Theorem climeq 14146

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 14083	. 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 2611	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐺‘𝑘))
8	1, 2, 6, 7	clim2 14083	. 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 270	1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℂcc 9813   < clt 9953   − cmin 10145  ℤcz 11254  ℤ_≥cuz 11563  ℝ⁺crp 11708  abscabs 13822   ⇝ cli 14063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-pre-lttri 9889  ax-pre-lttrn 9890

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-neg 10148  df-z 11255  df-uz 11564  df-clim 14067

This theorem is referenced by:  climmpt  14150  climres  14154  climshft  14155  climshft2  14161  isumclim3  14332  iprodclim3  14570  logtayl  24206  dfef2  24497  climexp  38672  climeldmeq  38732  climfveq  38736  stirlinglem14  38980  fourierdlem112  39111  vonioolem1  39571