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| Description: Two functions that are eventually equal to one another have the same limit. (Contributed by Mario Carneiro, 5-Nov-2013.) (Revised by Mario Carneiro, 31-Jan-2014.) | 
| Ref | Expression | 
|---|---|
| climeq.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) | 
| climeq.2 | ⊢ (𝜑 → 𝐹 ∈ 𝑉) | 
| climeq.3 | ⊢ (𝜑 → 𝐺 ∈ 𝑊) | 
| climeq.5 | ⊢ (𝜑 → 𝑀 ∈ ℤ) | 
| climeq.6 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘)) | 
| Ref | Expression | 
|---|---|
| climeq | ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | climeq.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
| 2 | climeq.5 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
| 3 | climeq.2 | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
| 4 | climeq.6 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘)) | |
| 5 | 1, 2, 3, 4 | clim2 15541 | . 2 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑦)((𝐺‘𝑘) ∈ ℂ ∧ (abs‘((𝐺‘𝑘) − 𝐴)) < 𝑥)))) | 
| 6 | climeq.3 | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑊) | |
| 7 | eqidd 2737 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐺‘𝑘)) | |
| 8 | 1, 2, 6, 7 | clim2 15541 | . 2 ⊢ (𝜑 → (𝐺 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑦)((𝐺‘𝑘) ∈ ℂ ∧ (abs‘((𝐺‘𝑘) − 𝐴)) < 𝑥)))) | 
| 9 | 5, 8 | bitr4d 282 | 1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 ℂcc 11154 < clt 11296 − cmin 11493 ℤcz 12615 ℤ≥cuz 12879 ℝ+crp 13035 abscabs 15274 ⇝ cli 15521 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-pre-lttri 11230 ax-pre-lttrn 11231 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-neg 11496 df-z 12616 df-uz 12880 df-clim 15525 | 
| This theorem is referenced by: climmpt 15608 climres 15612 climshft 15613 climshft2 15619 isumclim3 15796 iprodclim3 16037 logtayl 26703 dfef2 27015 climexp 45625 climeldmeq 45685 climfveq 45689 climfveqf 45700 climeqf 45708 stirlinglem14 46107 fourierdlem112 46238 vonioolem1 46700 | 
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