ellimc3ap - Intuitionistic Logic Explorer

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Theorem ellimc3ap 14897

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 2339	. 2 ⊢ Ⅎ𝑧𝐹
5	1, 2, 3, 4	ellimc3apf 14896	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 2167 ∀wral 2475 ∃wrex 2476 ⊆ wss 3157 class class class wbr 4033 ⟶wf 5254 ‘cfv 5258 (class class class)co 5922 ℂcc 7877 < clt 8061 − cmin 8197 # cap 8608 ℝ⁺crp 9728 abscabs 11162 lim_ℂ climc 14890

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 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4151 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-cnex 7970

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-ral 2480 df-rex 2481 df-rab 2484 df-v 2765 df-sbc 2990 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-br 4034 df-opab 4095 df-id 4328 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-fv 5266 df-ov 5925 df-oprab 5926 df-mpo 5927 df-pm 6710 df-limced 14892

This theorem is referenced by: limcdifap 14898 limcimolemlt 14900 limcimo 14901 limcresi 14902 cnplimcim 14903 cnplimclemr 14905 limccnpcntop 14911 dveflem 14962

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