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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 27860 | . . . . 5 ⊢ 0ℎ ∈ ℋ | |
2 | snssi 4339 | . . . . 5 ⊢ (0ℎ ∈ ℋ → {0ℎ} ⊆ ℋ) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ {0ℎ} ⊆ ℋ |
4 | 1 | elexi 3213 | . . . . 5 ⊢ 0ℎ ∈ V |
5 | 4 | snid 4208 | . . . 4 ⊢ 0ℎ ∈ {0ℎ} |
6 | 3, 5 | pm3.2i 471 | . . 3 ⊢ ({0ℎ} ⊆ ℋ ∧ 0ℎ ∈ {0ℎ}) |
7 | velsn 4193 | . . . . . 6 ⊢ (𝑥 ∈ {0ℎ} ↔ 𝑥 = 0ℎ) | |
8 | velsn 4193 | . . . . . 6 ⊢ (𝑦 ∈ {0ℎ} ↔ 𝑦 = 0ℎ) | |
9 | oveq12 6659 | . . . . . . . 8 ⊢ ((𝑥 = 0ℎ ∧ 𝑦 = 0ℎ) → (𝑥 +ℎ 𝑦) = (0ℎ +ℎ 0ℎ)) | |
10 | 1 | hvaddid2i 27886 | . . . . . . . 8 ⊢ (0ℎ +ℎ 0ℎ) = 0ℎ |
11 | 9, 10 | syl6eq 2672 | . . . . . . 7 ⊢ ((𝑥 = 0ℎ ∧ 𝑦 = 0ℎ) → (𝑥 +ℎ 𝑦) = 0ℎ) |
12 | ovex 6678 | . . . . . . . 8 ⊢ (𝑥 +ℎ 𝑦) ∈ V | |
13 | 12 | elsn 4192 | . . . . . . 7 ⊢ ((𝑥 +ℎ 𝑦) ∈ {0ℎ} ↔ (𝑥 +ℎ 𝑦) = 0ℎ) |
14 | 11, 13 | sylibr 224 | . . . . . 6 ⊢ ((𝑥 = 0ℎ ∧ 𝑦 = 0ℎ) → (𝑥 +ℎ 𝑦) ∈ {0ℎ}) |
15 | 7, 8, 14 | syl2anb 496 | . . . . 5 ⊢ ((𝑥 ∈ {0ℎ} ∧ 𝑦 ∈ {0ℎ}) → (𝑥 +ℎ 𝑦) ∈ {0ℎ}) |
16 | 15 | rgen2a 2977 | . . . 4 ⊢ ∀𝑥 ∈ {0ℎ}∀𝑦 ∈ {0ℎ} (𝑥 +ℎ 𝑦) ∈ {0ℎ} |
17 | oveq2 6658 | . . . . . . . 8 ⊢ (𝑦 = 0ℎ → (𝑥 ·ℎ 𝑦) = (𝑥 ·ℎ 0ℎ)) | |
18 | hvmul0 27881 | . . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (𝑥 ·ℎ 0ℎ) = 0ℎ) | |
19 | 17, 18 | sylan9eqr 2678 | . . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 = 0ℎ) → (𝑥 ·ℎ 𝑦) = 0ℎ) |
20 | ovex 6678 | . . . . . . . 8 ⊢ (𝑥 ·ℎ 𝑦) ∈ V | |
21 | 20 | elsn 4192 | . . . . . . 7 ⊢ ((𝑥 ·ℎ 𝑦) ∈ {0ℎ} ↔ (𝑥 ·ℎ 𝑦) = 0ℎ) |
22 | 19, 21 | sylibr 224 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 = 0ℎ) → (𝑥 ·ℎ 𝑦) ∈ {0ℎ}) |
23 | 8, 22 | sylan2b 492 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ {0ℎ}) → (𝑥 ·ℎ 𝑦) ∈ {0ℎ}) |
24 | 23 | rgen2 2975 | . . . 4 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ {0ℎ} (𝑥 ·ℎ 𝑦) ∈ {0ℎ} |
25 | 16, 24 | pm3.2i 471 | . . 3 ⊢ (∀𝑥 ∈ {0ℎ}∀𝑦 ∈ {0ℎ} (𝑥 +ℎ 𝑦) ∈ {0ℎ} ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ {0ℎ} (𝑥 ·ℎ 𝑦) ∈ {0ℎ}) |
26 | issh2 28066 | . . 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 955 | . 2 ⊢ {0ℎ} ∈ Sℋ |
28 | 4 | fconst2 6470 | . . . . . 6 ⊢ (𝑓:ℕ⟶{0ℎ} ↔ 𝑓 = (ℕ × {0ℎ})) |
29 | hlim0 28092 | . . . . . . 7 ⊢ (ℕ × {0ℎ}) ⇝𝑣 0ℎ | |
30 | breq1 4656 | . . . . . . 7 ⊢ (𝑓 = (ℕ × {0ℎ}) → (𝑓 ⇝𝑣 0ℎ ↔ (ℕ × {0ℎ}) ⇝𝑣 0ℎ)) | |
31 | 29, 30 | mpbiri 248 | . . . . . 6 ⊢ (𝑓 = (ℕ × {0ℎ}) → 𝑓 ⇝𝑣 0ℎ) |
32 | 28, 31 | sylbi 207 | . . . . 5 ⊢ (𝑓:ℕ⟶{0ℎ} → 𝑓 ⇝𝑣 0ℎ) |
33 | hlimuni 28095 | . . . . . 6 ⊢ ((𝑓 ⇝𝑣 0ℎ ∧ 𝑓 ⇝𝑣 𝑥) → 0ℎ = 𝑥) | |
34 | 33 | eleq1d 2686 | . . . . 5 ⊢ ((𝑓 ⇝𝑣 0ℎ ∧ 𝑓 ⇝𝑣 𝑥) → (0ℎ ∈ {0ℎ} ↔ 𝑥 ∈ {0ℎ})) |
35 | 32, 34 | sylan 488 | . . . 4 ⊢ ((𝑓:ℕ⟶{0ℎ} ∧ 𝑓 ⇝𝑣 𝑥) → (0ℎ ∈ {0ℎ} ↔ 𝑥 ∈ {0ℎ})) |
36 | 5, 35 | mpbii 223 | . . 3 ⊢ ((𝑓:ℕ⟶{0ℎ} ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ {0ℎ}) |
37 | 36 | gen2 1723 | . 2 ⊢ ∀𝑓∀𝑥((𝑓:ℕ⟶{0ℎ} ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ {0ℎ}) |
38 | isch2 28080 | . 2 ⊢ ({0ℎ} ∈ Cℋ ↔ ({0ℎ} ∈ Sℋ ∧ ∀𝑓∀𝑥((𝑓:ℕ⟶{0ℎ} ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ {0ℎ}))) | |
39 | 27, 37, 38 | mpbir2an 955 | 1 ⊢ {0ℎ} ∈ Cℋ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∀wal 1481 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 {csn 4177 class class class wbr 4653 × cxp 5112 ⟶wf 5884 (class class class)co 6650 ℂcc 9934 ℕcn 11020 ℋchil 27776 +ℎ cva 27777 ·ℎ csm 27778 0ℎc0v 27781 ⇝𝑣 chli 27784 Sℋ csh 27785 Cℋ cch 27786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 ax-hilex 27856 ax-hfvadd 27857 ax-hvcom 27858 ax-hvass 27859 ax-hv0cl 27860 ax-hvaddid 27861 ax-hfvmul 27862 ax-hvmulid 27863 ax-hvmulass 27864 ax-hvdistr1 27865 ax-hvdistr2 27866 ax-hvmul0 27867 ax-hfi 27936 ax-his1 27939 ax-his2 27940 ax-his3 27941 ax-his4 27942 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-icc 12182 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-bases 20750 df-lm 21033 df-haus 21119 df-grpo 27347 df-gid 27348 df-ginv 27349 df-gdiv 27350 df-ablo 27399 df-vc 27414 df-nv 27447 df-va 27450 df-ba 27451 df-sm 27452 df-0v 27453 df-vs 27454 df-nmcv 27455 df-ims 27456 df-hnorm 27825 df-hvsub 27828 df-hlim 27829 df-sh 28064 df-ch 28078 |
This theorem is referenced by: h0elch 28112 h1de2ctlem 28414 |
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