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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 14613 | . 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 2827 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐺‘𝑘)) | |
8 | 1, 2, 6, 7 | clim2 14613 | . 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 274 | 1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∀wral 3118 ∃wrex 3119 class class class wbr 4874 ‘cfv 6124 (class class class)co 6906 ℂcc 10251 < clt 10392 − cmin 10586 ℤcz 11705 ℤ≥cuz 11969 ℝ+crp 12113 abscabs 14352 ⇝ cli 14593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-pre-lttri 10327 ax-pre-lttrn 10328 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-po 5264 df-so 5265 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-ov 6909 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-neg 10589 df-z 11706 df-uz 11970 df-clim 14597 |
This theorem is referenced by: climmpt 14680 climres 14684 climshft 14685 climshft2 14691 isumclim3 14866 iprodclim3 15104 logtayl 24806 dfef2 25111 climexp 40633 climeldmeq 40693 climfveq 40697 climfveqf 40708 climeqf 40716 stirlinglem14 41099 fourierdlem112 41230 vonioolem1 41689 |
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