Theorem climeq 15474

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 = (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))

Assertion

Ref	Expression
climeq	⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴))

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.

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 15411	. 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 2730	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐺‘𝑘))
8	1, 2, 6, 7	clim2 15411	. 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))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   class class class wbr 5092  ‘cfv 6482  (class class class)co 7349  ℂcc 11007   < clt 11149   − cmin 11347  ℤcz 12471  ℤ_≥cuz 12735  ℝ⁺crp 12893  abscabs 15141   ⇝ cli 15391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-pre-lttri 11083  ax-pre-lttrn 11084

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-po 5527  df-so 5528  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-neg 11350  df-z 12472  df-uz 12736  df-clim 15395

This theorem is referenced by:  climmpt  15478  climres  15482  climshft  15483  climshft2  15489  isumclim3  15666  iprodclim3  15907  logtayl  26567  dfef2  26879  climexp  45586  climeldmeq  45646  climfveq  45650  climfveqf  45661  climeqf  45669  stirlinglem14  46068  fourierdlem112  46199  vonioolem1  46661