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Mirrors > Home > MPE Home > Th. List > climrecl | Structured version Visualization version GIF version |
Description: The limit of a convergent real sequence is real. Corollary 12-2.5 of [Gleason] p. 172. (Contributed by NM, 10-Sep-2005.) (Proof shortened by Mario Carneiro, 10-May-2016.) |
Ref | Expression |
---|---|
climshft2.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climshft2.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climrecl.3 | ⊢ (𝜑 → 𝐹 ⇝ 𝐴) |
climrecl.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) |
Ref | Expression |
---|---|
climrecl | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climshft2.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | climshft2.1 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | 2 | uzsup 12964 | . . 3 ⊢ (𝑀 ∈ ℤ → sup(𝑍, ℝ*, < ) = +∞) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝜑 → sup(𝑍, ℝ*, < ) = +∞) |
5 | climrecl.3 | . . . 4 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) | |
6 | climrel 14607 | . . . . . . 7 ⊢ Rel ⇝ | |
7 | 6 | brrelex1i 5397 | . . . . . 6 ⊢ (𝐹 ⇝ 𝐴 → 𝐹 ∈ V) |
8 | 5, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ V) |
9 | eqid 2825 | . . . . . 6 ⊢ (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) | |
10 | 2, 9 | climmpt 14686 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ V) → (𝐹 ⇝ 𝐴 ↔ (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ⇝ 𝐴)) |
11 | 1, 8, 10 | syl2anc 579 | . . . 4 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ⇝ 𝐴)) |
12 | 5, 11 | mpbid 224 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ⇝ 𝐴) |
13 | climrecl.4 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) | |
14 | 13 | recnd 10392 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
15 | 14 | fmpttd 6639 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)):𝑍⟶ℂ) |
16 | 2, 1, 15 | rlimclim 14661 | . . 3 ⊢ (𝜑 → ((𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ⇝𝑟 𝐴 ↔ (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ⇝ 𝐴)) |
17 | 12, 16 | mpbird 249 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ⇝𝑟 𝐴) |
18 | 4, 17, 13 | rlimrecl 14695 | 1 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1656 ∈ wcel 2164 Vcvv 3414 class class class wbr 4875 ↦ cmpt 4954 ‘cfv 6127 supcsup 8621 ℂcc 10257 ℝcr 10258 +∞cpnf 10395 ℝ*cxr 10397 < clt 10398 ℤcz 11711 ℤ≥cuz 11975 ⇝ cli 14599 ⇝𝑟 crli 14600 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-pm 8130 df-en 8229 df-dom 8230 df-sdom 8231 df-sup 8623 df-inf 8624 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-n0 11626 df-z 11712 df-uz 11976 df-rp 12120 df-fl 12895 df-seq 13103 df-exp 13162 df-cj 14223 df-re 14224 df-im 14225 df-sqrt 14359 df-abs 14360 df-clim 14603 df-rlim 14604 |
This theorem is referenced by: climle 14754 climsqz 14755 climsqz2 14756 isumrecl 14878 iprodrecl 15112 prmreclem6 16003 mbflimlem 23840 emcllem7 25148 regamcl 25207 relgamcl 25208 rge0scvg 30536 esumpcvgval 30681 climlec3 32157 rrncmslem 34172 cvgdvgrat 39351 radcnvrat 39352 climreeq 40638 climreclf 40689 fnlimfvre 40699 climliminflimsupd 40826 climliminflimsup 40833 climxlim 40845 sge0isum 41433 |
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