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Mirrors > Home > HSE Home > Th. List > nlfnval | Structured version Visualization version GIF version |
Description: Value of the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nlfnval | ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) = (◡𝑇 “ {0})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnex 11234 | . . 3 ⊢ ℂ ∈ V | |
2 | ax-hilex 31028 | . . 3 ⊢ ℋ ∈ V | |
3 | 1, 2 | elmap 8910 | . 2 ⊢ (𝑇 ∈ (ℂ ↑m ℋ) ↔ 𝑇: ℋ⟶ℂ) |
4 | cnvexg 7947 | . . . 4 ⊢ (𝑇 ∈ (ℂ ↑m ℋ) → ◡𝑇 ∈ V) | |
5 | imaexg 7936 | . . . 4 ⊢ (◡𝑇 ∈ V → (◡𝑇 “ {0}) ∈ V) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝑇 ∈ (ℂ ↑m ℋ) → (◡𝑇 “ {0}) ∈ V) |
7 | cnveq 5887 | . . . . 5 ⊢ (𝑡 = 𝑇 → ◡𝑡 = ◡𝑇) | |
8 | 7 | imaeq1d 6079 | . . . 4 ⊢ (𝑡 = 𝑇 → (◡𝑡 “ {0}) = (◡𝑇 “ {0})) |
9 | df-nlfn 31875 | . . . 4 ⊢ null = (𝑡 ∈ (ℂ ↑m ℋ) ↦ (◡𝑡 “ {0})) | |
10 | 8, 9 | fvmptg 7014 | . . 3 ⊢ ((𝑇 ∈ (ℂ ↑m ℋ) ∧ (◡𝑇 “ {0}) ∈ V) → (null‘𝑇) = (◡𝑇 “ {0})) |
11 | 6, 10 | mpdan 687 | . 2 ⊢ (𝑇 ∈ (ℂ ↑m ℋ) → (null‘𝑇) = (◡𝑇 “ {0})) |
12 | 3, 11 | sylbir 235 | 1 ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) = (◡𝑇 “ {0})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 {csn 4631 ◡ccnv 5688 “ cima 5692 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 ℂcc 11151 0cc0 11153 ℋchba 30948 nullcnl 30981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-hilex 31028 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 df-nlfn 31875 |
This theorem is referenced by: elnlfn 31957 nlelshi 32089 |
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