Theorem nlfnval 28740

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 10017	. . 3 ⊢ ℂ ∈ V
2		ax-hilex 27856	. . 3 ⊢ ℋ ∈ V
3	1, 2	elmap 7886	. 2 ⊢ (𝑇 ∈ (ℂ ↑𝑚 ℋ) ↔ 𝑇: ℋ⟶ℂ)
4		cnvexg 7112	. . . 4 ⊢ (𝑇 ∈ (ℂ ↑𝑚 ℋ) → ◡𝑇 ∈ V)
5		imaexg 7103	. . . 4 ⊢ (◡𝑇 ∈ V → (◡𝑇 “ {0}) ∈ V)
6	4, 5	syl 17	. . 3 ⊢ (𝑇 ∈ (ℂ ↑𝑚 ℋ) → (◡𝑇 “ {0}) ∈ V)
7		cnveq 5296	. . . . 5 ⊢ (𝑡 = 𝑇 → ◡𝑡 = ◡𝑇)
8	7	imaeq1d 5465	. . . 4 ⊢ (𝑡 = 𝑇 → (◡𝑡 “ {0}) = (◡𝑇 “ {0}))
9		df-nlfn 28705	. . . 4 ⊢ null = (𝑡 ∈ (ℂ ↑𝑚 ℋ) ↦ (◡𝑡 “ {0}))
10	8, 9	fvmptg 6280	. . 3 ⊢ ((𝑇 ∈ (ℂ ↑𝑚 ℋ) ∧ (◡𝑇 “ {0}) ∈ V) → (null‘𝑇) = (◡𝑇 “ {0}))
11	6, 10	mpdan 702	. 2 ⊢ (𝑇 ∈ (ℂ ↑𝑚 ℋ) → (null‘𝑇) = (◡𝑇 “ {0}))
12	3, 11	sylbir 225	1 ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) = (◡𝑇 “ {0}))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  Vcvv 3200  {csn 4177  ^◡ccnv 5113   “ cima 5117  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  ℂcc 9934  0cc0 9936   ℋchil 27776  nullcnl 27809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-hilex 27856

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-nlfn 28705

This theorem is referenced by:  elnlfn  28787  nlelshi  28919