Theorem nlfnval 31970

Description: Value of the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
nlfnval	⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) = (◡𝑇 “ {0}))

Proof of Theorem nlfnval

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnex 11110	. . 3 ⊢ ℂ ∈ V
2		ax-hilex 31088	. . 3 ⊢ ℋ ∈ V
3	1, 2	elmap 8809	. 2 ⊢ (𝑇 ∈ (ℂ ↑m ℋ) ↔ 𝑇: ℋ⟶ℂ)
4		cnvexg 7864	. . . 4 ⊢ (𝑇 ∈ (ℂ ↑m ℋ) → ◡𝑇 ∈ V)
5		imaexg 7853	. . . 4 ⊢ (◡𝑇 ∈ V → (◡𝑇 “ {0}) ∈ V)
6	4, 5	syl 17	. . 3 ⊢ (𝑇 ∈ (ℂ ↑m ℋ) → (◡𝑇 “ {0}) ∈ V)
7		cnveq 5815	. . . . 5 ⊢ (𝑡 = 𝑇 → ◡𝑡 = ◡𝑇)
8	7	imaeq1d 6011	. . . 4 ⊢ (𝑡 = 𝑇 → (◡𝑡 “ {0}) = (◡𝑇 “ {0}))
9		df-nlfn 31935	. . . 4 ⊢ null = (𝑡 ∈ (ℂ ↑m ℋ) ↦ (◡𝑡 “ {0}))
10	8, 9	fvmptg 6933	. . 3 ⊢ ((𝑇 ∈ (ℂ ↑m ℋ) ∧ (◡𝑇 “ {0}) ∈ V) → (null‘𝑇) = (◡𝑇 “ {0}))
11	6, 10	mpdan 693	. 2 ⊢ (𝑇 ∈ (ℂ ↑m ℋ) → (null‘𝑇) = (◡𝑇 “ {0}))
12	3, 11	sylbir 236	1 ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) = (◡𝑇 “ {0}))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  Vcvv 3431  {csn 4555  ^◡ccnv 5617   “ cima 5621  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356   ↑_m cmap 8763  ℂcc 11027  0cc0 11029   ℋchba 31008  nullcnl 31041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-hilex 31088

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-map 8765  df-nlfn 31935

This theorem is referenced by:  elnlfn  32017  nlelshi  32149