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Theorem rlimcl 14863

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 14861	. . . 4 ⊢ (𝐹 ⇝_𝑟 𝐴 → 𝐹:dom 𝐹⟶ℂ)
2		rlimss 14862	. . . 4 ⊢ (𝐹 ⇝_𝑟 𝐴 → dom 𝐹 ⊆ ℝ)
3		eqidd 2825	. . . 4 ⊢ ((𝐹 ⇝_𝑟 𝐴 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = (𝐹‘𝑥))
4	1, 2, 3	rlim 14855	. . 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 497	1 ⊢ (𝐹 ⇝_𝑟 𝐴 → 𝐴 ∈ ℂ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2113 ∀wral 3141 ∃wrex 3142 class class class wbr 5069 dom cdm 5558 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 ℝcr 10539 < clt 10678 ≤ cle 10679 − cmin 10873 ℝ⁺crp 12392 abscabs 14596 ⇝_𝑟 crli 14845

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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-pm 8412 df-rlim 14849

This theorem is referenced by: rlimi 14873 rlimclim1 14905 rlimuni 14910 rlimresb 14925 rlimcld2 14938 rlimabs 14968 rlimcj 14969 rlimre 14970 rlimim 14971 rlimo1 14976 rlimadd 15002 rlimsub 15003 rlimmul 15004 rlimdiv 15005 rlimsqzlem 15008 fsumrlim 15169 dchrisum0lem2a 26096 mulog2sumlem2 26114 mulog2sumlem3 26115

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