Theorem elnlfn 32351

Description: Membership in the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
elnlfn	⊢ (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ (𝑇‘𝐴) = 0)))

Proof of Theorem elnlfn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nlfnval 32304	. . . . . 6 ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) = (◡𝑇 “ {0}))
2		cnvimass 6086	. . . . . 6 ⊢ (◡𝑇 “ {0}) ⊆ dom 𝑇
3	1, 2	eqsstrdi 3982	. . . . 5 ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) ⊆ dom 𝑇)
4		fdm 6719	. . . . 5 ⊢ (𝑇: ℋ⟶ℂ → dom 𝑇 = ℋ)
5	3, 4	sseqtrd 3974	. . . 4 ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) ⊆ ℋ)
6	5	sseld 3937	. . 3 ⊢ (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) → 𝐴 ∈ ℋ))
7	6	pm4.71rd 572	. 2 ⊢ (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ∈ (null‘𝑇))))
8	1	eleq2d 2851	. . . . 5 ⊢ (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ 𝐴 ∈ (◡𝑇 “ {0})))
9	8	adantr 486	. . . 4 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (null‘𝑇) ↔ 𝐴 ∈ (◡𝑇 “ {0})))
10		ffn 6709	. . . . 5 ⊢ (𝑇: ℋ⟶ℂ → 𝑇 Fn ℋ)
11		eleq1 2853	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (◡𝑇 “ {0}) ↔ 𝐴 ∈ (◡𝑇 “ {0})))
12		fveqeq2 6894	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝑇‘𝑥) = 0 ↔ (𝑇‘𝐴) = 0))
13	11, 12	bibi12d 348	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝑥) = 0) ↔ (𝐴 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝐴) = 0)))
14	13	imbi2d 343	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑇 Fn ℋ → (𝑥 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝑥) = 0)) ↔ (𝑇 Fn ℋ → (𝐴 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝐴) = 0))))
15		0cn 11213	. . . . . . . . 9 ⊢ 0 ∈ ℂ
16		vex 3461	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
17	16	eliniseg 6098	. . . . . . . . 9 ⊢ (0 ∈ ℂ → (𝑥 ∈ (◡𝑇 “ {0}) ↔ 𝑥𝑇0))
18	15, 17	ax-mp 5	. . . . . . . 8 ⊢ (𝑥 ∈ (◡𝑇 “ {0}) ↔ 𝑥𝑇0)
19		fnbrfvb 6935	. . . . . . . 8 ⊢ ((𝑇 Fn ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) = 0 ↔ 𝑥𝑇0))
20	18, 19	bitr4id 293	. . . . . . 7 ⊢ ((𝑇 Fn ℋ ∧ 𝑥 ∈ ℋ) → (𝑥 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝑥) = 0))
21	20	expcom 419	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑇 Fn ℋ → (𝑥 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝑥) = 0)))
22	14, 21	vtoclga 3543	. . . . 5 ⊢ (𝐴 ∈ ℋ → (𝑇 Fn ℋ → (𝐴 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝐴) = 0)))
23	10, 22	mpan9 516	. . . 4 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝐴) = 0))
24	9, 23	bitrd 282	. . 3 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (null‘𝑇) ↔ (𝑇‘𝐴) = 0))
25	24	pm5.32da 590	. 2 ⊢ (𝑇: ℋ⟶ℂ → ((𝐴 ∈ ℋ ∧ 𝐴 ∈ (null‘𝑇)) ↔ (𝐴 ∈ ℋ ∧ (𝑇‘𝐴) = 0)))
26	7, 25	bitrd 282	1 ⊢ (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ (𝑇‘𝐴) = 0)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {csn 4591   class class class wbr 5111  ^◡ccnv 5662  dom cdm 5663   “ cima 5666   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  ℂcc 11113  0cc0 11115   ℋchba 31342  nullcnl 31375

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11171  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-mulcl 11177  ax-i2m1 11183  ax-hilex 31422

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-map 8832  df-nlfn 32269

This theorem is used by:  elnlfn2  32352  nlelshi  32483  nlelchi  32484  riesz3i  32485