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Theorem rlimrecl 15536

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 11089	. . 3 ⊢ ℝ ⊆ ℂ
4	3	a1i 11	. 2 ⊢ (𝜑 → ℝ ⊆ ℂ)
5		eldifi 4072	. . . . . 6 ⊢ (𝑦 ∈ (ℂ ∖ ℝ) → 𝑦 ∈ ℂ)
6	5	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → 𝑦 ∈ ℂ)
7	6	imcld 15151	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → (ℑ‘𝑦) ∈ ℝ)
8	7	recnd 11167	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → (ℑ‘𝑦) ∈ ℂ)
9		eldifn 4073	. . . . 5 ⊢ (𝑦 ∈ (ℂ ∖ ℝ) → ¬ 𝑦 ∈ ℝ)
10	9	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → ¬ 𝑦 ∈ ℝ)
11		reim0b 15075	. . . . . 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 15407	. 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 11167	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℂ)
19	16, 18	subcld 11499	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (𝑦 − 𝑧) ∈ ℂ)
20		absimle 15265	. . . 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 15173	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (ℑ‘(𝑦 − 𝑧)) = ((ℑ‘𝑦) − (ℑ‘𝑧)))
23		reim0 15074	. . . . . . 7 ⊢ (𝑧 ∈ ℝ → (ℑ‘𝑧) = 0)
24	23	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (ℑ‘𝑧) = 0)
25	24	oveq2d 7377	. . . . 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 11488	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → ((ℑ‘𝑦) − 0) = (ℑ‘𝑦))
28	22, 25, 27	3eqtrrd 2777	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (ℑ‘𝑦) = (ℑ‘(𝑦 − 𝑧)))
29	28	fveq2d 6839	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (abs‘(ℑ‘𝑦)) = (abs‘(ℑ‘(𝑦 − 𝑧))))
30	18, 16	abssubd 15412	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (abs‘(𝑧 − 𝑦)) = (abs‘(𝑦 − 𝑧)))
31	21, 29, 30	3brtr4d 5118	. 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 15534	1 ⊢ (𝜑 → 𝐶 ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∖ cdif 3887 ⊆ wss 3890 class class class wbr 5086 ↦ cmpt 5167 ‘cfv 6493 (class class class)co 7361 supcsup 9347 ℂcc 11030 ℝcr 11031 0cc0 11032 +∞cpnf 11170 ℝ^*cxr 11172 < clt 11173 ≤ cle 11174 − cmin 11371 ℑcim 15054 abscabs 15190 ⇝_𝑟 crli 15441

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-pre-sup 11110

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-2nd 7937 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-er 8637 df-pm 8770 df-en 8888 df-dom 8889 df-sdom 8890 df-sup 9349 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-3 12239 df-n0 12432 df-z 12519 df-uz 12783 df-rp 12937 df-seq 13958 df-exp 14018 df-cj 15055 df-re 15056 df-im 15057 df-sqrt 15191 df-abs 15192 df-rlim 15445

This theorem is referenced by: rlimge0 15537 climrecl 15539 rlimle 15604 divsqrtsumo1 26964 mulog2sumlem1 27514

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