ellimc3ap - Intuitionistic Logic Explorer

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Theorem ellimc3ap 15248

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 2350	. 2 ⊢ Ⅎ𝑧𝐹
5	1, 2, 3, 4	ellimc3apf 15247	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 2178 ∀wral 2486 ∃wrex 2487 ⊆ wss 3174 class class class wbr 4059 ⟶wf 5286 ‘cfv 5290 (class class class)co 5967 ℂcc 7958 < clt 8142 − cmin 8278 # cap 8689 ℝ⁺crp 9810 abscabs 11423 lim_ℂ climc 15241

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 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-sep 4178 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-setind 4603 ax-cnex 8051

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-ral 2491 df-rex 2492 df-rab 2495 df-v 2778 df-sbc 3006 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-br 4060 df-opab 4122 df-id 4358 df-xp 4699 df-rel 4700 df-cnv 4701 df-co 4702 df-dm 4703 df-rn 4704 df-iota 5251 df-fun 5292 df-fn 5293 df-f 5294 df-fv 5298 df-ov 5970 df-oprab 5971 df-mpo 5972 df-pm 6761 df-limced 15243

This theorem is referenced by: limcdifap 15249 limcimolemlt 15251 limcimo 15252 limcresi 15253 cnplimcim 15254 cnplimclemr 15256 limccnpcntop 15262 dveflem 15313

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