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Theorem rlimrecl 15501

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 11081	. . 3 ⊢ ℝ ⊆ ℂ
4	3	a1i 11	. 2 ⊢ (𝜑 → ℝ ⊆ ℂ)
5		eldifi 4081	. . . . . 6 ⊢ (𝑦 ∈ (ℂ ∖ ℝ) → 𝑦 ∈ ℂ)
6	5	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → 𝑦 ∈ ℂ)
7	6	imcld 15116	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → (ℑ‘𝑦) ∈ ℝ)
8	7	recnd 11158	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → (ℑ‘𝑦) ∈ ℂ)
9		eldifn 4082	. . . . 5 ⊢ (𝑦 ∈ (ℂ ∖ ℝ) → ¬ 𝑦 ∈ ℝ)
10	9	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) → ¬ 𝑦 ∈ ℝ)
11		reim0b 15040	. . . . . 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 2967	. . . 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 15372	. 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 11158	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℂ)
19	16, 18	subcld 11490	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (𝑦 − 𝑧) ∈ ℂ)
20		absimle 15230	. . . 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 15138	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (ℑ‘(𝑦 − 𝑧)) = ((ℑ‘𝑦) − (ℑ‘𝑧)))
23		reim0 15039	. . . . . . 7 ⊢ (𝑧 ∈ ℝ → (ℑ‘𝑧) = 0)
24	23	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (ℑ‘𝑧) = 0)
25	24	oveq2d 7372	. . . . 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 11479	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → ((ℑ‘𝑦) − 0) = (ℑ‘𝑦))
28	22, 25, 27	3eqtrrd 2774	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (ℑ‘𝑦) = (ℑ‘(𝑦 − 𝑧)))
29	28	fveq2d 6836	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (abs‘(ℑ‘𝑦)) = (abs‘(ℑ‘(𝑦 − 𝑧))))
30	18, 16	abssubd 15377	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ ℝ)) ∧ 𝑧 ∈ ℝ) → (abs‘(𝑧 − 𝑦)) = (abs‘(𝑦 − 𝑧)))
31	21, 29, 30	3brtr4d 5128	. 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 15499	1 ⊢ (𝜑 → 𝐶 ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2930 ∖ cdif 3896 ⊆ wss 3899 class class class wbr 5096 ↦ cmpt 5177 ‘cfv 6490 (class class class)co 7356 supcsup 9341 ℂcc 11022 ℝcr 11023 0cc0 11024 +∞cpnf 11161 ℝ^*cxr 11163 < clt 11164 ≤ cle 11165 − cmin 11362 ℑcim 15019 abscabs 15155 ⇝_𝑟 crli 15406

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8633 df-pm 8764 df-en 8882 df-dom 8883 df-sdom 8884 df-sup 9343 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-n0 12400 df-z 12487 df-uz 12750 df-rp 12904 df-seq 13923 df-exp 13983 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-rlim 15410

This theorem is referenced by: rlimge0 15502 climrecl 15504 rlimle 15569 divsqrtsumo1 26948 mulog2sumlem1 27499

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