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Mirrors > Home > HSE Home > Th. List > chlimi | Structured version Visualization version GIF version |
Description: The limit property of a closed subspace of a Hilbert space. (Contributed by NM, 14-Sep-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
chlim.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
chlimi | ⊢ ((𝐻 ∈ Cℋ ∧ 𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝐴) → 𝐴 ∈ 𝐻) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isch2 29000 | . . . 4 ⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ∀𝑓∀𝑥((𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻))) | |
2 | 1 | simprbi 499 | . . 3 ⊢ (𝐻 ∈ Cℋ → ∀𝑓∀𝑥((𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻)) |
3 | nnex 11644 | . . . . . . 7 ⊢ ℕ ∈ V | |
4 | fex 6989 | . . . . . . 7 ⊢ ((𝐹:ℕ⟶𝐻 ∧ ℕ ∈ V) → 𝐹 ∈ V) | |
5 | 3, 4 | mpan2 689 | . . . . . 6 ⊢ (𝐹:ℕ⟶𝐻 → 𝐹 ∈ V) |
6 | 5 | adantr 483 | . . . . 5 ⊢ ((𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝐴) → 𝐹 ∈ V) |
7 | feq1 6495 | . . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (𝑓:ℕ⟶𝐻 ↔ 𝐹:ℕ⟶𝐻)) | |
8 | breq1 5069 | . . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (𝑓 ⇝𝑣 𝑥 ↔ 𝐹 ⇝𝑣 𝑥)) | |
9 | 7, 8 | anbi12d 632 | . . . . . . . . 9 ⊢ (𝑓 = 𝐹 → ((𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 𝑥) ↔ (𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝑥))) |
10 | 9 | imbi1d 344 | . . . . . . . 8 ⊢ (𝑓 = 𝐹 → (((𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻) ↔ ((𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻))) |
11 | 10 | albidv 1921 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (∀𝑥((𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻) ↔ ∀𝑥((𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻))) |
12 | 11 | spcgv 3595 | . . . . . 6 ⊢ (𝐹 ∈ V → (∀𝑓∀𝑥((𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻) → ∀𝑥((𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻))) |
13 | chlim.1 | . . . . . . 7 ⊢ 𝐴 ∈ V | |
14 | breq2 5070 | . . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝐹 ⇝𝑣 𝑥 ↔ 𝐹 ⇝𝑣 𝐴)) | |
15 | 14 | anbi2d 630 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝑥) ↔ (𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝐴))) |
16 | eleq1 2900 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐻 ↔ 𝐴 ∈ 𝐻)) | |
17 | 15, 16 | imbi12d 347 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (((𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻) ↔ ((𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝐴) → 𝐴 ∈ 𝐻))) |
18 | 13, 17 | spcv 3606 | . . . . . 6 ⊢ (∀𝑥((𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻) → ((𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝐴) → 𝐴 ∈ 𝐻)) |
19 | 12, 18 | syl6 35 | . . . . 5 ⊢ (𝐹 ∈ V → (∀𝑓∀𝑥((𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻) → ((𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝐴) → 𝐴 ∈ 𝐻))) |
20 | 6, 19 | syl 17 | . . . 4 ⊢ ((𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝐴) → (∀𝑓∀𝑥((𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻) → ((𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝐴) → 𝐴 ∈ 𝐻))) |
21 | 20 | pm2.43b 55 | . . 3 ⊢ (∀𝑓∀𝑥((𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻) → ((𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝐴) → 𝐴 ∈ 𝐻)) |
22 | 2, 21 | syl 17 | . 2 ⊢ (𝐻 ∈ Cℋ → ((𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝐴) → 𝐴 ∈ 𝐻)) |
23 | 22 | 3impib 1112 | 1 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝐴) → 𝐴 ∈ 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∀wal 1535 = wceq 1537 ∈ wcel 2114 Vcvv 3494 class class class wbr 5066 ⟶wf 6351 ℕcn 11638 ⇝𝑣 chli 28704 Sℋ csh 28705 Cℋ cch 28706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-1cn 10595 ax-addcl 10597 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-map 8408 df-nn 11639 df-ch 28998 |
This theorem is referenced by: hhsscms 29055 chintcli 29108 chscllem4 29417 |
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