Theorem islfl 36211

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 36210	. . 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 2898	. 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 6669	. . . . . . 7 ⊢ (𝑓 = 𝐺 → (𝑓‘((𝑟 · 𝑥) + 𝑦)) = (𝐺‘((𝑟 · 𝑥) + 𝑦)))
12		fveq1 6669	. . . . . . . . 9 ⊢ (𝑓 = 𝐺 → (𝑓‘𝑥) = (𝐺‘𝑥))
13	12	oveq2d 7172	. . . . . . . 8 ⊢ (𝑓 = 𝐺 → (𝑟 × (𝑓‘𝑥)) = (𝑟 × (𝐺‘𝑥)))
14		fveq1 6669	. . . . . . . 8 ⊢ (𝑓 = 𝐺 → (𝑓‘𝑦) = (𝐺‘𝑦))
15	13, 14	oveq12d 7174	. . . . . . 7 ⊢ (𝑓 = 𝐺 → ((𝑟 × (𝑓‘𝑥)) ⨣ (𝑓‘𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)))
16	11, 15	eqeq12d 2837	. . . . . 6 ⊢ (𝑓 = 𝐺 → ((𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓‘𝑥)) ⨣ (𝑓‘𝑦)) ↔ (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦))))
17	16	2ralbidv 3199	. . . . 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 3197	. . . 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 3680	. . 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 6684	. . . . 5 ⊢ 𝐾 ∈ V
21	1	fvexi 6684	. . . . 5 ⊢ 𝑉 ∈ V
22	20, 21	elmap 8435	. . . 4 ⊢ (𝐺 ∈ (𝐾 ↑m 𝑉) ↔ 𝐺:𝑉⟶𝐾)
23	22	anbi1i 625	. . 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 277	. 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	syl6bb 289	1 ⊢ (𝑊 ∈ 𝑋 → (𝐺 ∈ 𝐹 ↔ (𝐺:𝑉⟶𝐾 ∧ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  {crab 3142  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156   ↑_m cmap 8406  Basecbs 16483  +_gcplusg 16565  ._rcmulr 16566  Scalarcsca 16568   ·_𝑠 cvsca 16569  LFnlclfn 36208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  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 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-map 8408  df-lfl 36209

This theorem is referenced by:  lfli  36212  islfld  36213  lflf  36214  lfl0f  36220  lfladdcl  36222  lflnegcl  36226  lshpkrcl  36267