Theorem rlimcl 15521

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 15519	. . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹:dom 𝐹⟶ℂ)
2		rlimss 15520	. . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → dom 𝐹 ⊆ ℝ)
3		eqidd 2762	. . . 4 ⊢ ((𝐹 ⇝𝑟 𝐴 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = (𝐹‘𝑥))
4	1, 2, 3	rlim 15513	. . 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 269	. 2 ⊢ (𝐹 ⇝𝑟 𝐴 → (𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ ∀𝑥 ∈ dom 𝐹(𝑧 ≤ 𝑥 → (abs‘((𝐹‘𝑥) − 𝐴)) < 𝑦)))
6	5	simpld 498	1 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐴 ∈ ℂ)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085   class class class wbr 5097  dom cdm 5643  ‘cfv 6516  (class class class)co 7391  ℂcc 11065  ℝcr 11066   < clt 11210   ≤ cle 11211   − cmin 11408  ℝ⁺crp 12987  abscabs 15252   ⇝_𝑟 crli 15503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-cnex 11123  ax-resscn 11124

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-pm 8805  df-rlim 15507

This theorem is referenced by:  rlimi  15531  rlimclim1  15563  rlimuni  15568  rlimresb  15583  rlimcld2  15596  rlimabs  15627  rlimcj  15628  rlimre  15629  rlimim  15630  rlimo1  15635  rlimadd  15661  rlimsub  15662  rlimmul  15663  rlimdiv  15664  rlimsqzlem  15667  fsumrlim  15830  dchrisum0lem2a  27569  mulog2sumlem2  27587  mulog2sumlem3  27588