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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 27464 | . . . 4 ⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ∀𝑓∀𝑥((𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻))) | |
2 | 1 | simprbi 479 | . . 3 ⊢ (𝐻 ∈ Cℋ → ∀𝑓∀𝑥((𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻)) |
3 | nnex 10903 | . . . . . . 7 ⊢ ℕ ∈ V | |
4 | fex 6394 | . . . . . . 7 ⊢ ((𝐹:ℕ⟶𝐻 ∧ ℕ ∈ V) → 𝐹 ∈ V) | |
5 | 3, 4 | mpan2 703 | . . . . . 6 ⊢ (𝐹:ℕ⟶𝐻 → 𝐹 ∈ V) |
6 | 5 | adantr 480 | . . . . 5 ⊢ ((𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝐴) → 𝐹 ∈ V) |
7 | feq1 5939 | . . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (𝑓:ℕ⟶𝐻 ↔ 𝐹:ℕ⟶𝐻)) | |
8 | breq1 4586 | . . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (𝑓 ⇝𝑣 𝑥 ↔ 𝐹 ⇝𝑣 𝑥)) | |
9 | 7, 8 | anbi12d 743 | . . . . . . . . 9 ⊢ (𝑓 = 𝐹 → ((𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 𝑥) ↔ (𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝑥))) |
10 | 9 | imbi1d 330 | . . . . . . . 8 ⊢ (𝑓 = 𝐹 → (((𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻) ↔ ((𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻))) |
11 | 10 | albidv 1836 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (∀𝑥((𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻) ↔ ∀𝑥((𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻))) |
12 | 11 | spcgv 3266 | . . . . . 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 4587 | . . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝐹 ⇝𝑣 𝑥 ↔ 𝐹 ⇝𝑣 𝐴)) | |
15 | 14 | anbi2d 736 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝑥) ↔ (𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝐴))) |
16 | eleq1 2676 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐻 ↔ 𝐴 ∈ 𝐻)) | |
17 | 15, 16 | imbi12d 333 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (((𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻) ↔ ((𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝐴) → 𝐴 ∈ 𝐻))) |
18 | 13, 17 | spcv 3272 | . . . . . 6 ⊢ (∀𝑥((𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻) → ((𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝐴) → 𝐴 ∈ 𝐻)) |
19 | 12, 18 | syl6 34 | . . . . 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 53 | . . 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 1254 | 1 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝐴) → 𝐴 ∈ 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 ∀wal 1473 = wceq 1475 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 ⟶wf 5800 ℕcn 10897 ⇝𝑣 chli 27168 Sℋ csh 27169 Cℋ cch 27170 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-map 7746 df-nn 10898 df-ch 27462 |
This theorem is referenced by: hhsscms 27520 chintcli 27574 chscllem4 27883 |
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