Theorem lflset 37073

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 3450	. 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
2		lflset.f	. . 3 ⊢ 𝐹 = (LFnl‘𝑊)
3		fveq2 6774	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
4		lflset.d	. . . . . . . . 9 ⊢ 𝐷 = (Scalar‘𝑊)
5	3, 4	eqtr4di 2796	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐷)
6	5	fveq2d 6778	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐷))
7		lflset.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝐷)
8	6, 7	eqtr4di 2796	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
9		fveq2 6774	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
10		lflset.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑊)
11	9, 10	eqtr4di 2796	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
12	8, 11	oveq12d 7293	. . . . 5 ⊢ (𝑤 = 𝑊 → ((Base‘(Scalar‘𝑤)) ↑m (Base‘𝑤)) = (𝐾 ↑m 𝑉))
13		fveq2 6774	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (+g‘𝑤) = (+g‘𝑊))
14		lflset.a	. . . . . . . . . . . 12 ⊢ + = (+g‘𝑊)
15	13, 14	eqtr4di 2796	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (+g‘𝑤) = + )
16		fveq2 6774	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
17		lflset.s	. . . . . . . . . . . . 13 ⊢ · = ( ·𝑠 ‘𝑊)
18	16, 17	eqtr4di 2796	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = · )
19	18	oveqd 7292	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (𝑟( ·𝑠 ‘𝑤)𝑥) = (𝑟 · 𝑥))
20		eqidd 2739	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → 𝑦 = 𝑦)
21	15, 19, 20	oveq123d 7296	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ((𝑟( ·𝑠 ‘𝑤)𝑥)(+g‘𝑤)𝑦) = ((𝑟 · 𝑥) + 𝑦))
22	21	fveq2d 6778	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑓‘((𝑟( ·𝑠 ‘𝑤)𝑥)(+g‘𝑤)𝑦)) = (𝑓‘((𝑟 · 𝑥) + 𝑦)))
23	5	fveq2d 6778	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = (+g‘𝐷))
24		lflset.p	. . . . . . . . . . 11 ⊢ ⨣ = (+g‘𝐷)
25	23, 24	eqtr4di 2796	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = ⨣ )
26	5	fveq2d 6778	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = (.r‘𝐷))
27		lflset.t	. . . . . . . . . . . 12 ⊢ × = (.r‘𝐷)
28	26, 27	eqtr4di 2796	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = × )
29	28	oveqd 7292	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (𝑟(.r‘(Scalar‘𝑤))(𝑓‘𝑥)) = (𝑟 × (𝑓‘𝑥)))
30		eqidd 2739	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (𝑓‘𝑦) = (𝑓‘𝑦))
31	25, 29, 30	oveq123d 7296	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((𝑟(.r‘(Scalar‘𝑤))(𝑓‘𝑥))(+g‘(Scalar‘𝑤))(𝑓‘𝑦)) = ((𝑟 × (𝑓‘𝑥)) ⨣ (𝑓‘𝑦)))
32	22, 31	eqeq12d 2754	. . . . . . . 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 3336	. . . . . . 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 3336	. . . . . 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 3336	. . . . 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 3420	. . . 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 37072	. . . 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 7308	. . . . 5 ⊢ (𝐾 ↑m 𝑉) ∈ V
39	38	rabex 5256	. . . 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 6875	. . 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	eqtrid 2790	. 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 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068  Vcvv 3432  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615  Basecbs 16912  +_gcplusg 16962  ._rcmulr 16963  Scalarcsca 16965   ·_𝑠 cvsca 16966  LFnlclfn 37071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-lfl 37072

This theorem is referenced by:  islfl  37074