Theorem nlfnval 32492

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 11288	. . 3 ⊢ ℂ ∈ V
2		ax-hilex 31610	. . 3 ⊢ ℋ ∈ V
3	1, 2	elmap 8907	. 2 ⊢ (𝑇 ∈ (ℂ ↑m ℋ) ↔ 𝑇: ℋ⟶ℂ)
4		cnvexg 7940	. . . 4 ⊢ (𝑇 ∈ (ℂ ↑m ℋ) → ◡𝑇 ∈ V)
5		imaexg 7929	. . . 4 ⊢ (◡𝑇 ∈ V → (◡𝑇 “ {0}) ∈ V)
6	4, 5	syl 18	. . 3 ⊢ (𝑇 ∈ (ℂ ↑m ℋ) → (◡𝑇 “ {0}) ∈ V)
7		cnveq 5852	. . . . 5 ⊢ (𝑡 = 𝑇 → ◡𝑡 = ◡𝑇)
8	7	imaeq1d 6053	. . . 4 ⊢ (𝑡 = 𝑇 → (◡𝑡 “ {0}) = (◡𝑇 “ {0}))
9		df-nlfn 32457	. . . 4 ⊢ null = (𝑡 ∈ (ℂ ↑m ℋ) ↦ (◡𝑡 “ {0}))
10	8, 9	fvmptg 6996	. . 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 1569   ∈ wcel 2144  Vcvv 3450  {csn 4584  ^◡ccnv 5650   “ cima 5654  ⟶wf 6539  ‘cfv 6543  (class class class)co 7424   ↑_m cmap 8855  ℂcc 11205  0cc0 11207   ℋchba 31530  nullcnl 31563

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-cnex 11263  ax-hilex 31610

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-map 8857  df-nlfn 32457

This theorem is used by:  elnlfn  32539  nlelshi  32671