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Mirrors > Home > MPE Home > Th. List > lmclimf | Structured version Visualization version GIF version |
Description: Relate a limit on the metric space of complex numbers to our complex number limit notation. (Contributed by NM, 24-Jul-2007.) (Revised by Mario Carneiro, 1-May-2014.) |
Ref | Expression |
---|---|
lmclim.2 | ⊢ 𝐽 = (TopOpen‘ℂfld) |
lmclim.3 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
Ref | Expression |
---|---|
lmclimf | ⊢ ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℂ) → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹 ⇝ 𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 476 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℂ) → 𝐹:𝑍⟶ℂ) | |
2 | fdm 6089 | . . . 4 ⊢ (𝐹:𝑍⟶ℂ → dom 𝐹 = 𝑍) | |
3 | eqimss2 3691 | . . . 4 ⊢ (dom 𝐹 = 𝑍 → 𝑍 ⊆ dom 𝐹) | |
4 | 1, 2, 3 | 3syl 18 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℂ) → 𝑍 ⊆ dom 𝐹) |
5 | lmclim.2 | . . . 4 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
6 | lmclim.3 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
7 | 5, 6 | lmclim 23147 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑍 ⊆ dom 𝐹) → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝐹 ⇝ 𝑃))) |
8 | 4, 7 | syldan 486 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℂ) → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝐹 ⇝ 𝑃))) |
9 | uzssz 11745 | . . . . . 6 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
10 | zsscn 11423 | . . . . . 6 ⊢ ℤ ⊆ ℂ | |
11 | 9, 10 | sstri 3645 | . . . . 5 ⊢ (ℤ≥‘𝑀) ⊆ ℂ |
12 | 6, 11 | eqsstri 3668 | . . . 4 ⊢ 𝑍 ⊆ ℂ |
13 | cnex 10055 | . . . . 5 ⊢ ℂ ∈ V | |
14 | elpm2r 7917 | . . . . 5 ⊢ (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝑍⟶ℂ ∧ 𝑍 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑pm ℂ)) | |
15 | 13, 13, 14 | mpanl12 718 | . . . 4 ⊢ ((𝐹:𝑍⟶ℂ ∧ 𝑍 ⊆ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ)) |
16 | 1, 12, 15 | sylancl 695 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ)) |
17 | 16 | biantrurd 528 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℂ) → (𝐹 ⇝ 𝑃 ↔ (𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝐹 ⇝ 𝑃))) |
18 | 8, 17 | bitr4d 271 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℂ) → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹 ⇝ 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ⊆ wss 3607 class class class wbr 4685 dom cdm 5143 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 ↑pm cpm 7900 ℂcc 9972 ℤcz 11415 ℤ≥cuz 11725 ⇝ cli 14259 TopOpenctopn 16129 ℂfldccnfld 19794 ⇝𝑡clm 21078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-sup 8389 df-inf 8390 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-fz 12365 df-seq 12842 df-exp 12901 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-clim 14263 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-plusg 16001 df-mulr 16002 df-starv 16003 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-rest 16130 df-topn 16131 df-topgen 16151 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-cnfld 19795 df-top 20747 df-topon 20764 df-bases 20798 df-lm 21081 |
This theorem is referenced by: lmlim 30121 climreeq 40163 |
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