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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 2253 | . 2 ⊢ Ⅎ𝑧𝐹 | |
5 | 1, 2, 3, 4 | ellimc3apf 12579 | 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 1461 ∀wral 2388 ∃wrex 2389 ⊆ wss 3035 class class class wbr 3893 ⟶wf 5075 ‘cfv 5079 (class class class)co 5726 ℂcc 7539 < clt 7718 − cmin 7850 # cap 8255 ℝ+crp 9337 abscabs 10655 limℂ climc 12573 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-cnex 7630 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-ral 2393 df-rex 2394 df-rab 2397 df-v 2657 df-sbc 2877 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-opab 3948 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-fv 5087 df-ov 5729 df-oprab 5730 df-mpo 5731 df-pm 6497 df-limced 12575 |
This theorem is referenced by: limcdifap 12581 limcimolemlt 12583 limcimo 12584 limcresi 12585 cnplimcim 12586 cnplimclemr 12588 limccnpcntop 12594 |
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