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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 23382 | . 2 ⊢ (abs ∘ − ) ∈ (Met‘ℂ) | |
2 | eqid 2823 | . 2 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − )) | |
3 | cnxmet 23383 | . . . 4 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
4 | 2 | mopntopon 23051 | . . . 4 ⊢ ((abs ∘ − ) ∈ (∞Met‘ℂ) → (MetOpen‘(abs ∘ − )) ∈ (TopOn‘ℂ)) |
5 | cnfldbas 20551 | . . . . 5 ⊢ ℂ = (Base‘ℂfld) | |
6 | cnfldtset 20555 | . . . . 5 ⊢ (MetOpen‘(abs ∘ − )) = (TopSet‘ℂfld) | |
7 | 5, 6 | topontopn 21550 | . . . 4 ⊢ ((MetOpen‘(abs ∘ − )) ∈ (TopOn‘ℂ) → (MetOpen‘(abs ∘ − )) = (TopOpen‘ℂfld)) |
8 | 3, 4, 7 | mp2b 10 | . . 3 ⊢ (MetOpen‘(abs ∘ − )) = (TopOpen‘ℂfld) |
9 | absf 14699 | . . . . . 6 ⊢ abs:ℂ⟶ℝ | |
10 | subf 10890 | . . . . . 6 ⊢ − :(ℂ × ℂ)⟶ℂ | |
11 | fco 6533 | . . . . . 6 ⊢ ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ) | |
12 | 9, 10, 11 | mp2an 690 | . . . . 5 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℝ |
13 | ffn 6516 | . . . . 5 ⊢ ((abs ∘ − ):(ℂ × ℂ)⟶ℝ → (abs ∘ − ) Fn (ℂ × ℂ)) | |
14 | fnresdm 6468 | . . . . 5 ⊢ ((abs ∘ − ) Fn (ℂ × ℂ) → ((abs ∘ − ) ↾ (ℂ × ℂ)) = (abs ∘ − )) | |
15 | 12, 13, 14 | mp2b 10 | . . . 4 ⊢ ((abs ∘ − ) ↾ (ℂ × ℂ)) = (abs ∘ − ) |
16 | cnfldds 20557 | . . . . 5 ⊢ (abs ∘ − ) = (dist‘ℂfld) | |
17 | 16 | reseq1i 5851 | . . . 4 ⊢ ((abs ∘ − ) ↾ (ℂ × ℂ)) = ((dist‘ℂfld) ↾ (ℂ × ℂ)) |
18 | 15, 17 | eqtr3i 2848 | . . 3 ⊢ (abs ∘ − ) = ((dist‘ℂfld) ↾ (ℂ × ℂ)) |
19 | 8, 5, 18 | isms2 23062 | . 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 1537 ∈ wcel 2114 × cxp 5555 ↾ cres 5559 ∘ ccom 5561 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 ℂcc 10537 ℝcr 10538 − cmin 10872 abscabs 14595 distcds 16576 TopOpenctopn 16697 ∞Metcxmet 20532 Metcmet 20533 MetOpencmopn 20537 ℂfldccnfld 20547 TopOnctopon 21520 MetSpcms 22930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-fz 12896 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-plusg 16580 df-mulr 16581 df-starv 16582 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-rest 16698 df-topn 16699 df-topgen 16719 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-xms 22932 df-ms 22933 |
This theorem is referenced by: cnfldxms 23387 cnfldtps 23388 cnngp 23390 cncms 23960 cnpwstotbnd 35077 |
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