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Mirrors > Home > HSE Home > Th. List > h2hcau | Structured version Visualization version GIF version |
Description: The Cauchy sequences of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
h2hc.1 | ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 |
h2hc.2 | ⊢ 𝑈 ∈ NrmCVec |
h2hc.3 | ⊢ ℋ = (BaseSet‘𝑈) |
h2hc.4 | ⊢ 𝐷 = (IndMet‘𝑈) |
Ref | Expression |
---|---|
h2hcau | ⊢ Cauchy = ((Cau‘𝐷) ∩ ( ℋ ↑m ℕ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 3115 | . 2 ⊢ {𝑓 ∈ ( ℋ ↑m ℕ) ∣ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(normℎ‘((𝑓‘𝑗) −ℎ (𝑓‘𝑘))) < 𝑥} = {𝑓 ∣ (𝑓 ∈ ( ℋ ↑m ℕ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(normℎ‘((𝑓‘𝑗) −ℎ (𝑓‘𝑘))) < 𝑥)} | |
2 | df-hcau 28756 | . 2 ⊢ Cauchy = {𝑓 ∈ ( ℋ ↑m ℕ) ∣ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(normℎ‘((𝑓‘𝑗) −ℎ (𝑓‘𝑘))) < 𝑥} | |
3 | elin 3897 | . . . 4 ⊢ (𝑓 ∈ ((Cau‘𝐷) ∩ ( ℋ ↑m ℕ)) ↔ (𝑓 ∈ (Cau‘𝐷) ∧ 𝑓 ∈ ( ℋ ↑m ℕ))) | |
4 | ancom 464 | . . . 4 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓 ∈ ( ℋ ↑m ℕ)) ↔ (𝑓 ∈ ( ℋ ↑m ℕ) ∧ 𝑓 ∈ (Cau‘𝐷))) | |
5 | h2hc.3 | . . . . . . . 8 ⊢ ℋ = (BaseSet‘𝑈) | |
6 | 5 | hlex 28681 | . . . . . . 7 ⊢ ℋ ∈ V |
7 | nnex 11631 | . . . . . . 7 ⊢ ℕ ∈ V | |
8 | 6, 7 | elmap 8418 | . . . . . 6 ⊢ (𝑓 ∈ ( ℋ ↑m ℕ) ↔ 𝑓:ℕ⟶ ℋ) |
9 | nnuz 12269 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
10 | h2hc.2 | . . . . . . . . 9 ⊢ 𝑈 ∈ NrmCVec | |
11 | h2hc.4 | . . . . . . . . . 10 ⊢ 𝐷 = (IndMet‘𝑈) | |
12 | 5, 11 | imsxmet 28475 | . . . . . . . . 9 ⊢ (𝑈 ∈ NrmCVec → 𝐷 ∈ (∞Met‘ ℋ)) |
13 | 10, 12 | mp1i 13 | . . . . . . . 8 ⊢ (𝑓:ℕ⟶ ℋ → 𝐷 ∈ (∞Met‘ ℋ)) |
14 | 1zzd 12001 | . . . . . . . 8 ⊢ (𝑓:ℕ⟶ ℋ → 1 ∈ ℤ) | |
15 | eqidd 2799 | . . . . . . . 8 ⊢ ((𝑓:ℕ⟶ ℋ ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) = (𝑓‘𝑘)) | |
16 | eqidd 2799 | . . . . . . . 8 ⊢ ((𝑓:ℕ⟶ ℋ ∧ 𝑗 ∈ ℕ) → (𝑓‘𝑗) = (𝑓‘𝑗)) | |
17 | id 22 | . . . . . . . 8 ⊢ (𝑓:ℕ⟶ ℋ → 𝑓:ℕ⟶ ℋ) | |
18 | 9, 13, 14, 15, 16, 17 | iscauf 23884 | . . . . . . 7 ⊢ (𝑓:ℕ⟶ ℋ → (𝑓 ∈ (Cau‘𝐷) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑗)𝐷(𝑓‘𝑘)) < 𝑥)) |
19 | ffvelrn 6826 | . . . . . . . . . . . . 13 ⊢ ((𝑓:ℕ⟶ ℋ ∧ 𝑗 ∈ ℕ) → (𝑓‘𝑗) ∈ ℋ) | |
20 | 19 | adantr 484 | . . . . . . . . . . . 12 ⊢ (((𝑓:ℕ⟶ ℋ ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝑓‘𝑗) ∈ ℋ) |
21 | eluznn 12306 | . . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ ℕ) | |
22 | ffvelrn 6826 | . . . . . . . . . . . . . 14 ⊢ ((𝑓:ℕ⟶ ℋ ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ ℋ) | |
23 | 21, 22 | sylan2 595 | . . . . . . . . . . . . 13 ⊢ ((𝑓:ℕ⟶ ℋ ∧ (𝑗 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘𝑗))) → (𝑓‘𝑘) ∈ ℋ) |
24 | 23 | anassrs 471 | . . . . . . . . . . . 12 ⊢ (((𝑓:ℕ⟶ ℋ ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝑓‘𝑘) ∈ ℋ) |
25 | h2hc.1 | . . . . . . . . . . . . 13 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
26 | 25, 10, 5, 11 | h2hmetdval 28761 | . . . . . . . . . . . 12 ⊢ (((𝑓‘𝑗) ∈ ℋ ∧ (𝑓‘𝑘) ∈ ℋ) → ((𝑓‘𝑗)𝐷(𝑓‘𝑘)) = (normℎ‘((𝑓‘𝑗) −ℎ (𝑓‘𝑘)))) |
27 | 20, 24, 26 | syl2anc 587 | . . . . . . . . . . 11 ⊢ (((𝑓:ℕ⟶ ℋ ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ((𝑓‘𝑗)𝐷(𝑓‘𝑘)) = (normℎ‘((𝑓‘𝑗) −ℎ (𝑓‘𝑘)))) |
28 | 27 | breq1d 5040 | . . . . . . . . . 10 ⊢ (((𝑓:ℕ⟶ ℋ ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (((𝑓‘𝑗)𝐷(𝑓‘𝑘)) < 𝑥 ↔ (normℎ‘((𝑓‘𝑗) −ℎ (𝑓‘𝑘))) < 𝑥)) |
29 | 28 | ralbidva 3161 | . . . . . . . . 9 ⊢ ((𝑓:ℕ⟶ ℋ ∧ 𝑗 ∈ ℕ) → (∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑗)𝐷(𝑓‘𝑘)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(normℎ‘((𝑓‘𝑗) −ℎ (𝑓‘𝑘))) < 𝑥)) |
30 | 29 | rexbidva 3255 | . . . . . . . 8 ⊢ (𝑓:ℕ⟶ ℋ → (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑗)𝐷(𝑓‘𝑘)) < 𝑥 ↔ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(normℎ‘((𝑓‘𝑗) −ℎ (𝑓‘𝑘))) < 𝑥)) |
31 | 30 | ralbidv 3162 | . . . . . . 7 ⊢ (𝑓:ℕ⟶ ℋ → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑗)𝐷(𝑓‘𝑘)) < 𝑥 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(normℎ‘((𝑓‘𝑗) −ℎ (𝑓‘𝑘))) < 𝑥)) |
32 | 18, 31 | bitrd 282 | . . . . . 6 ⊢ (𝑓:ℕ⟶ ℋ → (𝑓 ∈ (Cau‘𝐷) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(normℎ‘((𝑓‘𝑗) −ℎ (𝑓‘𝑘))) < 𝑥)) |
33 | 8, 32 | sylbi 220 | . . . . 5 ⊢ (𝑓 ∈ ( ℋ ↑m ℕ) → (𝑓 ∈ (Cau‘𝐷) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(normℎ‘((𝑓‘𝑗) −ℎ (𝑓‘𝑘))) < 𝑥)) |
34 | 33 | pm5.32i 578 | . . . 4 ⊢ ((𝑓 ∈ ( ℋ ↑m ℕ) ∧ 𝑓 ∈ (Cau‘𝐷)) ↔ (𝑓 ∈ ( ℋ ↑m ℕ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(normℎ‘((𝑓‘𝑗) −ℎ (𝑓‘𝑘))) < 𝑥)) |
35 | 3, 4, 34 | 3bitri 300 | . . 3 ⊢ (𝑓 ∈ ((Cau‘𝐷) ∩ ( ℋ ↑m ℕ)) ↔ (𝑓 ∈ ( ℋ ↑m ℕ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(normℎ‘((𝑓‘𝑗) −ℎ (𝑓‘𝑘))) < 𝑥)) |
36 | 35 | abbi2i 2929 | . 2 ⊢ ((Cau‘𝐷) ∩ ( ℋ ↑m ℕ)) = {𝑓 ∣ (𝑓 ∈ ( ℋ ↑m ℕ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(normℎ‘((𝑓‘𝑗) −ℎ (𝑓‘𝑘))) < 𝑥)} |
37 | 1, 2, 36 | 3eqtr4i 2831 | 1 ⊢ Cauchy = ((Cau‘𝐷) ∩ ( ℋ ↑m ℕ)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {cab 2776 ∀wral 3106 ∃wrex 3107 {crab 3110 ∩ cin 3880 〈cop 4531 class class class wbr 5030 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 1c1 10527 < clt 10664 ℕcn 11625 ℤ≥cuz 12231 ℝ+crp 12377 ∞Metcxmet 20076 Cauccau 23857 NrmCVeccnv 28367 BaseSetcba 28369 IndMetcims 28374 ℋchba 28702 +ℎ cva 28703 ·ℎ csm 28704 normℎcno 28706 −ℎ cmv 28708 Cauchyccauold 28709 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-xneg 12495 df-xadd 12496 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-cau 23860 df-grpo 28276 df-gid 28277 df-ginv 28278 df-gdiv 28279 df-ablo 28328 df-vc 28342 df-nv 28375 df-va 28378 df-ba 28379 df-sm 28380 df-0v 28381 df-vs 28382 df-nmcv 28383 df-ims 28384 df-hvsub 28754 df-hcau 28756 |
This theorem is referenced by: axhcompl-zf 28781 hhcau 28981 |
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