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| Mirrors > Home > ILE Home > Th. List > climeq | 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 11513 | . 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 2205 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐺‘𝑘)) | |
| 8 | 1, 2, 6, 7 | clim2 11513 | . 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 191 | 1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1372 ∈ wcel 2175 ∀wral 2483 ∃wrex 2484 class class class wbr 4043 ‘cfv 5268 (class class class)co 5934 ℂcc 7905 < clt 8089 − cmin 8225 ℤcz 9354 ℤ≥cuz 9630 ℝ+crp 9757 abscabs 11227 ⇝ cli 11508 |
| 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 615 ax-in2 616 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pow 4217 ax-pr 4252 ax-un 4478 ax-setind 4583 ax-cnex 7998 ax-resscn 7999 ax-1cn 8000 ax-1re 8001 ax-icn 8002 ax-addcl 8003 ax-addrcl 8004 ax-mulcl 8005 ax-addcom 8007 ax-addass 8009 ax-distr 8011 ax-i2m1 8012 ax-0lt1 8013 ax-0id 8015 ax-rnegex 8016 ax-cnre 8018 ax-pre-ltirr 8019 ax-pre-ltwlin 8020 ax-pre-lttrn 8021 ax-pre-apti 8022 ax-pre-ltadd 8023 |
| This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rab 2492 df-v 2773 df-sbc 2998 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-if 3571 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-br 4044 df-opab 4105 df-mpt 4106 df-id 4338 df-xp 4679 df-rel 4680 df-cnv 4681 df-co 4682 df-dm 4683 df-rn 4684 df-res 4685 df-ima 4686 df-iota 5229 df-fun 5270 df-fn 5271 df-f 5272 df-fv 5276 df-riota 5889 df-ov 5937 df-oprab 5938 df-mpo 5939 df-pnf 8091 df-mnf 8092 df-xr 8093 df-ltxr 8094 df-le 8095 df-sub 8227 df-neg 8228 df-inn 9019 df-n0 9278 df-z 9355 df-uz 9631 df-clim 11509 |
| This theorem is referenced by: climmpt 11530 climres 11533 climshft 11534 climshft2 11536 isumclim3 11653 |
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