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GIF version
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Related theorems GIF version |
| Description: A Cauchy sequence on a Hilbert space converges. |
| Ref | Expression |
|---|---|
| cauchyconv | ⊢ (F ∈ Cauchy → ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ ∀w ∈ ℕ ((y ≤ z ∧ y ≤ w) → (norm ‘((F ‘z) −v (F ‘w))) < x))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hcauchy 5103 | . 2 ⊢ (F ∈ Cauchy ↔ (F:ℕ–→ ℋ ∧ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ ∀w ∈ ℕ ((y ≤ z ∧ y ≤ w) → (norm ‘((F ‘z) −v (F ‘w))) < x)))) | |
| 2 | 1 | pm3.27bd 263 | 1 ⊢ (F ∈ Cauchy → ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ ∀w ∈ ℕ ((y ≤ z ∧ y ≤ w) → (norm ‘((F ‘z) −v (F ‘w))) < x))) |
| Colors of variables: wff set class |
| Syntax hints: → wi 2 ∧ wa 196 ∈ wcel 1092 ∀wral 1201 ∃wrex 1202 class class class wbr 2054 –→wf 2418 ‘cfv 2422 (class class class)co 3001 ℝcr 4027 0cc0 4028 < clt 4033 ≤ cle 4092 ℕcn 4093 ℋ chil 4958 −v cmv 4962 normcno 4964 Cauchyccau 4965 |
| This theorem was proved from axioms: ax-1 3 ax-2 4 ax-3 5 ax-mp 6 ax-4 673 ax-5 674 ax-6 675 ax-7 676 ax-gen 677 ax-8 798 ax-9 799 ax-10 800 ax-11 801 ax-12 802 ax-13 804 ax-14 805 ax-16 922 ax-17 925 ax-ext 1074 ax-rep 1075 ax-un 1076 ax-pow 1077 ax-reg 1078 ax-inf 1079 |
| This theorem depends on definitions:
df-bi 128 df-or 197
df-an 198 df-3or 582 df-3an 583 df-ex 679
df-sb 853 df-eu 1009 df-mo 1010 df-clab 1093 df-cleq 1097 df-clel 1099 df-ne 1192 df-ral 1205 df-rex 1206 df-reu 1207 df-rab 1208 df-v 1349 df-sbc 1441 df-dif 1489 df-un 1490 df-in 1491 df-ss 1492 df-pss 1494 df-nul 1708 df-if 1777 df-pw 1799 df-sn 1811 df-pr 1812 df-tp 1814 df-op 1815 df-uni 1920 df-int 1966 df-iun 1996 df-tr 2042 df-br 2063 df-opab 2098 df-eprel 2122 df-id 2125 df-po 2128 df-so 2138 df-fr 2169 df-we 2186 df-ord 2202 df-on 2203 df-lim 2204 df-suc 2205 df-om 2373 df-xp 2424 df-rel 2425 df-cnv 2426 df-co 2427 df-dm 2428 df-rn 2429 df-res 2430 df-ima 2431 df-fun 2432 df-fn 2433 df-f 2434 df-f1 2435 df-fv |