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Mirrors > Home > HSE Home > Th. List > hsn0elch | Structured version Visualization version GIF version |
Description: The zero subspace belongs to the set of closed subspaces of Hilbert space. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hsn0elch | ⊢ {0ℎ} ∈ Cℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hv0cl 28783 | . . . . 5 ⊢ 0ℎ ∈ ℋ | |
2 | snssi 4744 | . . . . 5 ⊢ (0ℎ ∈ ℋ → {0ℎ} ⊆ ℋ) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ {0ℎ} ⊆ ℋ |
4 | 1 | elexi 3516 | . . . . 5 ⊢ 0ℎ ∈ V |
5 | 4 | snid 4604 | . . . 4 ⊢ 0ℎ ∈ {0ℎ} |
6 | 3, 5 | pm3.2i 473 | . . 3 ⊢ ({0ℎ} ⊆ ℋ ∧ 0ℎ ∈ {0ℎ}) |
7 | velsn 4586 | . . . . . 6 ⊢ (𝑥 ∈ {0ℎ} ↔ 𝑥 = 0ℎ) | |
8 | velsn 4586 | . . . . . 6 ⊢ (𝑦 ∈ {0ℎ} ↔ 𝑦 = 0ℎ) | |
9 | oveq12 7168 | . . . . . . . 8 ⊢ ((𝑥 = 0ℎ ∧ 𝑦 = 0ℎ) → (𝑥 +ℎ 𝑦) = (0ℎ +ℎ 0ℎ)) | |
10 | 1 | hvaddid2i 28809 | . . . . . . . 8 ⊢ (0ℎ +ℎ 0ℎ) = 0ℎ |
11 | 9, 10 | syl6eq 2875 | . . . . . . 7 ⊢ ((𝑥 = 0ℎ ∧ 𝑦 = 0ℎ) → (𝑥 +ℎ 𝑦) = 0ℎ) |
12 | ovex 7192 | . . . . . . . 8 ⊢ (𝑥 +ℎ 𝑦) ∈ V | |
13 | 12 | elsn 4585 | . . . . . . 7 ⊢ ((𝑥 +ℎ 𝑦) ∈ {0ℎ} ↔ (𝑥 +ℎ 𝑦) = 0ℎ) |
14 | 11, 13 | sylibr 236 | . . . . . 6 ⊢ ((𝑥 = 0ℎ ∧ 𝑦 = 0ℎ) → (𝑥 +ℎ 𝑦) ∈ {0ℎ}) |
15 | 7, 8, 14 | syl2anb 599 | . . . . 5 ⊢ ((𝑥 ∈ {0ℎ} ∧ 𝑦 ∈ {0ℎ}) → (𝑥 +ℎ 𝑦) ∈ {0ℎ}) |
16 | 15 | rgen2 3206 | . . . 4 ⊢ ∀𝑥 ∈ {0ℎ}∀𝑦 ∈ {0ℎ} (𝑥 +ℎ 𝑦) ∈ {0ℎ} |
17 | oveq2 7167 | . . . . . . . 8 ⊢ (𝑦 = 0ℎ → (𝑥 ·ℎ 𝑦) = (𝑥 ·ℎ 0ℎ)) | |
18 | hvmul0 28804 | . . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (𝑥 ·ℎ 0ℎ) = 0ℎ) | |
19 | 17, 18 | sylan9eqr 2881 | . . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 = 0ℎ) → (𝑥 ·ℎ 𝑦) = 0ℎ) |
20 | ovex 7192 | . . . . . . . 8 ⊢ (𝑥 ·ℎ 𝑦) ∈ V | |
21 | 20 | elsn 4585 | . . . . . . 7 ⊢ ((𝑥 ·ℎ 𝑦) ∈ {0ℎ} ↔ (𝑥 ·ℎ 𝑦) = 0ℎ) |
22 | 19, 21 | sylibr 236 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 = 0ℎ) → (𝑥 ·ℎ 𝑦) ∈ {0ℎ}) |
23 | 8, 22 | sylan2b 595 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ {0ℎ}) → (𝑥 ·ℎ 𝑦) ∈ {0ℎ}) |
24 | 23 | rgen2 3206 | . . . 4 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ {0ℎ} (𝑥 ·ℎ 𝑦) ∈ {0ℎ} |
25 | 16, 24 | pm3.2i 473 | . . 3 ⊢ (∀𝑥 ∈ {0ℎ}∀𝑦 ∈ {0ℎ} (𝑥 +ℎ 𝑦) ∈ {0ℎ} ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ {0ℎ} (𝑥 ·ℎ 𝑦) ∈ {0ℎ}) |
26 | issh2 28989 | . . 3 ⊢ ({0ℎ} ∈ Sℋ ↔ (({0ℎ} ⊆ ℋ ∧ 0ℎ ∈ {0ℎ}) ∧ (∀𝑥 ∈ {0ℎ}∀𝑦 ∈ {0ℎ} (𝑥 +ℎ 𝑦) ∈ {0ℎ} ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ {0ℎ} (𝑥 ·ℎ 𝑦) ∈ {0ℎ}))) | |
27 | 6, 25, 26 | mpbir2an 709 | . 2 ⊢ {0ℎ} ∈ Sℋ |
28 | 4 | fconst2 6970 | . . . . . 6 ⊢ (𝑓:ℕ⟶{0ℎ} ↔ 𝑓 = (ℕ × {0ℎ})) |
29 | hlim0 29015 | . . . . . . 7 ⊢ (ℕ × {0ℎ}) ⇝𝑣 0ℎ | |
30 | breq1 5072 | . . . . . . 7 ⊢ (𝑓 = (ℕ × {0ℎ}) → (𝑓 ⇝𝑣 0ℎ ↔ (ℕ × {0ℎ}) ⇝𝑣 0ℎ)) | |
31 | 29, 30 | mpbiri 260 | . . . . . 6 ⊢ (𝑓 = (ℕ × {0ℎ}) → 𝑓 ⇝𝑣 0ℎ) |
32 | 28, 31 | sylbi 219 | . . . . 5 ⊢ (𝑓:ℕ⟶{0ℎ} → 𝑓 ⇝𝑣 0ℎ) |
33 | hlimuni 29018 | . . . . . 6 ⊢ ((𝑓 ⇝𝑣 0ℎ ∧ 𝑓 ⇝𝑣 𝑥) → 0ℎ = 𝑥) | |
34 | 33 | eleq1d 2900 | . . . . 5 ⊢ ((𝑓 ⇝𝑣 0ℎ ∧ 𝑓 ⇝𝑣 𝑥) → (0ℎ ∈ {0ℎ} ↔ 𝑥 ∈ {0ℎ})) |
35 | 32, 34 | sylan 582 | . . . 4 ⊢ ((𝑓:ℕ⟶{0ℎ} ∧ 𝑓 ⇝𝑣 𝑥) → (0ℎ ∈ {0ℎ} ↔ 𝑥 ∈ {0ℎ})) |
36 | 5, 35 | mpbii 235 | . . 3 ⊢ ((𝑓:ℕ⟶{0ℎ} ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ {0ℎ}) |
37 | 36 | gen2 1796 | . 2 ⊢ ∀𝑓∀𝑥((𝑓:ℕ⟶{0ℎ} ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ {0ℎ}) |
38 | isch2 29003 | . 2 ⊢ ({0ℎ} ∈ Cℋ ↔ ({0ℎ} ∈ Sℋ ∧ ∀𝑓∀𝑥((𝑓:ℕ⟶{0ℎ} ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ {0ℎ}))) | |
39 | 27, 37, 38 | mpbir2an 709 | 1 ⊢ {0ℎ} ∈ Cℋ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1534 = wceq 1536 ∈ wcel 2113 ∀wral 3141 ⊆ wss 3939 {csn 4570 class class class wbr 5069 × cxp 5556 ⟶wf 6354 (class class class)co 7159 ℂcc 10538 ℕcn 11641 ℋchba 28699 +ℎ cva 28700 ·ℎ csm 28701 0ℎc0v 28704 ⇝𝑣 chli 28707 Sℋ csh 28708 Cℋ cch 28709 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 ax-hilex 28779 ax-hfvadd 28780 ax-hvcom 28781 ax-hvass 28782 ax-hv0cl 28783 ax-hvaddid 28784 ax-hfvmul 28785 ax-hvmulid 28786 ax-hvmulass 28787 ax-hvdistr1 28788 ax-hvdistr2 28789 ax-hvmul0 28790 ax-hfi 28859 ax-his1 28862 ax-his2 28863 ax-his3 28864 ax-his4 28865 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-map 8411 df-pm 8412 df-en 8513 df-dom 8514 df-sdom 8515 df-sup 8909 df-inf 8910 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-icc 12748 df-seq 13373 df-exp 13433 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-topgen 16720 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-top 21505 df-topon 21522 df-bases 21557 df-lm 21840 df-haus 21926 df-grpo 28273 df-gid 28274 df-ginv 28275 df-gdiv 28276 df-ablo 28325 df-vc 28339 df-nv 28372 df-va 28375 df-ba 28376 df-sm 28377 df-0v 28378 df-vs 28379 df-nmcv 28380 df-ims 28381 df-hnorm 28748 df-hvsub 28751 df-hlim 28752 df-sh 28987 df-ch 29001 |
This theorem is referenced by: h0elch 29035 h1de2ctlem 29335 |
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