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| Mirrors > Home > HSE Home > Th. List > elnlfn | Structured version Visualization version GIF version | ||
| Description: Membership in the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| elnlfn | ⊢ (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ (𝑇‘𝐴) = 0))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nlfnval 31900 | . . . . . 6 ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) = (◡𝑇 “ {0})) | |
| 2 | cnvimass 6100 | . . . . . 6 ⊢ (◡𝑇 “ {0}) ⊆ dom 𝑇 | |
| 3 | 1, 2 | eqsstrdi 4028 | . . . . 5 ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) ⊆ dom 𝑇) |
| 4 | fdm 6745 | . . . . 5 ⊢ (𝑇: ℋ⟶ℂ → dom 𝑇 = ℋ) | |
| 5 | 3, 4 | sseqtrd 4020 | . . . 4 ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) ⊆ ℋ) |
| 6 | 5 | sseld 3982 | . . 3 ⊢ (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) → 𝐴 ∈ ℋ)) |
| 7 | 6 | pm4.71rd 562 | . 2 ⊢ (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ∈ (null‘𝑇)))) |
| 8 | 1 | eleq2d 2827 | . . . . 5 ⊢ (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ 𝐴 ∈ (◡𝑇 “ {0}))) |
| 9 | 8 | adantr 480 | . . . 4 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (null‘𝑇) ↔ 𝐴 ∈ (◡𝑇 “ {0}))) |
| 10 | ffn 6736 | . . . . 5 ⊢ (𝑇: ℋ⟶ℂ → 𝑇 Fn ℋ) | |
| 11 | eleq1 2829 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (◡𝑇 “ {0}) ↔ 𝐴 ∈ (◡𝑇 “ {0}))) | |
| 12 | fveqeq2 6915 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝑇‘𝑥) = 0 ↔ (𝑇‘𝐴) = 0)) | |
| 13 | 11, 12 | bibi12d 345 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝑥) = 0) ↔ (𝐴 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝐴) = 0))) |
| 14 | 13 | imbi2d 340 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑇 Fn ℋ → (𝑥 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝑥) = 0)) ↔ (𝑇 Fn ℋ → (𝐴 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝐴) = 0)))) |
| 15 | 0cn 11253 | . . . . . . . . 9 ⊢ 0 ∈ ℂ | |
| 16 | vex 3484 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
| 17 | 16 | eliniseg 6112 | . . . . . . . . 9 ⊢ (0 ∈ ℂ → (𝑥 ∈ (◡𝑇 “ {0}) ↔ 𝑥𝑇0)) |
| 18 | 15, 17 | ax-mp 5 | . . . . . . . 8 ⊢ (𝑥 ∈ (◡𝑇 “ {0}) ↔ 𝑥𝑇0) |
| 19 | fnbrfvb 6959 | . . . . . . . 8 ⊢ ((𝑇 Fn ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) = 0 ↔ 𝑥𝑇0)) | |
| 20 | 18, 19 | bitr4id 290 | . . . . . . 7 ⊢ ((𝑇 Fn ℋ ∧ 𝑥 ∈ ℋ) → (𝑥 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝑥) = 0)) |
| 21 | 20 | expcom 413 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑇 Fn ℋ → (𝑥 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝑥) = 0))) |
| 22 | 14, 21 | vtoclga 3577 | . . . . 5 ⊢ (𝐴 ∈ ℋ → (𝑇 Fn ℋ → (𝐴 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝐴) = 0))) |
| 23 | 10, 22 | mpan9 506 | . . . 4 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝐴) = 0)) |
| 24 | 9, 23 | bitrd 279 | . . 3 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (null‘𝑇) ↔ (𝑇‘𝐴) = 0)) |
| 25 | 24 | pm5.32da 579 | . 2 ⊢ (𝑇: ℋ⟶ℂ → ((𝐴 ∈ ℋ ∧ 𝐴 ∈ (null‘𝑇)) ↔ (𝐴 ∈ ℋ ∧ (𝑇‘𝐴) = 0))) |
| 26 | 7, 25 | bitrd 279 | 1 ⊢ (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ (𝑇‘𝐴) = 0))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {csn 4626 class class class wbr 5143 ◡ccnv 5684 dom cdm 5685 “ cima 5688 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 ℂcc 11153 0cc0 11155 ℋchba 30938 nullcnl 30971 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-mulcl 11217 ax-i2m1 11223 ax-hilex 31018 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-nlfn 31865 |
| This theorem is referenced by: elnlfn2 31948 nlelshi 32079 nlelchi 32080 riesz3i 32081 |
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