Theorem islfld 39938

Description: Properties that determine a linear functional. TODO: use this in place of islfl 39936 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 6883	. . . . . 6 ⊢ (𝜑 → (Base‘𝐷) = (Base‘(Scalar‘𝑊)))
7	4, 6	eqtrd 2795	. . . . 5 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝑊)))
8	3, 7	feq23d 6698	. . . 4 ⊢ (𝜑 → (𝐺:𝑉⟶𝐾 ↔ 𝐺:(Base‘𝑊)⟶(Base‘(Scalar‘𝑊))))
9	2, 8	mpbid 235	. . 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 3208	. . . 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 7431	. . . . . . . . . 10 ⊢ (𝜑 → (𝑟 · 𝑥) = (𝑟( ·𝑠 ‘𝑊)𝑥))
15		eqidd 2761	. . . . . . . . . 10 ⊢ (𝜑 → 𝑦 = 𝑦)
16	12, 14, 15	oveq123d 7435	. . . . . . . . 9 ⊢ (𝜑 → ((𝑟 · 𝑥) + 𝑦) = ((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦))
17	16	fveq2d 6883	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘((𝑟 · 𝑥) + 𝑦)) = (𝐺‘((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)))
18		islfld.p	. . . . . . . . . 10 ⊢ (𝜑 → ⨣ = (+g‘𝐷))
19	5	fveq2d 6883	. . . . . . . . . 10 ⊢ (𝜑 → (+g‘𝐷) = (+g‘(Scalar‘𝑊)))
20	18, 19	eqtrd 2795	. . . . . . . . 9 ⊢ (𝜑 → ⨣ = (+g‘(Scalar‘𝑊)))
21		islfld.t	. . . . . . . . . . 11 ⊢ (𝜑 → × = (.r‘𝐷))
22	5	fveq2d 6883	. . . . . . . . . . 11 ⊢ (𝜑 → (.r‘𝐷) = (.r‘(Scalar‘𝑊)))
23	21, 22	eqtrd 2795	. . . . . . . . . 10 ⊢ (𝜑 → × = (.r‘(Scalar‘𝑊)))
24	23	oveqd 7431	. . . . . . . . 9 ⊢ (𝜑 → (𝑟 × (𝐺‘𝑥)) = (𝑟(.r‘(Scalar‘𝑊))(𝐺‘𝑥)))
25		eqidd 2761	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘𝑦) = (𝐺‘𝑦))
26	20, 24, 25	oveq123d 7435	. . . . . . . 8 ⊢ (𝜑 → ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)) = ((𝑟(.r‘(Scalar‘𝑊))(𝐺‘𝑥))(+g‘(Scalar‘𝑊))(𝐺‘𝑦)))
27	17, 26	eqeq12d 2776	. . . . . . 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 3334	. . . . . 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 3334	. . . . 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 3334	. . . 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 235	. . 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 2760	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
33		eqid 2760	. . . . 5 ⊢ (+g‘𝑊) = (+g‘𝑊)
34		eqid 2760	. . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
35		eqid 2760	. . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
36		eqid 2760	. . . . 5 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
37		eqid 2760	. . . . 5 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
38		eqid 2760	. . . . 5 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
39		eqid 2760	. . . . 5 ⊢ (LFnl‘𝑊) = (LFnl‘𝑊)
40	32, 33, 34, 35, 36, 37, 38, 39	islfl 39936	. . . 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 483	. . 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 850	. 2 ⊢ (𝜑 → 𝐺 ∈ (LFnl‘𝑊))
43		islfld.f	. 2 ⊢ (𝜑 → 𝐹 = (LFnl‘𝑊))
44	42, 43	eleqtrrd 2863	1 ⊢ (𝜑 → 𝐺 ∈ 𝐹)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ⟶wf 6529  ‘cfv 6533  (class class class)co 7414  Basecbs 17304  +_gcplusg 17345  ._rcmulr 17346  Scalarcsca 17348   ·_𝑠 cvsca 17349  LFnlclfn 39933

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8831  df-lfl 39934

This theorem is used by:  lflvscl  39953