqbidv 3129	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (∀𝑦 ∈ (Base‘𝑤)(𝑓‘((𝑟( ·𝑠 ‘𝑤)𝑥)(+g‘𝑤)𝑦)) = ((𝑟(.r‘(Scalar‘𝑤))(𝑓‘𝑥))(+g‘(Scalar‘𝑤))(𝑓‘𝑦)) ↔ ∀𝑦 ∈ 𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓‘𝑥)) ⨣ (𝑓‘𝑦))))
34	11, 33	raleqbidv 3129	. . . . . 6 ⊢ (𝑤 = 𝑊 → (∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)(𝑓‘((𝑟( ·𝑠 ‘𝑤)𝑥)(+g‘𝑤)𝑦)) = ((𝑟(.r‘(Scalar‘𝑤))(𝑓‘𝑥))(+g‘(Scalar‘𝑤))(𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓‘𝑥)) ⨣ (𝑓‘𝑦))))
35	8, 34	raleqbidv 3129	. . . . 5 ⊢ (𝑤 = 𝑊 → (∀𝑟 ∈ (Base‘(Scalar‘𝑤))∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)(𝑓‘((𝑟( ·𝑠 ‘𝑤)𝑥)(+g‘𝑤)𝑦)) = ((𝑟(.r‘(Scalar‘𝑤))(𝑓‘𝑥))(+g‘(Scalar‘𝑤))(𝑓‘𝑦)) ↔ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓‘𝑥)) ⨣ (𝑓‘𝑦))))
36	12, 35	rabeqbidv 3168	. . . 4 ⊢ (𝑤 = 𝑊 → {𝑓 ∈ ((Base‘(Scalar‘𝑤)) ↑𝑚 (Base‘𝑤)) ∣ ∀𝑟 ∈ (Base‘(Scalar‘𝑤))∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)(𝑓‘((𝑟( ·𝑠 ‘𝑤)𝑥)(+g‘𝑤)𝑦)) = ((𝑟(.r‘(Scalar‘𝑤))(𝑓‘𝑥))(+g‘(Scalar‘𝑤))(𝑓‘𝑦))} = {𝑓 ∈ (𝐾 ↑𝑚 𝑉) ∣ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓‘𝑥)) ⨣ (𝑓‘𝑦))})
37		df-lfl 33363	. . . 4 ⊢ LFnl = (𝑤 ∈ V ↦ {𝑓 ∈ ((Base‘(Scalar‘𝑤)) ↑𝑚 (Base‘𝑤)) ∣ ∀𝑟 ∈ (Base‘(Scalar‘𝑤))∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)(𝑓‘((𝑟( ·𝑠 ‘𝑤)𝑥)(+g‘𝑤)𝑦)) = ((𝑟(.r‘(Scalar‘𝑤))(𝑓‘𝑥))(+g‘(Scalar‘𝑤))(𝑓‘𝑦))})
38		ovex 6577	. . . . 5 ⊢ (𝐾 ↑𝑚 𝑉) ∈ V
39	38	rabex 4740	. . . 4 ⊢ {𝑓 ∈ (𝐾 ↑𝑚 𝑉) ∣ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓‘𝑥)) ⨣ (𝑓‘𝑦))} ∈ V
40	36, 37, 39	fvmpt 6191	. . 3 ⊢ (𝑊 ∈ V → (LFnl‘𝑊) = {𝑓 ∈ (𝐾 ↑𝑚 𝑉) ∣ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓‘𝑥)) ⨣ (𝑓‘𝑦))})
41	2, 40	syl5eq 2656	. 2 ⊢ (𝑊 ∈ V → 𝐹 = {𝑓 ∈ (𝐾 ↑𝑚 𝑉) ∣ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓‘𝑥)) ⨣ (𝑓‘𝑦))})
42	1, 41	syl 17	1 ⊢ (𝑊 ∈ 𝑋 → 𝐹 = {𝑓 ∈ (𝐾 ↑𝑚 𝑉) ∣ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓‘𝑥)) ⨣ (𝑓‘𝑦))})

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173  ‘cfv 5804  (class class class)co 6549   ↑_𝑚 cmap 7744  Basecbs 15695  +_gcplusg 15768  ._rcmulr 15769  Scalarcsca 15771   ·_𝑠 cvsca 15772  LFnlclfn 33362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-lfl 33363

This theorem is referenced by:  islfl  33365