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Mirrors > Home > MPE Home > Th. List > climeq | Structured version Visualization version GIF version |
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 15506 | . 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 2727 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐺‘𝑘)) | |
8 | 1, 2, 6, 7 | clim2 15506 | . 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 281 | 1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ∃wrex 3060 class class class wbr 5153 ‘cfv 6554 (class class class)co 7424 ℂcc 11156 < clt 11298 − cmin 11494 ℤcz 12610 ℤ≥cuz 12874 ℝ+crp 13028 abscabs 15239 ⇝ cli 15486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-pre-lttri 11232 ax-pre-lttrn 11233 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-neg 11497 df-z 12611 df-uz 12875 df-clim 15490 |
This theorem is referenced by: climmpt 15573 climres 15577 climshft 15578 climshft2 15584 isumclim3 15763 iprodclim3 16002 logtayl 26687 dfef2 26999 climexp 45226 climeldmeq 45286 climfveq 45290 climfveqf 45301 climeqf 45309 stirlinglem14 45708 fourierdlem112 45839 vonioolem1 46301 |
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