Theorem islfld 36192

Description: Properties that determine a linear functional. TODO: use this in place of islfl 36190 when it shortens the proof. (Contributed by NM, 19-Oct-2014.)

Hypotheses

Ref	Expression
islfld.v	⊢ (𝜑 → 𝑉 = (Base‘𝑊))
islfld.a	⊢ (𝜑 → + = (+g‘𝑊))
islfld.d	⊢ (𝜑 → 𝐷 = (Scalar‘𝑊))
islfld.s	⊢ (𝜑 → · = ( ·𝑠 ‘𝑊))
islfld.k	⊢ (𝜑 → 𝐾 = (Base‘𝐷))
islfld.p	⊢ (𝜑 → ⨣ = (+g‘𝐷))
islfld.t	⊢ (𝜑 → × = (.r‘𝐷))
islfld.f	⊢ (𝜑 → 𝐹 = (LFnl‘𝑊))
islfld.u	⊢ (𝜑 → 𝐺:𝑉⟶𝐾)
islfld.l	⊢ ((𝜑 ∧ (𝑟 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)))
islfld.w	⊢ (𝜑 → 𝑊 ∈ 𝑋)

Assertion

Ref	Expression
islfld	⊢ (𝜑 → 𝐺 ∈ 𝐹)

Distinct variable groups:   𝑥,𝑟,𝑦,𝐺   𝐾,𝑟,𝑥,𝑦   𝑥,𝑉,𝑦   𝑊,𝑟,𝑥,𝑦   𝜑,𝑟,𝑥,𝑦

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

Proof of Theorem islfld

Step	Hyp	Ref	Expression
1		islfld.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ 𝑋)
2		islfld.u	. . . 4 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾)
3		islfld.v	. . . . 5 ⊢ (𝜑 → 𝑉 = (Base‘𝑊))
4		islfld.k	. . . . . 6 ⊢ (𝜑 → 𝐾 = (Base‘𝐷))
5		islfld.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 = (Scalar‘𝑊))
6	5	fveq2d 6669	. . . . . 6 ⊢ (𝜑 → (Base‘𝐷) = (Base‘(Scalar‘𝑊)))
7	4, 6	eqtrd 2856	. . . . 5 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝑊)))
8	3, 7	feq23d 6504	. . . 4 ⊢ (𝜑 → (𝐺:𝑉⟶𝐾 ↔ 𝐺:(Base‘𝑊)⟶(Base‘(Scalar‘𝑊))))
9	2, 8	mpbid 234	. . 3 ⊢ (𝜑 → 𝐺:(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)))
10		islfld.l	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)))
11	10	ralrimivvva 3192	. . . 4 ⊢ (𝜑 → ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)))
12		islfld.a	. . . . . . . . . 10 ⊢ (𝜑 → + = (+g‘𝑊))
13		islfld.s	. . . . . . . . . . 11 ⊢ (𝜑 → · = ( ·𝑠 ‘𝑊))
14	13	oveqd 7167	. . . . . . . . . 10 ⊢ (𝜑 → (𝑟 · 𝑥) = (𝑟( ·𝑠 ‘𝑊)𝑥))
15		eqidd 2822	. . . . . . . . . 10 ⊢ (𝜑 → 𝑦 = 𝑦)
16	12, 14, 15	oveq123d 7171	. . . . . . . . 9 ⊢ (𝜑 → ((𝑟 · 𝑥) + 𝑦) = ((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦))
17	16	fveq2d 6669	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘((𝑟 · 𝑥) + 𝑦)) = (𝐺‘((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)))
18		islfld.p	. . . . . . . . . 10 ⊢ (𝜑 → ⨣ = (+g‘𝐷))
19	5	fveq2d 6669	. . . . . . . . . 10 ⊢ (𝜑 → (+g‘𝐷) = (+g‘(Scalar‘𝑊)))
20	18, 19	eqtrd 2856	. . . . . . . . 9 ⊢ (𝜑 → ⨣ = (+g‘(Scalar‘𝑊)))
21		islfld.t	. . . . . . . . . . 11 ⊢ (𝜑 → × = (.r‘𝐷))
22	5	fveq2d 6669	. . . . . . . . . . 11 ⊢ (𝜑 → (.r‘𝐷) = (.r‘(Scalar‘𝑊)))
23	21, 22	eqtrd 2856	. . . . . . . . . 10 ⊢ (𝜑 → × = (.r‘(Scalar‘𝑊)))
24	23	oveqd 7167	. . . . . . . . 9 ⊢ (𝜑 → (𝑟 × (𝐺‘𝑥)) = (𝑟(.r‘(Scalar‘𝑊))(𝐺‘𝑥)))
25		eqidd 2822	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘𝑦) = (𝐺‘𝑦))
26	20, 24, 25	oveq123d 7171	. . . . . . . 8 ⊢ (𝜑 → ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)) = ((𝑟(.r‘(Scalar‘𝑊))(𝐺‘𝑥))(+g‘(Scalar‘𝑊))(𝐺‘𝑦)))
27	17, 26	eqeq12d 2837	. . . . . . 7 ⊢ (𝜑 → ((𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)) ↔ (𝐺‘((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)) = ((𝑟(.r‘(Scalar‘𝑊))(𝐺‘𝑥))(+g‘(Scalar‘𝑊))(𝐺‘𝑦))))
28	3, 27	raleqbidv 3402	. . . . . 6 ⊢ (𝜑 → (∀𝑦 ∈ 𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)) ↔ ∀𝑦 ∈ (Base‘𝑊)(𝐺‘((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)) = ((𝑟(.r‘(Scalar‘𝑊))(𝐺‘𝑥))(+g‘(Scalar‘𝑊))(𝐺‘𝑦))))
29	3, 28	raleqbidv 3402	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)) ↔ ∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)(𝐺‘((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)) = ((𝑟(.r‘(Scalar‘𝑊))(𝐺‘𝑥))(+g‘(Scalar‘𝑊))(𝐺‘𝑦))))
30	7, 29	raleqbidv 3402	. . . 4 ⊢ (𝜑 → (∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)) ↔ ∀𝑟 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)(𝐺‘((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)) = ((𝑟(.r‘(Scalar‘𝑊))(𝐺‘𝑥))(+g‘(Scalar‘𝑊))(𝐺‘𝑦))))
31	11, 30	mpbid 234	. . 3 ⊢ (𝜑 → ∀𝑟 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)(𝐺‘((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)) = ((𝑟(.r‘(Scalar‘𝑊))(𝐺‘𝑥))(+g‘(Scalar‘𝑊))(𝐺‘𝑦)))
32		eqid 2821	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
33		eqid 2821	. . . . 5 ⊢ (+g‘𝑊) = (+g‘𝑊)
34		eqid 2821	. . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
35		eqid 2821	. . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
36		eqid 2821	. . . . 5 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
37		eqid 2821	. . . . 5 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
38		eqid 2821	. . . . 5 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
39		eqid 2821	. . . . 5 ⊢ (LFnl‘𝑊) = (LFnl‘𝑊)
40	32, 33, 34, 35, 36, 37, 38, 39	islfl 36190	. . . 4 ⊢ (𝑊 ∈ 𝑋 → (𝐺 ∈ (LFnl‘𝑊) ↔ (𝐺:(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)) ∧ ∀𝑟 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)(𝐺‘((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)) = ((𝑟(.r‘(Scalar‘𝑊))(𝐺‘𝑥))(+g‘(Scalar‘𝑊))(𝐺‘𝑦)))))
41	40	biimpar 480	. . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ (𝐺:(Base‘𝑊)⟶(Base‘(Scalar‘𝑊)) ∧ ∀𝑟 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)(𝐺‘((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)) = ((𝑟(.r‘(Scalar‘𝑊))(𝐺‘𝑥))(+g‘(Scalar‘𝑊))(𝐺‘𝑦)))) → 𝐺 ∈ (LFnl‘𝑊))
42	1, 9, 31, 41	syl12anc 834	. 2 ⊢ (𝜑 → 𝐺 ∈ (LFnl‘𝑊))
43		islfld.f	. 2 ⊢ (𝜑 → 𝐹 = (LFnl‘𝑊))
44	42, 43	eleqtrrd 2916	1 ⊢ (𝜑 → 𝐺 ∈ 𝐹)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ⟶wf 6346  ‘cfv 6350  (class class class)co 7150  Basecbs 16477  +_gcplusg 16559  ._rcmulr 16560  Scalarcsca 16562   ·_𝑠 cvsca 16563  LFnlclfn 36187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-map 8402  df-lfl 36188

This theorem is referenced by:  lflvscl  36207