Theorem islfl 40383

Description: The predicate "is a linear functional". (Contributed by NM, 15-Apr-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
islfl	⊢ (𝑊 ∈ 𝑋 → (𝐺 ∈ 𝐹 ↔ (𝐺:𝑉⟶𝐾 ∧ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)))))

Distinct variable groups:   𝐾,𝑟   𝑥,𝑦,𝑉   𝑥,𝑟,𝑦,𝑊   𝐺,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)   𝐾(𝑥, 𝑦)   𝑉(𝑟)   𝑋(𝑥, 𝑦, 𝑟)

Proof of Theorem islfl

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lflset.v	. . . 4 ⊢ 𝑉 = (Base‘𝑊)
2		lflset.a	. . . 4 ⊢ + = (+g‘𝑊)
3		lflset.d	. . . 4 ⊢ 𝐷 = (Scalar‘𝑊)
4		lflset.s	. . . 4 ⊢ · = ( ·𝑠 ‘𝑊)
5		lflset.k	. . . 4 ⊢ 𝐾 = (Base‘𝐷)
6		lflset.p	. . . 4 ⊢ ⨣ = (+g‘𝐷)
7		lflset.t	. . . 4 ⊢ × = (.r‘𝐷)
8		lflset.f	. . . 4 ⊢ 𝐹 = (LFnl‘𝑊)
9	1, 2, 3, 4, 5, 6, 7, 8	lflset 40382	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝐹 = {𝑓 ∈ (𝐾 ↑m 𝑉) ∣ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓‘𝑥)) ⨣ (𝑓‘𝑦))})
10	9	eleq2d 2850	. 2 ⊢ (𝑊 ∈ 𝑋 → (𝐺 ∈ 𝐹 ↔ 𝐺 ∈ {𝑓 ∈ (𝐾 ↑m 𝑉) ∣ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓‘𝑥)) ⨣ (𝑓‘𝑦))}))
11		fveq1 6915	. . . . . . 7 ⊢ (𝑓 = 𝐺 → (𝑓‘((𝑟 · 𝑥) + 𝑦)) = (𝐺‘((𝑟 · 𝑥) + 𝑦)))
12		fveq1 6915	. . . . . . . . 9 ⊢ (𝑓 = 𝐺 → (𝑓‘𝑥) = (𝐺‘𝑥))
13	12	oveq2d 7467	. . . . . . . 8 ⊢ (𝑓 = 𝐺 → (𝑟 × (𝑓‘𝑥)) = (𝑟 × (𝐺‘𝑥)))
14		fveq1 6915	. . . . . . . 8 ⊢ (𝑓 = 𝐺 → (𝑓‘𝑦) = (𝐺‘𝑦))
15	13, 14	oveq12d 7469	. . . . . . 7 ⊢ (𝑓 = 𝐺 → ((𝑟 × (𝑓‘𝑥)) ⨣ (𝑓‘𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)))
16	11, 15	eqeq12d 2779	. . . . . 6 ⊢ (𝑓 = 𝐺 → ((𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓‘𝑥)) ⨣ (𝑓‘𝑦)) ↔ (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦))))
17	16	2ralbidv 3230	. . . . 5 ⊢ (𝑓 = 𝐺 → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓‘𝑥)) ⨣ (𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦))))
18	17	ralbidv 3189	. . . 4 ⊢ (𝑓 = 𝐺 → (∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓‘𝑥)) ⨣ (𝑓‘𝑦)) ↔ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦))))
19	18	elrab 3648	. . 3 ⊢ (𝐺 ∈ {𝑓 ∈ (𝐾 ↑m 𝑉) ∣ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓‘𝑥)) ⨣ (𝑓‘𝑦))} ↔ (𝐺 ∈ (𝐾 ↑m 𝑉) ∧ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦))))
20	5	fvexi 6930	. . . . 5 ⊢ 𝐾 ∈ V
21	1	fvexi 6930	. . . . 5 ⊢ 𝑉 ∈ V
22	20, 21	elmap 8937	. . . 4 ⊢ (𝐺 ∈ (𝐾 ↑m 𝑉) ↔ 𝐺:𝑉⟶𝐾)
23	22	anbi1i 636	. . 3 ⊢ ((𝐺 ∈ (𝐾 ↑m 𝑉) ∧ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦))) ↔ (𝐺:𝑉⟶𝐾 ∧ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦))))
24	19, 23	bitri 278	. 2 ⊢ (𝐺 ∈ {𝑓 ∈ (𝐾 ↑m 𝑉) ∣ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓‘𝑥)) ⨣ (𝑓‘𝑦))} ↔ (𝐺:𝑉⟶𝐾 ∧ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦))))
25	10, 24	bitrdi 290	1 ⊢ (𝑊 ∈ 𝑋 → (𝐺 ∈ 𝐹 ↔ (𝐺:𝑉⟶𝐾 ∧ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451   ↑_m cmap 8885  Basecbs 17431  +_gcplusg 17472  ._rcmulr 17473  Scalarcsca 17475   ·_𝑠 cvsca 17476  LFnlclfn 40380

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-map 8887  df-lfl 40381

This theorem is used by:  lfli  40384  islfld  40385  lflf  40386  lfl0f  40392  lfladdcl  40394  lflnegcl  40398  lshpkrcl  40439