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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 2319 | . 2 ⊢ Ⅎ𝑧𝐹 | |
5 | 1, 2, 3, 4 | ellimc3apf 14317 | 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 104 ↔ wb 105 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 ⊆ wss 3131 class class class wbr 4005 ⟶wf 5214 ‘cfv 5218 (class class class)co 5878 ℂcc 7812 < clt 7995 − cmin 8131 # cap 8541 ℝ+crp 9656 abscabs 11009 limℂ climc 14311 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7905 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fv 5226 df-ov 5881 df-oprab 5882 df-mpo 5883 df-pm 6654 df-limced 14313 |
This theorem is referenced by: limcdifap 14319 limcimolemlt 14321 limcimo 14322 limcresi 14323 cnplimcim 14324 cnplimclemr 14326 limccnpcntop 14332 dveflem 14375 |
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