Theorem nlfnval 29433

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 10412	. . 3 ⊢ ℂ ∈ V
2		ax-hilex 28549	. . 3 ⊢ ℋ ∈ V
3	1, 2	elmap 8231	. 2 ⊢ (𝑇 ∈ (ℂ ↑𝑚 ℋ) ↔ 𝑇: ℋ⟶ℂ)
4		cnvexg 7442	. . . 4 ⊢ (𝑇 ∈ (ℂ ↑𝑚 ℋ) → ◡𝑇 ∈ V)
5		imaexg 7433	. . . 4 ⊢ (◡𝑇 ∈ V → (◡𝑇 “ {0}) ∈ V)
6	4, 5	syl 17	. . 3 ⊢ (𝑇 ∈ (ℂ ↑𝑚 ℋ) → (◡𝑇 “ {0}) ∈ V)
7		cnveq 5591	. . . . 5 ⊢ (𝑡 = 𝑇 → ◡𝑡 = ◡𝑇)
8	7	imaeq1d 5767	. . . 4 ⊢ (𝑡 = 𝑇 → (◡𝑡 “ {0}) = (◡𝑇 “ {0}))
9		df-nlfn 29398	. . . 4 ⊢ null = (𝑡 ∈ (ℂ ↑𝑚 ℋ) ↦ (◡𝑡 “ {0}))
10	8, 9	fvmptg 6591	. . 3 ⊢ ((𝑇 ∈ (ℂ ↑𝑚 ℋ) ∧ (◡𝑇 “ {0}) ∈ V) → (null‘𝑇) = (◡𝑇 “ {0}))
11	6, 10	mpdan 674	. 2 ⊢ (𝑇 ∈ (ℂ ↑𝑚 ℋ) → (null‘𝑇) = (◡𝑇 “ {0}))
12	3, 11	sylbir 227	1 ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) = (◡𝑇 “ {0}))

Syntax hints:   → wi 4   = wceq 1507   ∈ wcel 2048  Vcvv 3412  {csn 4439  ^◡ccnv 5403   “ cima 5407  ⟶wf 6182  ‘cfv 6186  (class class class)co 6974   ↑_𝑚 cmap 8202  ℂcc 10329  0cc0 10331   ℋchba 28469  nullcnl 28502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2747  ax-sep 5058  ax-nul 5065  ax-pow 5117  ax-pr 5184  ax-un 7277  ax-cnex 10387  ax-hilex 28549

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2756  df-cleq 2768  df-clel 2843  df-nfc 2915  df-ral 3090  df-rex 3091  df-rab 3094  df-v 3414  df-sbc 3681  df-dif 3831  df-un 3833  df-in 3835  df-ss 3842  df-nul 4178  df-if 4349  df-pw 4422  df-sn 4440  df-pr 4442  df-op 4446  df-uni 4711  df-br 4928  df-opab 4990  df-mpt 5007  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-fv 6194  df-ov 6977  df-oprab 6978  df-mpo 6979  df-map 8204  df-nlfn 29398

This theorem is referenced by:  elnlfn  29480  nlelshi  29612