MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  climeq Structured version   Visualization version   GIF version

Theorem climeq 14912
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.)
Hypotheses
Ref Expression
climeq.1 𝑍 = (ℤ𝑀)
climeq.2 (𝜑𝐹𝑉)
climeq.3 (𝜑𝐺𝑊)
climeq.5 (𝜑𝑀 ∈ ℤ)
climeq.6 ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐺𝑘))
Assertion
Ref Expression
climeq (𝜑 → (𝐹𝐴𝐺𝐴))
Distinct variable groups:   𝐴,𝑘   𝑘,𝐹   𝑘,𝐺   𝜑,𝑘   𝑘,𝑍
Allowed substitution hints:   𝑀(𝑘)   𝑉(𝑘)   𝑊(𝑘)

Proof of Theorem climeq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 climeq.1 . . 3 𝑍 = (ℤ𝑀)
2 climeq.5 . . 3 (𝜑𝑀 ∈ ℤ)
3 climeq.2 . . 3 (𝜑𝐹𝑉)
4 climeq.6 . . 3 ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐺𝑘))
51, 2, 3, 4clim2 14849 . 2 (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦𝑍𝑘 ∈ (ℤ𝑦)((𝐺𝑘) ∈ ℂ ∧ (abs‘((𝐺𝑘) − 𝐴)) < 𝑥))))
6 climeq.3 . . 3 (𝜑𝐺𝑊)
7 eqidd 2819 . . 3 ((𝜑𝑘𝑍) → (𝐺𝑘) = (𝐺𝑘))
81, 2, 6, 7clim2 14849 . 2 (𝜑 → (𝐺𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦𝑍𝑘 ∈ (ℤ𝑦)((𝐺𝑘) ∈ ℂ ∧ (abs‘((𝐺𝑘) − 𝐴)) < 𝑥))))
95, 8bitr4d 283 1 (𝜑 → (𝐹𝐴𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136   class class class wbr 5057  cfv 6348  (class class class)co 7145  cc 10523   < clt 10663  cmin 10858  cz 11969  cuz 12231  +crp 12377  abscabs 14581  cli 14829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-pre-lttri 10599  ax-pre-lttrn 10600
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-neg 10861  df-z 11970  df-uz 12232  df-clim 14833
This theorem is referenced by:  climmpt  14916  climres  14920  climshft  14921  climshft2  14927  isumclim3  15102  iprodclim3  15342  logtayl  25170  dfef2  25475  climexp  41762  climeldmeq  41822  climfveq  41826  climfveqf  41837  climeqf  41845  stirlinglem14  42249  fourierdlem112  42380  vonioolem1  42839
  Copyright terms: Public domain W3C validator