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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climreeq | Structured version Visualization version GIF version |
Description: If 𝐹 is a real function, then 𝐹 converges to 𝐴 with respect to the standard topology on the reals if and only if it converges to 𝐴 with respect to the standard topology on complex numbers. In the theorem, 𝑅 is defined to be convergence w.r.t. the standard topology on the reals and then 𝐹𝑅𝐴 represents the statement "𝐹 converges to 𝐴, with respect to the standard topology on the reals". Notice that there is no need for the hypothesis that 𝐴 is a real number. (Contributed by Glauco Siliprandi, 2-Jul-2017.) |
Ref | Expression |
---|---|
climreeq.1 | ⊢ 𝑅 = (⇝𝑡‘(topGen‘ran (,))) |
climreeq.2 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climreeq.3 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climreeq.4 | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) |
Ref | Expression |
---|---|
climreeq | ⊢ (𝜑 → (𝐹𝑅𝐴 ↔ 𝐹 ⇝ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climreeq.1 | . . 3 ⊢ 𝑅 = (⇝𝑡‘(topGen‘ran (,))) | |
2 | 1 | breqi 5111 | . 2 ⊢ (𝐹𝑅𝐴 ↔ 𝐹(⇝𝑡‘(topGen‘ran (,)))𝐴) |
3 | climreeq.3 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
4 | climreeq.4 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) | |
5 | ax-resscn 11107 | . . . . . 6 ⊢ ℝ ⊆ ℂ | |
6 | 5 | a1i 11 | . . . . 5 ⊢ (𝜑 → ℝ ⊆ ℂ) |
7 | 4, 6 | fssd 6686 | . . . 4 ⊢ (𝜑 → 𝐹:𝑍⟶ℂ) |
8 | eqid 2736 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
9 | climreeq.2 | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
10 | 8, 9 | lmclimf 24666 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℂ) → (𝐹(⇝𝑡‘(TopOpen‘ℂfld))𝐴 ↔ 𝐹 ⇝ 𝐴)) |
11 | 3, 7, 10 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐹(⇝𝑡‘(TopOpen‘ℂfld))𝐴 ↔ 𝐹 ⇝ 𝐴)) |
12 | 8 | tgioo2 24164 | . . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
13 | reex 11141 | . . . . . . 7 ⊢ ℝ ∈ V | |
14 | 13 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → ℝ ∈ V) |
15 | 8 | cnfldtop 24145 | . . . . . . 7 ⊢ (TopOpen‘ℂfld) ∈ Top |
16 | 15 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → (TopOpen‘ℂfld) ∈ Top) |
17 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐴 ∈ ℝ) | |
18 | 3 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝑀 ∈ ℤ) |
19 | 4 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐹:𝑍⟶ℝ) |
20 | 12, 9, 14, 16, 17, 18, 19 | lmss 22647 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → (𝐹(⇝𝑡‘(TopOpen‘ℂfld))𝐴 ↔ 𝐹(⇝𝑡‘(topGen‘ran (,)))𝐴)) |
21 | 20 | pm5.32da 579 | . . . 4 ⊢ (𝜑 → ((𝐴 ∈ ℝ ∧ 𝐹(⇝𝑡‘(TopOpen‘ℂfld))𝐴) ↔ (𝐴 ∈ ℝ ∧ 𝐹(⇝𝑡‘(topGen‘ran (,)))𝐴))) |
22 | simpr 485 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐹(⇝𝑡‘(TopOpen‘ℂfld))𝐴) → 𝐹(⇝𝑡‘(TopOpen‘ℂfld))𝐴) | |
23 | 3 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘(TopOpen‘ℂfld))𝐴) → 𝑀 ∈ ℤ) |
24 | 11 | biimpa 477 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘(TopOpen‘ℂfld))𝐴) → 𝐹 ⇝ 𝐴) |
25 | 4 | ffvelcdmda 7034 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ ℝ) |
26 | 25 | adantlr 713 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹(⇝𝑡‘(TopOpen‘ℂfld))𝐴) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ ℝ) |
27 | 9, 23, 24, 26 | climrecl 15464 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘(TopOpen‘ℂfld))𝐴) → 𝐴 ∈ ℝ) |
28 | 27 | ex 413 | . . . . . 6 ⊢ (𝜑 → (𝐹(⇝𝑡‘(TopOpen‘ℂfld))𝐴 → 𝐴 ∈ ℝ)) |
29 | 28 | ancrd 552 | . . . . 5 ⊢ (𝜑 → (𝐹(⇝𝑡‘(TopOpen‘ℂfld))𝐴 → (𝐴 ∈ ℝ ∧ 𝐹(⇝𝑡‘(TopOpen‘ℂfld))𝐴))) |
30 | 22, 29 | impbid2 225 | . . . 4 ⊢ (𝜑 → ((𝐴 ∈ ℝ ∧ 𝐹(⇝𝑡‘(TopOpen‘ℂfld))𝐴) ↔ 𝐹(⇝𝑡‘(TopOpen‘ℂfld))𝐴)) |
31 | simpr 485 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐹(⇝𝑡‘(topGen‘ran (,)))𝐴) → 𝐹(⇝𝑡‘(topGen‘ran (,)))𝐴) | |
32 | retopon 24125 | . . . . . . . . 9 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ) | |
33 | 32 | a1i 11 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘(topGen‘ran (,)))𝐴) → (topGen‘ran (,)) ∈ (TopOn‘ℝ)) |
34 | simpr 485 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘(topGen‘ran (,)))𝐴) → 𝐹(⇝𝑡‘(topGen‘ran (,)))𝐴) | |
35 | lmcl 22646 | . . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ 𝐹(⇝𝑡‘(topGen‘ran (,)))𝐴) → 𝐴 ∈ ℝ) | |
36 | 33, 34, 35 | syl2anc 584 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘(topGen‘ran (,)))𝐴) → 𝐴 ∈ ℝ) |
37 | 36 | ex 413 | . . . . . 6 ⊢ (𝜑 → (𝐹(⇝𝑡‘(topGen‘ran (,)))𝐴 → 𝐴 ∈ ℝ)) |
38 | 37 | ancrd 552 | . . . . 5 ⊢ (𝜑 → (𝐹(⇝𝑡‘(topGen‘ran (,)))𝐴 → (𝐴 ∈ ℝ ∧ 𝐹(⇝𝑡‘(topGen‘ran (,)))𝐴))) |
39 | 31, 38 | impbid2 225 | . . . 4 ⊢ (𝜑 → ((𝐴 ∈ ℝ ∧ 𝐹(⇝𝑡‘(topGen‘ran (,)))𝐴) ↔ 𝐹(⇝𝑡‘(topGen‘ran (,)))𝐴)) |
40 | 21, 30, 39 | 3bitr3d 308 | . . 3 ⊢ (𝜑 → (𝐹(⇝𝑡‘(TopOpen‘ℂfld))𝐴 ↔ 𝐹(⇝𝑡‘(topGen‘ran (,)))𝐴)) |
41 | 11, 40 | bitr3d 280 | . 2 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐹(⇝𝑡‘(topGen‘ran (,)))𝐴)) |
42 | 2, 41 | bitr4id 289 | 1 ⊢ (𝜑 → (𝐹𝑅𝐴 ↔ 𝐹 ⇝ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3445 ⊆ wss 3910 class class class wbr 5105 ran crn 5634 ⟶wf 6492 ‘cfv 6496 ℂcc 11048 ℝcr 11049 ℤcz 12498 ℤ≥cuz 12762 (,)cioo 13263 ⇝ cli 15365 TopOpenctopn 17302 topGenctg 17318 ℂfldccnfld 20794 Topctop 22240 TopOnctopon 22257 ⇝𝑡clm 22575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 ax-pre-sup 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-er 8647 df-map 8766 df-pm 8767 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-fi 9346 df-sup 9377 df-inf 9378 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-div 11812 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-7 12220 df-8 12221 df-9 12222 df-n0 12413 df-z 12499 df-dec 12618 df-uz 12763 df-q 12873 df-rp 12915 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-ioo 13267 df-fz 13424 df-fl 13696 df-seq 13906 df-exp 13967 df-cj 14983 df-re 14984 df-im 14985 df-sqrt 15119 df-abs 15120 df-clim 15369 df-rlim 15370 df-struct 17018 df-slot 17053 df-ndx 17065 df-base 17083 df-plusg 17145 df-mulr 17146 df-starv 17147 df-tset 17151 df-ple 17152 df-ds 17154 df-unif 17155 df-rest 17303 df-topn 17304 df-topgen 17324 df-psmet 20786 df-xmet 20787 df-met 20788 df-bl 20789 df-mopn 20790 df-cnfld 20795 df-top 22241 df-topon 22258 df-topsp 22280 df-bases 22294 df-lm 22578 df-xms 23671 df-ms 23672 |
This theorem is referenced by: xlimclim 44037 stirlingr 44303 |
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