Theorem lflset 36759

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 3416	. 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
2		lflset.f	. . 3 ⊢ 𝐹 = (LFnl‘𝑊)
3		fveq2 6695	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
4		lflset.d	. . . . . . . . 9 ⊢ 𝐷 = (Scalar‘𝑊)
5	3, 4	eqtr4di 2789	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐷)
6	5	fveq2d 6699	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐷))
7		lflset.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝐷)
8	6, 7	eqtr4di 2789	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
9		fveq2 6695	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
10		lflset.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑊)
11	9, 10	eqtr4di 2789	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
12	8, 11	oveq12d 7209	. . . . 5 ⊢ (𝑤 = 𝑊 → ((Base‘(Scalar‘𝑤)) ↑m (Base‘𝑤)) = (𝐾 ↑m 𝑉))
13		fveq2 6695	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (+g‘𝑤) = (+g‘𝑊))
14		lflset.a	. . . . . . . . . . . 12 ⊢ + = (+g‘𝑊)
15	13, 14	eqtr4di 2789	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (+g‘𝑤) = + )
16		fveq2 6695	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
17		lflset.s	. . . . . . . . . . . . 13 ⊢ · = ( ·𝑠 ‘𝑊)
18	16, 17	eqtr4di 2789	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = · )
19	18	oveqd 7208	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (𝑟( ·𝑠 ‘𝑤)𝑥) = (𝑟 · 𝑥))
20		eqidd 2737	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → 𝑦 = 𝑦)
21	15, 19, 20	oveq123d 7212	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ((𝑟( ·𝑠 ‘𝑤)𝑥)(+g‘𝑤)𝑦) = ((𝑟 · 𝑥) + 𝑦))
22	21	fveq2d 6699	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑓‘((𝑟( ·𝑠 ‘𝑤)𝑥)(+g‘𝑤)𝑦)) = (𝑓‘((𝑟 · 𝑥) + 𝑦)))
23	5	fveq2d 6699	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = (+g‘𝐷))
24		lflset.p	. . . . . . . . . . 11 ⊢ ⨣ = (+g‘𝐷)
25	23, 24	eqtr4di 2789	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = ⨣ )
26	5	fveq2d 6699	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = (.r‘𝐷))
27		lflset.t	. . . . . . . . . . . 12 ⊢ × = (.r‘𝐷)
28	26, 27	eqtr4di 2789	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = × )
29	28	oveqd 7208	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (𝑟(.r‘(Scalar‘𝑤))(𝑓‘𝑥)) = (𝑟 × (𝑓‘𝑥)))
30		eqidd 2737	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (𝑓‘𝑦) = (𝑓‘𝑦))
31	25, 29, 30	oveq123d 7212	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((𝑟(.r‘(Scalar‘𝑤))(𝑓‘𝑥))(+g‘(Scalar‘𝑤))(𝑓‘𝑦)) = ((𝑟 × (𝑓‘𝑥)) ⨣ (𝑓‘𝑦)))
32	22, 31	eqeq12d 2752	. . . . . . . 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 3303	. . . . . . 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 3303	. . . . . 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 3303	. . . . 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 3386	. . . 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 36758	. . . 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 7224	. . . . 5 ⊢ (𝐾 ↑m 𝑉) ∈ V
39	38	rabex 5210	. . . 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 6796	. . 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 2783	. 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 1543   ∈ wcel 2112  ∀wral 3051  {crab 3055  Vcvv 3398  ‘cfv 6358  (class class class)co 7191   ↑_m cmap 8486  Basecbs 16666  +_gcplusg 16749  ._rcmulr 16750  Scalarcsca 16752   ·_𝑠 cvsca 16753  LFnlclfn 36757

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-iota 6316  df-fun 6360  df-fv 6366  df-ov 7194  df-lfl 36758

This theorem is referenced by:  islfl  36760