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Mirrors > Home > HSE Home > Th. List > hcaucvg | Structured version Visualization version GIF version |
Description: A Cauchy sequence on a Hilbert space converges. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hcaucvg | ⊢ ((𝐹 ∈ Cauchy ∧ 𝐴 ∈ ℝ+) → ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑦) −ℎ (𝐹‘𝑧))) < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hcau 30941 | . . 3 ⊢ (𝐹 ∈ Cauchy ↔ (𝐹:ℕ⟶ ℋ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑦) −ℎ (𝐹‘𝑧))) < 𝑥)) | |
2 | 1 | simprbi 496 | . 2 ⊢ (𝐹 ∈ Cauchy → ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑦) −ℎ (𝐹‘𝑧))) < 𝑥) |
3 | breq2 5145 | . . . 4 ⊢ (𝑥 = 𝐴 → ((normℎ‘((𝐹‘𝑦) −ℎ (𝐹‘𝑧))) < 𝑥 ↔ (normℎ‘((𝐹‘𝑦) −ℎ (𝐹‘𝑧))) < 𝐴)) | |
4 | 3 | rexralbidv 3214 | . . 3 ⊢ (𝑥 = 𝐴 → (∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑦) −ℎ (𝐹‘𝑧))) < 𝑥 ↔ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑦) −ℎ (𝐹‘𝑧))) < 𝐴)) |
5 | 4 | rspccva 3605 | . 2 ⊢ ((∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑦) −ℎ (𝐹‘𝑧))) < 𝑥 ∧ 𝐴 ∈ ℝ+) → ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑦) −ℎ (𝐹‘𝑧))) < 𝐴) |
6 | 2, 5 | sylan 579 | 1 ⊢ ((𝐹 ∈ Cauchy ∧ 𝐴 ∈ ℝ+) → ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑦) −ℎ (𝐹‘𝑧))) < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ∃wrex 3064 class class class wbr 5141 ⟶wf 6532 ‘cfv 6536 (class class class)co 7404 < clt 11249 ℕcn 12213 ℤ≥cuz 12823 ℝ+crp 12977 ℋchba 30676 normℎcno 30680 −ℎ cmv 30682 Cauchyccauold 30683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-1cn 11167 ax-addcl 11169 ax-hilex 30756 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-map 8821 df-nn 12214 df-hcau 30730 |
This theorem is referenced by: chscllem2 31395 |
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