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Theorem rlimrecl 15601

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 11191	. . 3 ⊢ ℝ ⊆ ℂ
4	3	a1i 11	. 2 ⊢ (𝜑 → ℝ ⊆ ℂ)
5		eldifi 4111	. . . . . 6 ⊢ (𝑦 ∈ (ℂ ∖ ℝ) → 𝑦 ∈ ℂ)
6	5	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → 𝑦 ∈ ℂ)
7	6	imcld 15219	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → (ℑ‘𝑦) ∈ ℝ)
8	7	recnd 11268	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → (ℑ‘𝑦) ∈ ℂ)
9		eldifn 4112	. . . . 5 ⊢ (𝑦 ∈ (ℂ ∖ ℝ) → ¬ 𝑦 ∈ ℝ)
10	9	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → ¬ 𝑦 ∈ ℝ)
11		reim0b 15143	. . . . . 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 2970	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → (¬ 𝑦 ∈ ℝ ↔ (ℑ‘𝑦) ≠ 0))
14	10, 13	mpbid 232	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → (ℑ‘𝑦) ≠ 0)
15	8, 14	absrpcld 15472	. 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 11268	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℂ)
19	16, 18	subcld 11599	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (𝑦 − 𝑧) ∈ ℂ)
20		absimle 15333	. . . 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 15241	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (ℑ‘(𝑦 − 𝑧)) = ((ℑ‘𝑦) − (ℑ‘𝑧)))
23		reim0 15142	. . . . . . 7 ⊢ (𝑧 ∈ ℝ → (ℑ‘𝑧) = 0)
24	23	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (ℑ‘𝑧) = 0)
25	24	oveq2d 7426	. . . . 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 11588	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → ((ℑ‘𝑦) − 0) = (ℑ‘𝑦))
28	22, 25, 27	3eqtrrd 2776	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (ℑ‘𝑦) = (ℑ‘(𝑦 − 𝑧)))
29	28	fveq2d 6885	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (abs‘(ℑ‘𝑦)) = (abs‘(ℑ‘(𝑦 − 𝑧))))
30	18, 16	abssubd 15477	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (abs‘(𝑧 − 𝑦)) = (abs‘(𝑦 − 𝑧)))
31	21, 29, 30	3brtr4d 5156	. 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 15599	1 ⊢ (𝜑 → 𝐶 ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2933 ∖ cdif 3928 ⊆ wss 3931 class class class wbr 5124 ↦ cmpt 5206 ‘cfv 6536 (class class class)co 7410 supcsup 9457 ℂcc 11132 ℝcr 11133 0cc0 11134 +∞cpnf 11271 ℝ^*cxr 11273 < clt 11274 ≤ cle 11275 − cmin 11471 ℑcim 15122 abscabs 15258 ⇝_𝑟 crli 15506

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 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8724 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9459 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-n0 12507 df-z 12594 df-uz 12858 df-rp 13014 df-seq 14025 df-exp 14085 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-rlim 15510

This theorem is referenced by: rlimge0 15602 climrecl 15604 rlimle 15669 divsqrtsumo1 26951 mulog2sumlem1 27502

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