Theorem nlfnval 32540

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 11320	. . 3 ⊢ ℂ ∈ V
2		ax-hilex 31658	. . 3 ⊢ ℋ ∈ V
3	1, 2	elmap 8937	. 2 ⊢ (𝑇 ∈ (ℂ ↑m ℋ) ↔ 𝑇: ℋ⟶ℂ)
4		cnvexg 7970	. . . 4 ⊢ (𝑇 ∈ (ℂ ↑m ℋ) → ◡𝑇 ∈ V)
5		imaexg 7959	. . . 4 ⊢ (◡𝑇 ∈ V → (◡𝑇 “ {0}) ∈ V)
6	4, 5	syl 18	. . 3 ⊢ (𝑇 ∈ (ℂ ↑m ℋ) → (◡𝑇 “ {0}) ∈ V)
7		cnveq 5863	. . . . 5 ⊢ (𝑡 = 𝑇 → ◡𝑡 = ◡𝑇)
8	7	imaeq1d 6069	. . . 4 ⊢ (𝑡 = 𝑇 → (◡𝑡 “ {0}) = (◡𝑇 “ {0}))
9		df-nlfn 32505	. . . 4 ⊢ null = (𝑡 ∈ (ℂ ↑m ℋ) ↦ (◡𝑡 “ {0}))
10	8, 9	fvmptg 7022	. . 3 ⊢ ((𝑇 ∈ (ℂ ↑m ℋ) ∧ (◡𝑇 “ {0}) ∈ V) → (null‘𝑇) = (◡𝑇 “ {0}))
11	6, 10	mpdan 700	. 2 ⊢ (𝑇 ∈ (ℂ ↑m ℋ) → (null‘𝑇) = (◡𝑇 “ {0}))
12	3, 11	sylbir 238	1 ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) = (◡𝑇 “ {0}))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454  {csn 4588  ^◡ccnv 5657   “ cima 5661  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451   ↑_m cmap 8885  ℂcc 11237  0cc0 11239   ℋchba 31578  nullcnl 31611

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-hilex 31658

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-map 8887  df-nlfn 32505

This theorem is used by:  elnlfn  32587  nlelshi  32719