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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 13234 | . . 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 14852 | . . . . . . 7 ⊢ Rel ⇝ | |
7 | 6 | brrelex1i 5611 | . . . . . 6 ⊢ (𝐹 ⇝ 𝐴 → 𝐹 ∈ V) |
8 | 5, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ V) |
9 | eqid 2824 | . . . . . 6 ⊢ (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) | |
10 | 2, 9 | climmpt 14931 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ V) → (𝐹 ⇝ 𝐴 ↔ (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ⇝ 𝐴)) |
11 | 1, 8, 10 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ⇝ 𝐴)) |
12 | 5, 11 | mpbid 234 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ⇝ 𝐴) |
13 | climrecl.4 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) | |
14 | 13 | recnd 10672 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
15 | 14 | fmpttd 6882 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)):𝑍⟶ℂ) |
16 | 2, 1, 15 | rlimclim 14906 | . . 3 ⊢ (𝜑 → ((𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ⇝𝑟 𝐴 ↔ (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ⇝ 𝐴)) |
17 | 12, 16 | mpbird 259 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ⇝𝑟 𝐴) |
18 | 4, 17, 13 | rlimrecl 14940 | 1 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3497 class class class wbr 5069 ↦ cmpt 5149 ‘cfv 6358 supcsup 8907 ℂcc 10538 ℝcr 10539 +∞cpnf 10675 ℝ*cxr 10677 < clt 10678 ℤcz 11984 ℤ≥cuz 12246 ⇝ cli 14844 ⇝𝑟 crli 14845 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-pm 8412 df-en 8513 df-dom 8514 df-sdom 8515 df-sup 8909 df-inf 8910 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fl 13165 df-seq 13373 df-exp 13433 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-clim 14848 df-rlim 14849 |
This theorem is referenced by: climle 14999 climsqz 15000 climsqz2 15001 isumrecl 15123 iprodrecl 15359 prmreclem6 16260 mbflimlem 24271 emcllem7 25582 regamcl 25641 relgamcl 25642 rge0scvg 31196 esumpcvgval 31341 climlec3 32969 rrncmslem 35114 cvgdvgrat 40651 radcnvrat 40652 climreeq 41900 climreclf 41951 fnlimfvre 41961 climliminflimsupd 42088 climliminflimsup 42095 climxlim 42113 sge0isum 42716 |
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