Theorem rlimcl 15456

Description: Closure of the limit of a sequence of complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
rlimcl	⊢ (𝐹 ⇝𝑟 𝐴 → 𝐴 ∈ ℂ)

Proof of Theorem rlimcl

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rlimf 15454	. . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹:dom 𝐹⟶ℂ)
2		rlimss 15455	. . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → dom 𝐹 ⊆ ℝ)
3		eqidd 2740	. . . 4 ⊢ ((𝐹 ⇝𝑟 𝐴 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = (𝐹‘𝑥))
4	1, 2, 3	rlim 15448	. . 3 ⊢ (𝐹 ⇝𝑟 𝐴 → (𝐹 ⇝𝑟 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ ∀𝑥 ∈ dom 𝐹(𝑧 ≤ 𝑥 → (abs‘((𝐹‘𝑥) − 𝐴)) < 𝑦))))
5	4	ibi 268	. 2 ⊢ (𝐹 ⇝𝑟 𝐴 → (𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ ∀𝑥 ∈ dom 𝐹(𝑧 ≤ 𝑥 → (abs‘((𝐹‘𝑥) − 𝐴)) < 𝑦)))
6	5	simpld 495	1 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐴 ∈ ℂ)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063   class class class wbr 5072  dom cdm 5618  ‘cfv 6485  (class class class)co 7356  ℂcc 11027  ℝcr 11028   < clt 11170   ≤ cle 11171   − cmin 11368  ℝ⁺crp 12933  abscabs 15187   ⇝_𝑟 crli 15438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-pm 8766  df-rlim 15442

This theorem is referenced by:  rlimi  15466  rlimclim1  15498  rlimuni  15503  rlimresb  15518  rlimcld2  15531  rlimabs  15562  rlimcj  15563  rlimre  15564  rlimim  15565  rlimo1  15570  rlimadd  15596  rlimsub  15597  rlimmul  15598  rlimdiv  15599  rlimsqzlem  15602  fsumrlim  15765  dchrisum0lem2a  27498  mulog2sumlem2  27516  mulog2sumlem3  27517