Theorem islfld 37003

Description: Properties that determine a linear functional. TODO: use this in place of islfl 37001 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 6760	. . . . . 6 ⊢ (𝜑 → (Base‘𝐷) = (Base‘(Scalar‘𝑊)))
7	4, 6	eqtrd 2778	. . . . 5 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝑊)))
8	3, 7	feq23d 6579	. . . 4 ⊢ (𝜑 → (𝐺:𝑉⟶𝐾 ↔ 𝐺:(Base‘𝑊)⟶(Base‘(Scalar‘𝑊))))
9	2, 8	mpbid 231	. . 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 3115	. . . 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 7272	. . . . . . . . . 10 ⊢ (𝜑 → (𝑟 · 𝑥) = (𝑟( ·𝑠 ‘𝑊)𝑥))
15		eqidd 2739	. . . . . . . . . 10 ⊢ (𝜑 → 𝑦 = 𝑦)
16	12, 14, 15	oveq123d 7276	. . . . . . . . 9 ⊢ (𝜑 → ((𝑟 · 𝑥) + 𝑦) = ((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦))
17	16	fveq2d 6760	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘((𝑟 · 𝑥) + 𝑦)) = (𝐺‘((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)))
18		islfld.p	. . . . . . . . . 10 ⊢ (𝜑 → ⨣ = (+g‘𝐷))
19	5	fveq2d 6760	. . . . . . . . . 10 ⊢ (𝜑 → (+g‘𝐷) = (+g‘(Scalar‘𝑊)))
20	18, 19	eqtrd 2778	. . . . . . . . 9 ⊢ (𝜑 → ⨣ = (+g‘(Scalar‘𝑊)))
21		islfld.t	. . . . . . . . . . 11 ⊢ (𝜑 → × = (.r‘𝐷))
22	5	fveq2d 6760	. . . . . . . . . . 11 ⊢ (𝜑 → (.r‘𝐷) = (.r‘(Scalar‘𝑊)))
23	21, 22	eqtrd 2778	. . . . . . . . . 10 ⊢ (𝜑 → × = (.r‘(Scalar‘𝑊)))
24	23	oveqd 7272	. . . . . . . . 9 ⊢ (𝜑 → (𝑟 × (𝐺‘𝑥)) = (𝑟(.r‘(Scalar‘𝑊))(𝐺‘𝑥)))
25		eqidd 2739	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘𝑦) = (𝐺‘𝑦))
26	20, 24, 25	oveq123d 7276	. . . . . . . 8 ⊢ (𝜑 → ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)) = ((𝑟(.r‘(Scalar‘𝑊))(𝐺‘𝑥))(+g‘(Scalar‘𝑊))(𝐺‘𝑦)))
27	17, 26	eqeq12d 2754	. . . . . . 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 3327	. . . . . 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 3327	. . . . 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 3327	. . . 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 231	. . 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 2738	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
33		eqid 2738	. . . . 5 ⊢ (+g‘𝑊) = (+g‘𝑊)
34		eqid 2738	. . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
35		eqid 2738	. . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
36		eqid 2738	. . . . 5 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
37		eqid 2738	. . . . 5 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
38		eqid 2738	. . . . 5 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
39		eqid 2738	. . . . 5 ⊢ (LFnl‘𝑊) = (LFnl‘𝑊)
40	32, 33, 34, 35, 36, 37, 38, 39	islfl 37001	. . . 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 477	. . 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 833	. 2 ⊢ (𝜑 → 𝐺 ∈ (LFnl‘𝑊))
43		islfld.f	. 2 ⊢ (𝜑 → 𝐹 = (LFnl‘𝑊))
44	42, 43	eleqtrrd 2842	1 ⊢ (𝜑 → 𝐺 ∈ 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  ._rcmulr 16889  Scalarcsca 16891   ·_𝑠 cvsca 16892  LFnlclfn 36998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-map 8575  df-lfl 36999

This theorem is referenced by:  lflvscl  37018