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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 485 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℂ) → 𝐹:𝑍⟶ℂ) | |
2 | lmclim.3 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | uzssz 12820 | . . . . 5 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
4 | zsscn 12543 | . . . . 5 ⊢ ℤ ⊆ ℂ | |
5 | 3, 4 | sstri 3982 | . . . 4 ⊢ (ℤ≥‘𝑀) ⊆ ℂ |
6 | 2, 5 | eqsstri 4007 | . . 3 ⊢ 𝑍 ⊆ ℂ |
7 | cnex 11168 | . . . 4 ⊢ ℂ ∈ V | |
8 | elpm2r 8817 | . . . 4 ⊢ (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝑍⟶ℂ ∧ 𝑍 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑pm ℂ)) | |
9 | 7, 7, 8 | mpanl12 700 | . . 3 ⊢ ((𝐹:𝑍⟶ℂ ∧ 𝑍 ⊆ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ)) |
10 | 1, 6, 9 | sylancl 586 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ)) |
11 | fdm 6708 | . . . 4 ⊢ (𝐹:𝑍⟶ℂ → dom 𝐹 = 𝑍) | |
12 | eqimss2 4032 | . . . 4 ⊢ (dom 𝐹 = 𝑍 → 𝑍 ⊆ dom 𝐹) | |
13 | 1, 11, 12 | 3syl 18 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℂ) → 𝑍 ⊆ dom 𝐹) |
14 | lmclim.2 | . . . 4 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
15 | 14, 2 | lmclim 24742 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑍 ⊆ dom 𝐹) → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝐹 ⇝ 𝑃))) |
16 | 13, 15 | syldan 591 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℂ) → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝐹 ⇝ 𝑃))) |
17 | 10, 16 | mpbirand 705 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℂ) → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹 ⇝ 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3469 ⊆ wss 3939 class class class wbr 5136 dom cdm 5664 ⟶wf 6523 ‘cfv 6527 (class class class)co 7388 ↑pm cpm 8799 ℂcc 11085 ℤcz 12535 ℤ≥cuz 12799 ⇝ cli 15405 TopOpenctopn 17344 ℂfldccnfld 20871 ⇝𝑡clm 22652 |
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 2702 ax-rep 5273 ax-sep 5287 ax-nul 5294 ax-pow 5351 ax-pr 5415 ax-un 7703 ax-cnex 11143 ax-resscn 11144 ax-1cn 11145 ax-icn 11146 ax-addcl 11147 ax-addrcl 11148 ax-mulcl 11149 ax-mulrcl 11150 ax-mulcom 11151 ax-addass 11152 ax-mulass 11153 ax-distr 11154 ax-i2m1 11155 ax-1ne0 11156 ax-1rid 11157 ax-rnegex 11158 ax-rrecex 11159 ax-cnre 11160 ax-pre-lttri 11161 ax-pre-lttrn 11162 ax-pre-ltadd 11163 ax-pre-mulgt0 11164 ax-pre-sup 11165 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3374 df-reu 3375 df-rab 3429 df-v 3471 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4314 df-if 4518 df-pw 4593 df-sn 4618 df-pr 4620 df-tp 4622 df-op 4624 df-uni 4897 df-iun 4987 df-br 5137 df-opab 5199 df-mpt 5220 df-tr 5254 df-id 5562 df-eprel 5568 df-po 5576 df-so 5577 df-fr 5619 df-we 5621 df-xp 5670 df-rel 5671 df-cnv 5672 df-co 5673 df-dm 5674 df-rn 5675 df-res 5676 df-ima 5677 df-pred 6284 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-riota 7344 df-ov 7391 df-oprab 7392 df-mpo 7393 df-om 7834 df-1st 7952 df-2nd 7953 df-frecs 8243 df-wrecs 8274 df-recs 8348 df-rdg 8387 df-1o 8443 df-er 8681 df-map 8800 df-pm 8801 df-en 8918 df-dom 8919 df-sdom 8920 df-fin 8921 df-sup 9414 df-inf 9415 df-pnf 11227 df-mnf 11228 df-xr 11229 df-ltxr 11230 df-le 11231 df-sub 11423 df-neg 11424 df-div 11849 df-nn 12190 df-2 12252 df-3 12253 df-4 12254 df-5 12255 df-6 12256 df-7 12257 df-8 12258 df-9 12259 df-n0 12450 df-z 12536 df-dec 12655 df-uz 12800 df-q 12910 df-rp 12952 df-xneg 13069 df-xadd 13070 df-xmul 13071 df-fz 13462 df-seq 13944 df-exp 14005 df-cj 15023 df-re 15024 df-im 15025 df-sqrt 15159 df-abs 15160 df-clim 15409 df-struct 17057 df-slot 17092 df-ndx 17104 df-base 17122 df-plusg 17187 df-mulr 17188 df-starv 17189 df-tset 17193 df-ple 17194 df-ds 17196 df-unif 17197 df-rest 17345 df-topn 17346 df-topgen 17366 df-psmet 20863 df-xmet 20864 df-met 20865 df-bl 20866 df-mopn 20867 df-cnfld 20872 df-top 22318 df-topon 22335 df-bases 22371 df-lm 22655 |
This theorem is referenced by: lmlim 32693 climreeq 44039 |
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