Theorem elnlfn 31830

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 31783	. . . . . 6 ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) = (◡𝑇 “ {0}))
2		cnvimass 6042	. . . . . 6 ⊢ (◡𝑇 “ {0}) ⊆ dom 𝑇
3	1, 2	eqsstrdi 3988	. . . . 5 ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) ⊆ dom 𝑇)
4		fdm 6679	. . . . 5 ⊢ (𝑇: ℋ⟶ℂ → dom 𝑇 = ℋ)
5	3, 4	sseqtrd 3980	. . . 4 ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) ⊆ ℋ)
6	5	sseld 3942	. . 3 ⊢ (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) → 𝐴 ∈ ℋ))
7	6	pm4.71rd 562	. 2 ⊢ (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ∈ (null‘𝑇))))
8	1	eleq2d 2814	. . . . 5 ⊢ (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ 𝐴 ∈ (◡𝑇 “ {0})))
9	8	adantr 480	. . . 4 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (null‘𝑇) ↔ 𝐴 ∈ (◡𝑇 “ {0})))
10		ffn 6670	. . . . 5 ⊢ (𝑇: ℋ⟶ℂ → 𝑇 Fn ℋ)
11		eleq1 2816	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (◡𝑇 “ {0}) ↔ 𝐴 ∈ (◡𝑇 “ {0})))
12		fveqeq2 6849	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝑇‘𝑥) = 0 ↔ (𝑇‘𝐴) = 0))
13	11, 12	bibi12d 345	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝑥) = 0) ↔ (𝐴 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝐴) = 0)))
14	13	imbi2d 340	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑇 Fn ℋ → (𝑥 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝑥) = 0)) ↔ (𝑇 Fn ℋ → (𝐴 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝐴) = 0))))
15		0cn 11142	. . . . . . . . 9 ⊢ 0 ∈ ℂ
16		vex 3448	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
17	16	eliniseg 6054	. . . . . . . . 9 ⊢ (0 ∈ ℂ → (𝑥 ∈ (◡𝑇 “ {0}) ↔ 𝑥𝑇0))
18	15, 17	ax-mp 5	. . . . . . . 8 ⊢ (𝑥 ∈ (◡𝑇 “ {0}) ↔ 𝑥𝑇0)
19		fnbrfvb 6893	. . . . . . . 8 ⊢ ((𝑇 Fn ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) = 0 ↔ 𝑥𝑇0))
20	18, 19	bitr4id 290	. . . . . . 7 ⊢ ((𝑇 Fn ℋ ∧ 𝑥 ∈ ℋ) → (𝑥 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝑥) = 0))
21	20	expcom 413	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑇 Fn ℋ → (𝑥 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝑥) = 0)))
22	14, 21	vtoclga 3540	. . . . 5 ⊢ (𝐴 ∈ ℋ → (𝑇 Fn ℋ → (𝐴 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝐴) = 0)))
23	10, 22	mpan9 506	. . . 4 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝐴) = 0))
24	9, 23	bitrd 279	. . 3 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (null‘𝑇) ↔ (𝑇‘𝐴) = 0))
25	24	pm5.32da 579	. 2 ⊢ (𝑇: ℋ⟶ℂ → ((𝐴 ∈ ℋ ∧ 𝐴 ∈ (null‘𝑇)) ↔ (𝐴 ∈ ℋ ∧ (𝑇‘𝐴) = 0)))
26	7, 25	bitrd 279	1 ⊢ (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ (𝑇‘𝐴) = 0)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {csn 4585   class class class wbr 5102  ^◡ccnv 5630  dom cdm 5631   “ cima 5634   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  ℂcc 11042  0cc0 11044   ℋchba 30821  nullcnl 30854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-mulcl 11106  ax-i2m1 11112  ax-hilex 30901

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-map 8778  df-nlfn 31748

This theorem is referenced by:  elnlfn2  31831  nlelshi  31962  nlelchi  31963  riesz3i  31964