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Theorem rlimrecl 15299

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 10938	. . 3 ⊢ ℝ ⊆ ℂ
4	3	a1i 11	. 2 ⊢ (𝜑 → ℝ ⊆ ℂ)
5		eldifi 4060	. . . . . 6 ⊢ (𝑦 ∈ (ℂ ∖ ℝ) → 𝑦 ∈ ℂ)
6	5	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → 𝑦 ∈ ℂ)
7	6	imcld 14916	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → (ℑ‘𝑦) ∈ ℝ)
8	7	recnd 11013	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → (ℑ‘𝑦) ∈ ℂ)
9		eldifn 4061	. . . . 5 ⊢ (𝑦 ∈ (ℂ ∖ ℝ) → ¬ 𝑦 ∈ ℝ)
10	9	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → ¬ 𝑦 ∈ ℝ)
11		reim0b 14840	. . . . . 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 2981	. . . 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 15170	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → (abs‘(ℑ‘𝑦)) ∈ ℝ⁺)
16	6	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → 𝑦 ∈ ℂ)
17		simpr 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℝ)
18	17	recnd 11013	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℂ)
19	16, 18	subcld 11342	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (𝑦 − 𝑧) ∈ ℂ)
20		absimle 15031	. . . 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 14938	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (ℑ‘(𝑦 − 𝑧)) = ((ℑ‘𝑦) − (ℑ‘𝑧)))
23		reim0 14839	. . . . . . 7 ⊢ (𝑧 ∈ ℝ → (ℑ‘𝑧) = 0)
24	23	adantl 482	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (ℑ‘𝑧) = 0)
25	24	oveq2d 7283	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → ((ℑ‘𝑦) − (ℑ‘𝑧)) = ((ℑ‘𝑦) − 0))
26	8	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (ℑ‘𝑦) ∈ ℂ)
27	26	subid1d 11331	. . . . 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 6770	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (abs‘(ℑ‘𝑦)) = (abs‘(ℑ‘(𝑦 − 𝑧))))
30	18, 16	abssubd 15175	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (abs‘(𝑧 − 𝑦)) = (abs‘(𝑦 − 𝑧)))
31	21, 29, 30	3brtr4d 5105	. 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 15297	1 ⊢ (𝜑 → 𝐶 ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∖ cdif 3883 ⊆ wss 3886 class class class wbr 5073 ↦ cmpt 5156 ‘cfv 6426 (class class class)co 7267 supcsup 9186 ℂcc 10879 ℝcr 10880 0cc0 10881 +∞cpnf 11016 ℝ^*cxr 11018 < clt 11019 ≤ cle 11020 − cmin 11215 ℑcim 14819 abscabs 14955 ⇝_𝑟 crli 15204

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 ax-pre-sup 10959

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-2nd 7821 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-er 8485 df-pm 8605 df-en 8721 df-dom 8722 df-sdom 8723 df-sup 9188 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-div 11643 df-nn 11984 df-2 12046 df-3 12047 df-n0 12244 df-z 12330 df-uz 12593 df-rp 12741 df-seq 13732 df-exp 13793 df-cj 14820 df-re 14821 df-im 14822 df-sqrt 14956 df-abs 14957 df-rlim 15208

This theorem is referenced by: rlimge0 15300 climrecl 15302 rlimle 15369 divsqrtsumo1 26143 mulog2sumlem1 26692

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