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Mirrors > Home > MPE Home > Th. List > cnfldms | Structured version Visualization version GIF version |
Description: The complex number field is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
cnfldms | ⊢ ℂfld ∈ MetSp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmet 23980 | . 2 ⊢ (abs ∘ − ) ∈ (Met‘ℂ) | |
2 | eqid 2736 | . 2 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − )) | |
3 | cnxmet 23981 | . . . 4 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
4 | 2 | mopntopon 23637 | . . . 4 ⊢ ((abs ∘ − ) ∈ (∞Met‘ℂ) → (MetOpen‘(abs ∘ − )) ∈ (TopOn‘ℂ)) |
5 | cnfldbas 20646 | . . . . 5 ⊢ ℂ = (Base‘ℂfld) | |
6 | cnfldtset 20650 | . . . . 5 ⊢ (MetOpen‘(abs ∘ − )) = (TopSet‘ℂfld) | |
7 | 5, 6 | topontopn 22134 | . . . 4 ⊢ ((MetOpen‘(abs ∘ − )) ∈ (TopOn‘ℂ) → (MetOpen‘(abs ∘ − )) = (TopOpen‘ℂfld)) |
8 | 3, 4, 7 | mp2b 10 | . . 3 ⊢ (MetOpen‘(abs ∘ − )) = (TopOpen‘ℂfld) |
9 | absf 15094 | . . . . . 6 ⊢ abs:ℂ⟶ℝ | |
10 | subf 11269 | . . . . . 6 ⊢ − :(ℂ × ℂ)⟶ℂ | |
11 | fco 6654 | . . . . . 6 ⊢ ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ) | |
12 | 9, 10, 11 | mp2an 690 | . . . . 5 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℝ |
13 | ffn 6630 | . . . . 5 ⊢ ((abs ∘ − ):(ℂ × ℂ)⟶ℝ → (abs ∘ − ) Fn (ℂ × ℂ)) | |
14 | fnresdm 6582 | . . . . 5 ⊢ ((abs ∘ − ) Fn (ℂ × ℂ) → ((abs ∘ − ) ↾ (ℂ × ℂ)) = (abs ∘ − )) | |
15 | 12, 13, 14 | mp2b 10 | . . . 4 ⊢ ((abs ∘ − ) ↾ (ℂ × ℂ)) = (abs ∘ − ) |
16 | cnfldds 20652 | . . . . 5 ⊢ (abs ∘ − ) = (dist‘ℂfld) | |
17 | 16 | reseq1i 5899 | . . . 4 ⊢ ((abs ∘ − ) ↾ (ℂ × ℂ)) = ((dist‘ℂfld) ↾ (ℂ × ℂ)) |
18 | 15, 17 | eqtr3i 2766 | . . 3 ⊢ (abs ∘ − ) = ((dist‘ℂfld) ↾ (ℂ × ℂ)) |
19 | 8, 5, 18 | isms2 23648 | . 2 ⊢ (ℂfld ∈ MetSp ↔ ((abs ∘ − ) ∈ (Met‘ℂ) ∧ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − )))) |
20 | 1, 2, 19 | mpbir2an 709 | 1 ⊢ ℂfld ∈ MetSp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ∈ wcel 2104 × cxp 5598 ↾ cres 5602 ∘ ccom 5604 Fn wfn 6453 ⟶wf 6454 ‘cfv 6458 ℂcc 10915 ℝcr 10916 − cmin 11251 abscabs 14990 distcds 17016 TopOpenctopn 17177 ∞Metcxmet 20627 Metcmet 20628 MetOpencmopn 20632 ℂfldccnfld 20642 TopOnctopon 22104 MetSpcms 23516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 ax-pre-sup 10995 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-1st 7863 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-er 8529 df-map 8648 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-sup 9245 df-inf 9246 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-div 11679 df-nn 12020 df-2 12082 df-3 12083 df-4 12084 df-5 12085 df-6 12086 df-7 12087 df-8 12088 df-9 12089 df-n0 12280 df-z 12366 df-dec 12484 df-uz 12629 df-q 12735 df-rp 12777 df-xneg 12894 df-xadd 12895 df-xmul 12896 df-fz 13286 df-seq 13768 df-exp 13829 df-cj 14855 df-re 14856 df-im 14857 df-sqrt 14991 df-abs 14992 df-struct 16893 df-slot 16928 df-ndx 16940 df-base 16958 df-plusg 17020 df-mulr 17021 df-starv 17022 df-tset 17026 df-ple 17027 df-ds 17029 df-unif 17030 df-rest 17178 df-topn 17179 df-topgen 17199 df-psmet 20634 df-xmet 20635 df-met 20636 df-bl 20637 df-mopn 20638 df-cnfld 20643 df-top 22088 df-topon 22105 df-topsp 22127 df-bases 22141 df-xms 23518 df-ms 23519 |
This theorem is referenced by: cnfldxms 23985 cnfldtps 23986 cnngp 23988 cncms 24564 cnpwstotbnd 35999 |
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