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Mirrors > Home > HSE Home > Th. List > isch2 | Structured version Visualization version GIF version |
Description: Closed subspace 𝐻 of a Hilbert space. Definition of [Beran] p. 107. (Contributed by NM, 17-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
isch2 | ⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ∀𝑓∀𝑥((𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isch 29993 | . 2 ⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ( ⇝𝑣 “ (𝐻 ↑m ℕ)) ⊆ 𝐻)) | |
2 | alcom 2157 | . . . . 5 ⊢ (∀𝑓∀𝑥((𝑓 ∈ (𝐻 ↑m ℕ) ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻) ↔ ∀𝑥∀𝑓((𝑓 ∈ (𝐻 ↑m ℕ) ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻)) | |
3 | 19.23v 1946 | . . . . . . . 8 ⊢ (∀𝑓((𝑓 ∈ (𝐻 ↑m ℕ) ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻) ↔ (∃𝑓(𝑓 ∈ (𝐻 ↑m ℕ) ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻)) | |
4 | vex 3448 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
5 | 4 | elima2 6018 | . . . . . . . . 9 ⊢ (𝑥 ∈ ( ⇝𝑣 “ (𝐻 ↑m ℕ)) ↔ ∃𝑓(𝑓 ∈ (𝐻 ↑m ℕ) ∧ 𝑓 ⇝𝑣 𝑥)) |
6 | 5 | imbi1i 350 | . . . . . . . 8 ⊢ ((𝑥 ∈ ( ⇝𝑣 “ (𝐻 ↑m ℕ)) → 𝑥 ∈ 𝐻) ↔ (∃𝑓(𝑓 ∈ (𝐻 ↑m ℕ) ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻)) |
7 | 3, 6 | bitr4i 278 | . . . . . . 7 ⊢ (∀𝑓((𝑓 ∈ (𝐻 ↑m ℕ) ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻) ↔ (𝑥 ∈ ( ⇝𝑣 “ (𝐻 ↑m ℕ)) → 𝑥 ∈ 𝐻)) |
8 | 7 | albii 1822 | . . . . . 6 ⊢ (∀𝑥∀𝑓((𝑓 ∈ (𝐻 ↑m ℕ) ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻) ↔ ∀𝑥(𝑥 ∈ ( ⇝𝑣 “ (𝐻 ↑m ℕ)) → 𝑥 ∈ 𝐻)) |
9 | dfss2 3929 | . . . . . 6 ⊢ (( ⇝𝑣 “ (𝐻 ↑m ℕ)) ⊆ 𝐻 ↔ ∀𝑥(𝑥 ∈ ( ⇝𝑣 “ (𝐻 ↑m ℕ)) → 𝑥 ∈ 𝐻)) | |
10 | 8, 9 | bitr4i 278 | . . . . 5 ⊢ (∀𝑥∀𝑓((𝑓 ∈ (𝐻 ↑m ℕ) ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻) ↔ ( ⇝𝑣 “ (𝐻 ↑m ℕ)) ⊆ 𝐻) |
11 | 2, 10 | bitri 275 | . . . 4 ⊢ (∀𝑓∀𝑥((𝑓 ∈ (𝐻 ↑m ℕ) ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻) ↔ ( ⇝𝑣 “ (𝐻 ↑m ℕ)) ⊆ 𝐻) |
12 | nnex 12118 | . . . . . . . 8 ⊢ ℕ ∈ V | |
13 | elmapg 8737 | . . . . . . . 8 ⊢ ((𝐻 ∈ Sℋ ∧ ℕ ∈ V) → (𝑓 ∈ (𝐻 ↑m ℕ) ↔ 𝑓:ℕ⟶𝐻)) | |
14 | 12, 13 | mpan2 690 | . . . . . . 7 ⊢ (𝐻 ∈ Sℋ → (𝑓 ∈ (𝐻 ↑m ℕ) ↔ 𝑓:ℕ⟶𝐻)) |
15 | 14 | anbi1d 631 | . . . . . 6 ⊢ (𝐻 ∈ Sℋ → ((𝑓 ∈ (𝐻 ↑m ℕ) ∧ 𝑓 ⇝𝑣 𝑥) ↔ (𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 𝑥))) |
16 | 15 | imbi1d 342 | . . . . 5 ⊢ (𝐻 ∈ Sℋ → (((𝑓 ∈ (𝐻 ↑m ℕ) ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻) ↔ ((𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻))) |
17 | 16 | 2albidv 1927 | . . . 4 ⊢ (𝐻 ∈ Sℋ → (∀𝑓∀𝑥((𝑓 ∈ (𝐻 ↑m ℕ) ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻) ↔ ∀𝑓∀𝑥((𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻))) |
18 | 11, 17 | bitr3id 285 | . . 3 ⊢ (𝐻 ∈ Sℋ → (( ⇝𝑣 “ (𝐻 ↑m ℕ)) ⊆ 𝐻 ↔ ∀𝑓∀𝑥((𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻))) |
19 | 18 | pm5.32i 576 | . 2 ⊢ ((𝐻 ∈ Sℋ ∧ ( ⇝𝑣 “ (𝐻 ↑m ℕ)) ⊆ 𝐻) ↔ (𝐻 ∈ Sℋ ∧ ∀𝑓∀𝑥((𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻))) |
20 | 1, 19 | bitri 275 | 1 ⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ∀𝑓∀𝑥((𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∀wal 1540 ∃wex 1782 ∈ wcel 2107 Vcvv 3444 ⊆ wss 3909 class class class wbr 5104 “ cima 5635 ⟶wf 6490 (class class class)co 7352 ↑m cmap 8724 ℕcn 12112 ⇝𝑣 chli 29698 Sℋ csh 29699 Cℋ cch 29700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-1cn 11068 ax-addcl 11070 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7796 df-2nd 7915 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-map 8726 df-nn 12113 df-ch 29992 |
This theorem is referenced by: chlimi 30005 isch3 30012 helch 30014 hsn0elch 30019 chintcli 30102 chscl 30412 nlelchi 30832 hmopidmchi 30922 |
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