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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 10621 | . . 3 ⊢ ℂ ∈ V | |
2 | ax-hilex 28779 | . . 3 ⊢ ℋ ∈ V | |
3 | 1, 2 | elmap 8438 | . 2 ⊢ (𝑇 ∈ (ℂ ↑m ℋ) ↔ 𝑇: ℋ⟶ℂ) |
4 | cnvexg 7632 | . . . 4 ⊢ (𝑇 ∈ (ℂ ↑m ℋ) → ◡𝑇 ∈ V) | |
5 | imaexg 7623 | . . . 4 ⊢ (◡𝑇 ∈ V → (◡𝑇 “ {0}) ∈ V) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝑇 ∈ (ℂ ↑m ℋ) → (◡𝑇 “ {0}) ∈ V) |
7 | cnveq 5747 | . . . . 5 ⊢ (𝑡 = 𝑇 → ◡𝑡 = ◡𝑇) | |
8 | 7 | imaeq1d 5931 | . . . 4 ⊢ (𝑡 = 𝑇 → (◡𝑡 “ {0}) = (◡𝑇 “ {0})) |
9 | df-nlfn 29626 | . . . 4 ⊢ null = (𝑡 ∈ (ℂ ↑m ℋ) ↦ (◡𝑡 “ {0})) | |
10 | 8, 9 | fvmptg 6769 | . . 3 ⊢ ((𝑇 ∈ (ℂ ↑m ℋ) ∧ (◡𝑇 “ {0}) ∈ V) → (null‘𝑇) = (◡𝑇 “ {0})) |
11 | 6, 10 | mpdan 685 | . 2 ⊢ (𝑇 ∈ (ℂ ↑m ℋ) → (null‘𝑇) = (◡𝑇 “ {0})) |
12 | 3, 11 | sylbir 237 | 1 ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) = (◡𝑇 “ {0})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 Vcvv 3497 {csn 4570 ◡ccnv 5557 “ cima 5561 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ↑m cmap 8409 ℂcc 10538 0cc0 10540 ℋchba 28699 nullcnl 28732 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-hilex 28779 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-map 8411 df-nlfn 29626 |
This theorem is referenced by: elnlfn 29708 nlelshi 29840 |
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