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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 29843 | . . . . 5 ⊢ 0ℎ ∈ ℋ | |
2 | snssi 4767 | . . . . 5 ⊢ (0ℎ ∈ ℋ → {0ℎ} ⊆ ℋ) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ {0ℎ} ⊆ ℋ |
4 | 1 | elexi 3463 | . . . . 5 ⊢ 0ℎ ∈ V |
5 | 4 | snid 4621 | . . . 4 ⊢ 0ℎ ∈ {0ℎ} |
6 | 3, 5 | pm3.2i 471 | . . 3 ⊢ ({0ℎ} ⊆ ℋ ∧ 0ℎ ∈ {0ℎ}) |
7 | velsn 4601 | . . . . . 6 ⊢ (𝑥 ∈ {0ℎ} ↔ 𝑥 = 0ℎ) | |
8 | velsn 4601 | . . . . . 6 ⊢ (𝑦 ∈ {0ℎ} ↔ 𝑦 = 0ℎ) | |
9 | oveq12 7363 | . . . . . . . 8 ⊢ ((𝑥 = 0ℎ ∧ 𝑦 = 0ℎ) → (𝑥 +ℎ 𝑦) = (0ℎ +ℎ 0ℎ)) | |
10 | 1 | hvaddid2i 29869 | . . . . . . . 8 ⊢ (0ℎ +ℎ 0ℎ) = 0ℎ |
11 | 9, 10 | eqtrdi 2792 | . . . . . . 7 ⊢ ((𝑥 = 0ℎ ∧ 𝑦 = 0ℎ) → (𝑥 +ℎ 𝑦) = 0ℎ) |
12 | ovex 7387 | . . . . . . . 8 ⊢ (𝑥 +ℎ 𝑦) ∈ V | |
13 | 12 | elsn 4600 | . . . . . . 7 ⊢ ((𝑥 +ℎ 𝑦) ∈ {0ℎ} ↔ (𝑥 +ℎ 𝑦) = 0ℎ) |
14 | 11, 13 | sylibr 233 | . . . . . 6 ⊢ ((𝑥 = 0ℎ ∧ 𝑦 = 0ℎ) → (𝑥 +ℎ 𝑦) ∈ {0ℎ}) |
15 | 7, 8, 14 | syl2anb 598 | . . . . 5 ⊢ ((𝑥 ∈ {0ℎ} ∧ 𝑦 ∈ {0ℎ}) → (𝑥 +ℎ 𝑦) ∈ {0ℎ}) |
16 | 15 | rgen2 3193 | . . . 4 ⊢ ∀𝑥 ∈ {0ℎ}∀𝑦 ∈ {0ℎ} (𝑥 +ℎ 𝑦) ∈ {0ℎ} |
17 | oveq2 7362 | . . . . . . . 8 ⊢ (𝑦 = 0ℎ → (𝑥 ·ℎ 𝑦) = (𝑥 ·ℎ 0ℎ)) | |
18 | hvmul0 29864 | . . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (𝑥 ·ℎ 0ℎ) = 0ℎ) | |
19 | 17, 18 | sylan9eqr 2798 | . . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 = 0ℎ) → (𝑥 ·ℎ 𝑦) = 0ℎ) |
20 | ovex 7387 | . . . . . . . 8 ⊢ (𝑥 ·ℎ 𝑦) ∈ V | |
21 | 20 | elsn 4600 | . . . . . . 7 ⊢ ((𝑥 ·ℎ 𝑦) ∈ {0ℎ} ↔ (𝑥 ·ℎ 𝑦) = 0ℎ) |
22 | 19, 21 | sylibr 233 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 = 0ℎ) → (𝑥 ·ℎ 𝑦) ∈ {0ℎ}) |
23 | 8, 22 | sylan2b 594 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ {0ℎ}) → (𝑥 ·ℎ 𝑦) ∈ {0ℎ}) |
24 | 23 | rgen2 3193 | . . . 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 30049 | . . 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 7151 | . . . . . 6 ⊢ (𝑓:ℕ⟶{0ℎ} ↔ 𝑓 = (ℕ × {0ℎ})) |
29 | hlim0 30075 | . . . . . . 7 ⊢ (ℕ × {0ℎ}) ⇝𝑣 0ℎ | |
30 | breq1 5107 | . . . . . . 7 ⊢ (𝑓 = (ℕ × {0ℎ}) → (𝑓 ⇝𝑣 0ℎ ↔ (ℕ × {0ℎ}) ⇝𝑣 0ℎ)) | |
31 | 29, 30 | mpbiri 257 | . . . . . 6 ⊢ (𝑓 = (ℕ × {0ℎ}) → 𝑓 ⇝𝑣 0ℎ) |
32 | 28, 31 | sylbi 216 | . . . . 5 ⊢ (𝑓:ℕ⟶{0ℎ} → 𝑓 ⇝𝑣 0ℎ) |
33 | hlimuni 30078 | . . . . . 6 ⊢ ((𝑓 ⇝𝑣 0ℎ ∧ 𝑓 ⇝𝑣 𝑥) → 0ℎ = 𝑥) | |
34 | 33 | eleq1d 2822 | . . . . 5 ⊢ ((𝑓 ⇝𝑣 0ℎ ∧ 𝑓 ⇝𝑣 𝑥) → (0ℎ ∈ {0ℎ} ↔ 𝑥 ∈ {0ℎ})) |
35 | 32, 34 | sylan 580 | . . . 4 ⊢ ((𝑓:ℕ⟶{0ℎ} ∧ 𝑓 ⇝𝑣 𝑥) → (0ℎ ∈ {0ℎ} ↔ 𝑥 ∈ {0ℎ})) |
36 | 5, 35 | mpbii 232 | . . 3 ⊢ ((𝑓:ℕ⟶{0ℎ} ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ {0ℎ}) |
37 | 36 | gen2 1798 | . 2 ⊢ ∀𝑓∀𝑥((𝑓:ℕ⟶{0ℎ} ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ {0ℎ}) |
38 | isch2 30063 | . 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 205 ∧ wa 396 ∀wal 1539 = wceq 1541 ∈ wcel 2106 ∀wral 3063 ⊆ wss 3909 {csn 4585 class class class wbr 5104 × cxp 5630 ⟶wf 6490 (class class class)co 7354 ℂcc 11046 ℕcn 12150 ℋchba 29759 +ℎ cva 29760 ·ℎ csm 29761 0ℎc0v 29764 ⇝𝑣 chli 29767 Sℋ csh 29768 Cℋ cch 29769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 ax-pre-sup 11126 ax-addf 11127 ax-mulf 11128 ax-hilex 29839 ax-hfvadd 29840 ax-hvcom 29841 ax-hvass 29842 ax-hv0cl 29843 ax-hvaddid 29844 ax-hfvmul 29845 ax-hvmulid 29846 ax-hvmulass 29847 ax-hvdistr1 29848 ax-hvdistr2 29849 ax-hvmul0 29850 ax-hfi 29919 ax-his1 29922 ax-his2 29923 ax-his3 29924 ax-his4 29925 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 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-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7800 df-1st 7918 df-2nd 7919 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-er 8645 df-map 8764 df-pm 8765 df-en 8881 df-dom 8882 df-sdom 8883 df-sup 9375 df-inf 9376 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-div 11810 df-nn 12151 df-2 12213 df-3 12214 df-4 12215 df-n0 12411 df-z 12497 df-uz 12761 df-q 12871 df-rp 12913 df-xneg 13030 df-xadd 13031 df-xmul 13032 df-icc 13268 df-seq 13904 df-exp 13965 df-cj 14981 df-re 14982 df-im 14983 df-sqrt 15117 df-abs 15118 df-topgen 17322 df-psmet 20784 df-xmet 20785 df-met 20786 df-bl 20787 df-mopn 20788 df-top 22239 df-topon 22256 df-bases 22292 df-lm 22576 df-haus 22662 df-grpo 29333 df-gid 29334 df-ginv 29335 df-gdiv 29336 df-ablo 29385 df-vc 29399 df-nv 29432 df-va 29435 df-ba 29436 df-sm 29437 df-0v 29438 df-vs 29439 df-nmcv 29440 df-ims 29441 df-hnorm 29808 df-hvsub 29811 df-hlim 29812 df-sh 30047 df-ch 30061 |
This theorem is referenced by: h0elch 30095 h1de2ctlem 30395 |
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