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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 44776 | . . . . 5 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ) |
3 | 2 | ad2antlr 726 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) → 𝑗 ∈ ℤ) |
4 | eqid 2728 | . . . 4 ⊢ (ℤ≥‘𝑗) = (ℤ≥‘𝑗) | |
5 | climxlim2.f | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) | |
6 | 5 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐹:𝑍⟶ℝ*) |
7 | 1 | uzssd3 44799 | . . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → (ℤ≥‘𝑗) ⊆ 𝑍) |
8 | 7 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (ℤ≥‘𝑗) ⊆ 𝑍) |
9 | 6, 8 | fssresd 6759 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ*) |
10 | 9 | adantr 480 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) → (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ*) |
11 | simpr 484 | . . . 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 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐹 ⇝ 𝐴) |
14 | 1 | fvexi 6906 | . . . . . . . . 9 ⊢ 𝑍 ∈ V |
15 | 14 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ V) |
16 | 5, 15 | fexd 7234 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ V) |
17 | climres 15546 | . . . . . . 7 ⊢ ((𝑗 ∈ ℤ ∧ 𝐹 ∈ V) → ((𝐹 ↾ (ℤ≥‘𝑗)) ⇝ 𝐴 ↔ 𝐹 ⇝ 𝐴)) | |
18 | 2, 16, 17 | syl2anr 596 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝐹 ↾ (ℤ≥‘𝑗)) ⇝ 𝐴 ↔ 𝐹 ⇝ 𝐴)) |
19 | 13, 18 | mpbird 257 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹 ↾ (ℤ≥‘𝑗)) ⇝ 𝐴) |
20 | 19 | adantr 480 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) → (𝐹 ↾ (ℤ≥‘𝑗)) ⇝ 𝐴) |
21 | 3, 4, 10, 11, 20 | climxlim2lem 45224 | . . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) → (𝐹 ↾ (ℤ≥‘𝑗))~~>*𝐴) |
22 | 1, 5 | fuzxrpmcn 45207 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℝ* ↑pm ℂ)) |
23 | 22 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐹 ∈ (ℝ* ↑pm ℂ)) |
24 | 2 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ ℤ) |
25 | 23, 24 | xlimres 45200 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹~~>*𝐴 ↔ (𝐹 ↾ (ℤ≥‘𝑗))~~>*𝐴)) |
26 | 25 | adantr 480 | . . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) → (𝐹~~>*𝐴 ↔ (𝐹 ↾ (ℤ≥‘𝑗))~~>*𝐴)) |
27 | 21, 26 | mpbird 257 | . 2 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) → 𝐹~~>*𝐴) |
28 | climxlim2.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
29 | 5 | ffnd 6718 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝑍) |
30 | climcl 15470 | . . . . 5 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) | |
31 | 12, 30 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
32 | breldmg 5907 | . . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐴 ∈ ℂ ∧ 𝐹 ⇝ 𝐴) → 𝐹 ∈ dom ⇝ ) | |
33 | 16, 31, 12, 32 | syl3anc 1369 | . . 3 ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) |
34 | 28, 1, 29, 33 | climrescn 45127 | . 2 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) |
35 | 27, 34 | r19.29a 3158 | 1 ⊢ (𝜑 → 𝐹~~>*𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3470 ⊆ wss 3945 class class class wbr 5143 dom cdm 5673 ↾ cres 5675 ⟶wf 6539 ‘cfv 6543 (class class class)co 7415 ↑pm cpm 8840 ℂcc 11131 ℝ*cxr 11272 ℤcz 12583 ℤ≥cuz 12847 ⇝ cli 15455 ~~>*clsxlim 45197 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-er 8719 df-map 8841 df-pm 8842 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-fi 9429 df-sup 9460 df-inf 9461 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-q 12958 df-rp 13002 df-xneg 13119 df-xadd 13120 df-xmul 13121 df-ioo 13355 df-ioc 13356 df-ico 13357 df-icc 13358 df-fz 13512 df-fl 13784 df-seq 13994 df-exp 14054 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-clim 15459 df-rlim 15460 df-struct 17110 df-slot 17145 df-ndx 17157 df-base 17175 df-plusg 17240 df-mulr 17241 df-starv 17242 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-rest 17398 df-topn 17399 df-topgen 17419 df-ordt 17477 df-ps 18552 df-tsr 18553 df-psmet 21265 df-xmet 21266 df-met 21267 df-bl 21268 df-mopn 21269 df-cnfld 21274 df-top 22790 df-topon 22807 df-topsp 22829 df-bases 22843 df-lm 23127 df-xms 24220 df-ms 24221 df-xlim 45198 |
This theorem is referenced by: dfxlim2v 45226 meaiuninc3v 45863 |
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