Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climxlim2 | Structured version Visualization version GIF version |
Description: A sequence of extended reals, converging w.r.t. the standard topology on the complex numbers is a converging sequence w.r.t. the standard topology on the extended reals. This is non-trivial, because +∞ and -∞ could, in principle, be complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
climxlim2.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climxlim2.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climxlim2.f | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
climxlim2.a | ⊢ (𝜑 → 𝐹 ⇝ 𝐴) |
Ref | Expression |
---|---|
climxlim2 | ⊢ (𝜑 → 𝐹~~>*𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climxlim2.z | . . . . . 6 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | 1 | eluzelz2 41683 | . . . . 5 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ) |
3 | 2 | ad2antlr 725 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) → 𝑗 ∈ ℤ) |
4 | eqid 2823 | . . . 4 ⊢ (ℤ≥‘𝑗) = (ℤ≥‘𝑗) | |
5 | climxlim2.f | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) | |
6 | 5 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐹:𝑍⟶ℝ*) |
7 | 1 | uzssd3 41707 | . . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → (ℤ≥‘𝑗) ⊆ 𝑍) |
8 | 7 | adantl 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (ℤ≥‘𝑗) ⊆ 𝑍) |
9 | 6, 8 | fssresd 6547 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ*) |
10 | 9 | adantr 483 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) → (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ*) |
11 | simpr 487 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) → (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) | |
12 | climxlim2.a | . . . . . . 7 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) | |
13 | 12 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐹 ⇝ 𝐴) |
14 | 1 | fvexi 6686 | . . . . . . . . 9 ⊢ 𝑍 ∈ V |
15 | 14 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ V) |
16 | 5, 15 | fexd 41386 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ V) |
17 | climres 14934 | . . . . . . 7 ⊢ ((𝑗 ∈ ℤ ∧ 𝐹 ∈ V) → ((𝐹 ↾ (ℤ≥‘𝑗)) ⇝ 𝐴 ↔ 𝐹 ⇝ 𝐴)) | |
18 | 2, 16, 17 | syl2anr 598 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝐹 ↾ (ℤ≥‘𝑗)) ⇝ 𝐴 ↔ 𝐹 ⇝ 𝐴)) |
19 | 13, 18 | mpbird 259 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹 ↾ (ℤ≥‘𝑗)) ⇝ 𝐴) |
20 | 19 | adantr 483 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) → (𝐹 ↾ (ℤ≥‘𝑗)) ⇝ 𝐴) |
21 | 3, 4, 10, 11, 20 | climxlim2lem 42133 | . . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) → (𝐹 ↾ (ℤ≥‘𝑗))~~>*𝐴) |
22 | 1, 5 | fuzxrpmcn 42116 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℝ* ↑pm ℂ)) |
23 | 22 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐹 ∈ (ℝ* ↑pm ℂ)) |
24 | 2 | adantl 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ ℤ) |
25 | 23, 24 | xlimres 42109 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹~~>*𝐴 ↔ (𝐹 ↾ (ℤ≥‘𝑗))~~>*𝐴)) |
26 | 25 | adantr 483 | . . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) → (𝐹~~>*𝐴 ↔ (𝐹 ↾ (ℤ≥‘𝑗))~~>*𝐴)) |
27 | 21, 26 | mpbird 259 | . 2 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) → 𝐹~~>*𝐴) |
28 | climxlim2.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
29 | 5 | ffnd 6517 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝑍) |
30 | climcl 14858 | . . . . 5 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) | |
31 | 12, 30 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
32 | breldmg 5780 | . . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐴 ∈ ℂ ∧ 𝐹 ⇝ 𝐴) → 𝐹 ∈ dom ⇝ ) | |
33 | 16, 31, 12, 32 | syl3anc 1367 | . . 3 ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) |
34 | 28, 1, 29, 33 | climrescn 42036 | . 2 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) |
35 | 27, 34 | r19.29a 3291 | 1 ⊢ (𝜑 → 𝐹~~>*𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 class class class wbr 5068 dom cdm 5557 ↾ cres 5559 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ↑pm cpm 8409 ℂcc 10537 ℝ*cxr 10676 ℤcz 11984 ℤ≥cuz 12246 ⇝ cli 14843 ~~>*clsxlim 42106 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fi 8877 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fl 13165 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-rlim 14848 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-plusg 16580 df-mulr 16581 df-starv 16582 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-rest 16698 df-topn 16699 df-topgen 16719 df-ordt 16776 df-ps 17812 df-tsr 17813 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-lm 21839 df-xms 22932 df-ms 22933 df-xlim 42107 |
This theorem is referenced by: dfxlim2v 42135 meaiuninc3v 42773 |
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