Theorem elnlfn 29707

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 29660	. . . . . 6 ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) = (◡𝑇 “ {0}))
2		cnvimass 5951	. . . . . 6 ⊢ (◡𝑇 “ {0}) ⊆ dom 𝑇
3	1, 2	eqsstrdi 4023	. . . . 5 ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) ⊆ dom 𝑇)
4		fdm 6524	. . . . 5 ⊢ (𝑇: ℋ⟶ℂ → dom 𝑇 = ℋ)
5	3, 4	sseqtrd 4009	. . . 4 ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) ⊆ ℋ)
6	5	sseld 3968	. . 3 ⊢ (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) → 𝐴 ∈ ℋ))
7	6	pm4.71rd 565	. 2 ⊢ (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ∈ (null‘𝑇))))
8	1	eleq2d 2900	. . . . 5 ⊢ (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ 𝐴 ∈ (◡𝑇 “ {0})))
9	8	adantr 483	. . . 4 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (null‘𝑇) ↔ 𝐴 ∈ (◡𝑇 “ {0})))
10		ffn 6516	. . . . 5 ⊢ (𝑇: ℋ⟶ℂ → 𝑇 Fn ℋ)
11		eleq1 2902	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (◡𝑇 “ {0}) ↔ 𝐴 ∈ (◡𝑇 “ {0})))
12		fveqeq2 6681	. . . . . . . 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		fnbrfvb 6720	. . . . . . . 8 ⊢ ((𝑇 Fn ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) = 0 ↔ 𝑥𝑇0))
16		0cn 10635	. . . . . . . . 9 ⊢ 0 ∈ ℂ
17		vex 3499	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
18	17	eliniseg 5960	. . . . . . . . 9 ⊢ (0 ∈ ℂ → (𝑥 ∈ (◡𝑇 “ {0}) ↔ 𝑥𝑇0))
19	16, 18	ax-mp 5	. . . . . . . 8 ⊢ (𝑥 ∈ (◡𝑇 “ {0}) ↔ 𝑥𝑇0)
20	15, 19	syl6rbbr 292	. . . . . . 7 ⊢ ((𝑇 Fn ℋ ∧ 𝑥 ∈ ℋ) → (𝑥 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝑥) = 0))
21	20	expcom 416	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑇 Fn ℋ → (𝑥 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝑥) = 0)))
22	14, 21	vtoclga 3576	. . . . 5 ⊢ (𝐴 ∈ ℋ → (𝑇 Fn ℋ → (𝐴 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝐴) = 0)))
23	10, 22	mpan9 509	. . . 4 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝐴) = 0))
24	9, 23	bitrd 281	. . 3 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (null‘𝑇) ↔ (𝑇‘𝐴) = 0))
25	24	pm5.32da 581	. 2 ⊢ (𝑇: ℋ⟶ℂ → ((𝐴 ∈ ℋ ∧ 𝐴 ∈ (null‘𝑇)) ↔ (𝐴 ∈ ℋ ∧ (𝑇‘𝐴) = 0)))
26	7, 25	bitrd 281	1 ⊢ (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ (𝑇‘𝐴) = 0)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {csn 4569   class class class wbr 5068  ^◡ccnv 5556  dom cdm 5557   “ cima 5560   Fn wfn 6352  ⟶wf 6353  ‘cfv 6357  ℂcc 10537  0cc0 10539   ℋchba 28698  nullcnl 28731

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 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-mulcl 10601  ax-i2m1 10607  ax-hilex 28778

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 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-map 8410  df-nlfn 29625

This theorem is referenced by:  elnlfn2  29708  nlelshi  29839  nlelchi  29840  riesz3i  29841