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Mirrors > Home > ILE Home > Th. List > ellimc3ap | GIF version |
Description: Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.) Use apartness. (Revised by Jim Kingdon, 3-Jun-2023.) |
Ref | Expression |
---|---|
ellimc3.f | ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
ellimc3.a | ⊢ (𝜑 → 𝐴 ⊆ ℂ) |
ellimc3.b | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
ellimc3ap | ⊢ (𝜑 → (𝐶 ∈ (𝐹 limℂ 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ 𝐴 ((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ellimc3.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
2 | ellimc3.a | . 2 ⊢ (𝜑 → 𝐴 ⊆ ℂ) | |
3 | ellimc3.b | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
4 | nfcv 2281 | . 2 ⊢ Ⅎ𝑧𝐹 | |
5 | 1, 2, 3, 4 | ellimc3apf 12808 | 1 ⊢ (𝜑 → (𝐶 ∈ (𝐹 limℂ 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ 𝐴 ((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1480 ∀wral 2416 ∃wrex 2417 ⊆ wss 3071 class class class wbr 3929 ⟶wf 5119 ‘cfv 5123 (class class class)co 5774 ℂcc 7625 < clt 7807 − cmin 7940 # cap 8350 ℝ+crp 9448 abscabs 10776 limℂ climc 12802 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7718 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pm 6545 df-limced 12804 |
This theorem is referenced by: limcdifap 12810 limcimolemlt 12812 limcimo 12813 limcresi 12814 cnplimcim 12815 cnplimclemr 12817 limccnpcntop 12823 dveflem 12865 |
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