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Theorem rlimrecl 15217

Description: The limit of a real sequence is real. (Contributed by Mario Carneiro, 9-May-2016.)

**Hypotheses**
Ref	Expression
rlimcld2.1	⊢ (𝜑 → sup(𝐴, ℝ^*, < ) = +∞)
rlimcld2.2	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ⇝_𝑟 𝐶)
rlimrecl.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)

**Assertion**
Ref	Expression
rlimrecl	⊢ (𝜑 → 𝐶 ∈ ℝ)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐶 𝜑,𝑥 Allowed substitution hint: 𝐵(𝑥)

Proof of Theorem rlimrecl Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rlimcld2.1	. 2 ⊢ (𝜑 → sup(𝐴, ℝ^*, < ) = +∞)
2		rlimcld2.2	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ⇝_𝑟 𝐶)
3		ax-resscn 10859	. . 3 ⊢ ℝ ⊆ ℂ
4	3	a1i 11	. 2 ⊢ (𝜑 → ℝ ⊆ ℂ)
5		eldifi 4057	. . . . . 6 ⊢ (𝑦 ∈ (ℂ ∖ ℝ) → 𝑦 ∈ ℂ)
6	5	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → 𝑦 ∈ ℂ)
7	6	imcld 14834	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → (ℑ‘𝑦) ∈ ℝ)
8	7	recnd 10934	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → (ℑ‘𝑦) ∈ ℂ)
9		eldifn 4058	. . . . 5 ⊢ (𝑦 ∈ (ℂ ∖ ℝ) → ¬ 𝑦 ∈ ℝ)
10	9	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → ¬ 𝑦 ∈ ℝ)
11		reim0b 14758	. . . . . 6 ⊢ (𝑦 ∈ ℂ → (𝑦 ∈ ℝ ↔ (ℑ‘𝑦) = 0))
12	6, 11	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → (𝑦 ∈ ℝ ↔ (ℑ‘𝑦) = 0))
13	12	necon3bbid 2980	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → (¬ 𝑦 ∈ ℝ ↔ (ℑ‘𝑦) ≠ 0))
14	10, 13	mpbid 231	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → (ℑ‘𝑦) ≠ 0)
15	8, 14	absrpcld 15088	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → (abs‘(ℑ‘𝑦)) ∈ ℝ⁺)
16	6	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → 𝑦 ∈ ℂ)
17		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℝ)
18	17	recnd 10934	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℂ)
19	16, 18	subcld 11262	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (𝑦 − 𝑧) ∈ ℂ)
20		absimle 14949	. . . 4 ⊢ ((𝑦 − 𝑧) ∈ ℂ → (abs‘(ℑ‘(𝑦 − 𝑧))) ≤ (abs‘(𝑦 − 𝑧)))
21	19, 20	syl 17	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (abs‘(ℑ‘(𝑦 − 𝑧))) ≤ (abs‘(𝑦 − 𝑧)))
22	16, 18	imsubd 14856	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (ℑ‘(𝑦 − 𝑧)) = ((ℑ‘𝑦) − (ℑ‘𝑧)))
23		reim0 14757	. . . . . . 7 ⊢ (𝑧 ∈ ℝ → (ℑ‘𝑧) = 0)
24	23	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (ℑ‘𝑧) = 0)
25	24	oveq2d 7271	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → ((ℑ‘𝑦) − (ℑ‘𝑧)) = ((ℑ‘𝑦) − 0))
26	8	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (ℑ‘𝑦) ∈ ℂ)
27	26	subid1d 11251	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → ((ℑ‘𝑦) − 0) = (ℑ‘𝑦))
28	22, 25, 27	3eqtrrd 2783	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (ℑ‘𝑦) = (ℑ‘(𝑦 − 𝑧)))
29	28	fveq2d 6760	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (abs‘(ℑ‘𝑦)) = (abs‘(ℑ‘(𝑦 − 𝑧))))
30	18, 16	abssubd 15093	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (abs‘(𝑧 − 𝑦)) = (abs‘(𝑦 − 𝑧)))
31	21, 29, 30	3brtr4d 5102	. 2 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (abs‘(ℑ‘𝑦)) ≤ (abs‘(𝑧 − 𝑦)))
32		rlimrecl.3	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
33	1, 2, 4, 15, 31, 32	rlimcld2 15215	1 ⊢ (𝜑 → 𝐶 ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∖ cdif 3880 ⊆ wss 3883 class class class wbr 5070 ↦ cmpt 5153 ‘cfv 6418 (class class class)co 7255 supcsup 9129 ℂcc 10800 ℝcr 10801 0cc0 10802 +∞cpnf 10937 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 − cmin 11135 ℑcim 14737 abscabs 14873 ⇝_𝑟 crli 15122

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-sup 9131 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-rlim 15126

This theorem is referenced by: rlimge0 15218 climrecl 15220 rlimle 15287 divsqrtsumo1 26038 mulog2sumlem1 26587

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