ellimc3ap - Intuitionistic Logic Explorer

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Theorem ellimc3ap 13270

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 2308	. 2 ⊢ Ⅎ𝑧𝐹
5	1, 2, 3, 4	ellimc3apf 13269	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 103 ↔ wb 104 ∈ wcel 2136 ∀wral 2444 ∃wrex 2445 ⊆ wss 3116 class class class wbr 3982 ⟶wf 5184 ‘cfv 5188 (class class class)co 5842 ℂcc 7751 < clt 7933 − cmin 8069 # cap 8479 ℝ⁺crp 9589 abscabs 10939 lim_ℂ climc 13263

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-ral 2449 df-rex 2450 df-rab 2453 df-v 2728 df-sbc 2952 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-fv 5196 df-ov 5845 df-oprab 5846 df-mpo 5847 df-pm 6617 df-limced 13265

This theorem is referenced by: limcdifap 13271 limcimolemlt 13273 limcimo 13274 limcresi 13275 cnplimcim 13276 cnplimclemr 13278 limccnpcntop 13284 dveflem 13327

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