ellimc3ap - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > ellimc3ap		GIF version

Theorem ellimc3ap 15472

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 2375	. 2 ⊢ Ⅎ𝑧𝐹
5	1, 2, 3, 4	ellimc3apf 15471	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 2511 ∃wrex 2512 ⊆ wss 3201 class class class wbr 4093 ⟶wf 5329 ‘cfv 5333 (class class class)co 6028 ℂcc 8090 < clt 8273 − cmin 8409 # cap 8820 ℝ⁺crp 9949 abscabs 11637 lim_ℂ climc 15465

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-cnex 8183

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-ral 2516 df-rex 2517 df-rab 2520 df-v 2805 df-sbc 3033 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-br 4094 df-opab 4156 df-id 4396 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-fv 5341 df-ov 6031 df-oprab 6032 df-mpo 6033 df-pm 6863 df-limced 15467

This theorem is referenced by: limcdifap 15473 limcimolemlt 15475 limcimo 15476 limcresi 15477 cnplimcim 15478 cnplimclemr 15480 limccnpcntop 15486 dveflem 15537

W3C validator