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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 29652 | . . . . . 6 ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) = (◡𝑇 “ {0})) | |
2 | cnvimass 5943 | . . . . . 6 ⊢ (◡𝑇 “ {0}) ⊆ dom 𝑇 | |
3 | 1, 2 | eqsstrdi 4020 | . . . . 5 ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) ⊆ dom 𝑇) |
4 | fdm 6516 | . . . . 5 ⊢ (𝑇: ℋ⟶ℂ → dom 𝑇 = ℋ) | |
5 | 3, 4 | sseqtrd 4006 | . . . 4 ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) ⊆ ℋ) |
6 | 5 | sseld 3965 | . . 3 ⊢ (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) → 𝐴 ∈ ℋ)) |
7 | 6 | pm4.71rd 565 | . 2 ⊢ (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ∈ (null‘𝑇)))) |
8 | 1 | eleq2d 2898 | . . . . 5 ⊢ (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ 𝐴 ∈ (◡𝑇 “ {0}))) |
9 | 8 | adantr 483 | . . . 4 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (null‘𝑇) ↔ 𝐴 ∈ (◡𝑇 “ {0}))) |
10 | ffn 6508 | . . . . 5 ⊢ (𝑇: ℋ⟶ℂ → 𝑇 Fn ℋ) | |
11 | eleq1 2900 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (◡𝑇 “ {0}) ↔ 𝐴 ∈ (◡𝑇 “ {0}))) | |
12 | fveqeq2 6673 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝑇‘𝑥) = 0 ↔ (𝑇‘𝐴) = 0)) | |
13 | 11, 12 | bibi12d 348 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝑥) = 0) ↔ (𝐴 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝐴) = 0))) |
14 | 13 | imbi2d 343 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑇 Fn ℋ → (𝑥 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝑥) = 0)) ↔ (𝑇 Fn ℋ → (𝐴 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝐴) = 0)))) |
15 | fnbrfvb 6712 | . . . . . . . 8 ⊢ ((𝑇 Fn ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) = 0 ↔ 𝑥𝑇0)) | |
16 | 0cn 10627 | . . . . . . . . 9 ⊢ 0 ∈ ℂ | |
17 | vex 3497 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
18 | 17 | eliniseg 5952 | . . . . . . . . 9 ⊢ (0 ∈ ℂ → (𝑥 ∈ (◡𝑇 “ {0}) ↔ 𝑥𝑇0)) |
19 | 16, 18 | ax-mp 5 | . . . . . . . 8 ⊢ (𝑥 ∈ (◡𝑇 “ {0}) ↔ 𝑥𝑇0) |
20 | 15, 19 | syl6rbbr 292 | . . . . . . 7 ⊢ ((𝑇 Fn ℋ ∧ 𝑥 ∈ ℋ) → (𝑥 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝑥) = 0)) |
21 | 20 | expcom 416 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑇 Fn ℋ → (𝑥 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝑥) = 0))) |
22 | 14, 21 | vtoclga 3573 | . . . . 5 ⊢ (𝐴 ∈ ℋ → (𝑇 Fn ℋ → (𝐴 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝐴) = 0))) |
23 | 10, 22 | mpan9 509 | . . . 4 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (◡𝑇 “ {0}) ↔ (𝑇‘𝐴) = 0)) |
24 | 9, 23 | bitrd 281 | . . 3 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (null‘𝑇) ↔ (𝑇‘𝐴) = 0)) |
25 | 24 | pm5.32da 581 | . 2 ⊢ (𝑇: ℋ⟶ℂ → ((𝐴 ∈ ℋ ∧ 𝐴 ∈ (null‘𝑇)) ↔ (𝐴 ∈ ℋ ∧ (𝑇‘𝐴) = 0))) |
26 | 7, 25 | bitrd 281 | 1 ⊢ (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ (𝑇‘𝐴) = 0))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {csn 4560 class class class wbr 5058 ◡ccnv 5548 dom cdm 5549 “ cima 5552 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 ℂcc 10529 0cc0 10531 ℋchba 28690 nullcnl 28723 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-mulcl 10593 ax-i2m1 10599 ax-hilex 28770 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-nlfn 29617 |
This theorem is referenced by: elnlfn2 29700 nlelshi 29831 nlelchi 29832 riesz3i 29833 |
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