ellimc3ap - Intuitionistic Logic Explorer

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Theorem ellimc3ap 15384

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.)

**Hypotheses**
Ref	Expression
ellimc3.f	⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
ellimc3.a	⊢ (𝜑 → 𝐴 ⊆ ℂ)
ellimc3.b	⊢ (𝜑 → 𝐵 ∈ ℂ)

**Assertion**
Ref	Expression
ellimc3ap	⊢ (𝜑 → (𝐶 ∈ (𝐹 lim_ℂ 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ 𝐴 ((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))))

Distinct variable groups: 𝑧,𝐴 𝑥,𝐵,𝑦,𝑧 𝑥,𝐶,𝑦,𝑧 𝑥,𝐹,𝑦 𝜑,𝑥,𝑦 𝑧,𝐹 Allowed substitution hints: 𝜑(𝑧) 𝐴(𝑥,𝑦)

**Proof of Theorem ellimc3ap**
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 2374	. 2 ⊢ Ⅎ𝑧𝐹
5	1, 2, 3, 4	ellimc3apf 15383	1 ⊢ (𝜑 → (𝐶 ∈ (𝐹 lim_ℂ 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 2202 ∀wral 2510 ∃wrex 2511 ⊆ wss 3200 class class class wbr 4088 ⟶wf 5322 ‘cfv 5326 (class class class)co 6017 ℂcc 8029 < clt 8213 − cmin 8349 # cap 8760 ℝ⁺crp 9887 abscabs 11557 lim_ℂ climc 15377

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-cnex 8122

This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-ral 2515 df-rex 2516 df-rab 2519 df-v 2804 df-sbc 3032 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-br 4089 df-opab 4151 df-id 4390 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-fv 5334 df-ov 6020 df-oprab 6021 df-mpo 6022 df-pm 6819 df-limced 15379

This theorem is referenced by: limcdifap 15385 limcimolemlt 15387 limcimo 15388 limcresi 15389 cnplimcim 15390 cnplimclemr 15392 limccnpcntop 15398 dveflem 15449

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