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Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimclim2 | Structured version Visualization version GIF version |
Description: Given a sequence of extended reals, it converges to a real number 𝐴 w.r.t. the standard topology on the reals (see climreeq 43114), if and only if it converges to 𝐴 w.r.t. to the standard topology on the extended reals. In order for the first part of the statement to even make sense, the sequence will of course eventually become (and stay) real: showing this, is the key step of the proof. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
xlimclim2.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
xlimclim2.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
xlimclim2.f | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
xlimclim2.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Ref | Expression |
---|---|
xlimclim2 | ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝐹~~>*𝐴) → 𝐹~~>*𝐴) | |
2 | xlimclim2.z | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | xlimclim2.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) | |
4 | 3 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐹~~>*𝐴) → 𝐹:𝑍⟶ℝ*) |
5 | xlimclim2.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
6 | 5 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐹~~>*𝐴) → 𝐴 ∈ ℝ) |
7 | xlimclim2.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
8 | 7 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹~~>*𝐴) → 𝑀 ∈ ℤ) |
9 | 8, 2, 4, 6, 1 | xlimxrre 43332 | . . . 4 ⊢ ((𝜑 ∧ 𝐹~~>*𝐴) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) |
10 | 2, 4, 6, 9 | xlimclim2lem 43340 | . . 3 ⊢ ((𝜑 ∧ 𝐹~~>*𝐴) → (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴)) |
11 | 1, 10 | mpbid 231 | . 2 ⊢ ((𝜑 ∧ 𝐹~~>*𝐴) → 𝐹 ⇝ 𝐴) |
12 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ⇝ 𝐴) → 𝐹 ⇝ 𝐴) | |
13 | 3 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ⇝ 𝐴) → 𝐹:𝑍⟶ℝ*) |
14 | 5 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ⇝ 𝐴) → 𝐴 ∈ ℝ) |
15 | 7 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ⇝ 𝐴) → 𝑀 ∈ ℤ) |
16 | 15, 2, 13, 14, 12 | climxrre 43251 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ⇝ 𝐴) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) |
17 | 2, 13, 14, 16 | xlimclim2lem 43340 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ⇝ 𝐴) → (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴)) |
18 | 12, 17 | mpbird 256 | . 2 ⊢ ((𝜑 ∧ 𝐹 ⇝ 𝐴) → 𝐹~~>*𝐴) |
19 | 11, 18 | impbida 798 | 1 ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 class class class wbr 5075 ⟶wf 6424 ‘cfv 6428 ℝcr 10859 ℝ*cxr 10997 ℤcz 12308 ℤ≥cuz 12571 ⇝ cli 15182 ~~>*clsxlim 43319 |
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-rep 5210 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7580 ax-cnex 10916 ax-resscn 10917 ax-1cn 10918 ax-icn 10919 ax-addcl 10920 ax-addrcl 10921 ax-mulcl 10922 ax-mulrcl 10923 ax-mulcom 10924 ax-addass 10925 ax-mulass 10926 ax-distr 10927 ax-i2m1 10928 ax-1ne0 10929 ax-1rid 10930 ax-rnegex 10931 ax-rrecex 10932 ax-cnre 10933 ax-pre-lttri 10934 ax-pre-lttrn 10935 ax-pre-ltadd 10936 ax-pre-mulgt0 10937 ax-pre-sup 10938 |
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-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5486 df-eprel 5492 df-po 5500 df-so 5501 df-fr 5541 df-we 5543 df-xp 5592 df-rel 5593 df-cnv 5594 df-co 5595 df-dm 5596 df-rn 5597 df-res 5598 df-ima 5599 df-pred 6197 df-ord 6264 df-on 6265 df-lim 6266 df-suc 6267 df-iota 6386 df-fun 6430 df-fn 6431 df-f 6432 df-f1 6433 df-fo 6434 df-f1o 6435 df-fv 6436 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-1st 7822 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-er 8487 df-map 8606 df-pm 8607 df-en 8723 df-dom 8724 df-sdom 8725 df-fin 8726 df-fi 9159 df-sup 9190 df-inf 9191 df-pnf 11000 df-mnf 11001 df-xr 11002 df-ltxr 11003 df-le 11004 df-sub 11196 df-neg 11197 df-div 11622 df-nn 11963 df-2 12025 df-3 12026 df-4 12027 df-5 12028 df-6 12029 df-7 12030 df-8 12031 df-9 12032 df-n0 12223 df-z 12309 df-dec 12427 df-uz 12572 df-q 12678 df-rp 12720 df-xneg 12837 df-xadd 12838 df-xmul 12839 df-ioo 13072 df-ioc 13073 df-ico 13074 df-icc 13075 df-fz 13229 df-fl 13501 df-seq 13711 df-exp 13772 df-cj 14799 df-re 14800 df-im 14801 df-sqrt 14935 df-abs 14936 df-clim 15186 df-rlim 15187 df-struct 16837 df-slot 16872 df-ndx 16884 df-base 16902 df-plusg 16964 df-mulr 16965 df-starv 16966 df-tset 16970 df-ple 16971 df-ds 16973 df-unif 16974 df-rest 17122 df-topn 17123 df-topgen 17143 df-ordt 17201 df-ps 18273 df-tsr 18274 df-psmet 20578 df-xmet 20579 df-met 20580 df-bl 20581 df-mopn 20582 df-cnfld 20587 df-top 22032 df-topon 22049 df-topsp 22071 df-bases 22085 df-lm 22369 df-xms 23462 df-ms 23463 df-xlim 43320 |
This theorem is referenced by: climxlim2lem 43346 dfxlim2v 43348 xlimclimdm 43355 |
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