Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > metcl | Structured version Visualization version Unicode version |
Description: Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.) |
Ref | Expression |
---|---|
metcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metf 22135 | . 2 | |
2 | fovrn 6804 | . 2 | |
3 | 1, 2 | syl3an1 1359 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 w3a 1037 wcel 1990 cxp 5112 wf 5884 cfv 5888 (class class class)co 6650 cr 9935 cme 19732 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 df-met 19740 |
This theorem is referenced by: mettri2 22146 metrtri 22162 prdsmet 22175 imasf1omet 22181 blpnf 22202 bl2in 22205 mscl 22266 metss2lem 22316 methaus 22325 nmf2 22397 metdsre 22656 iscmet3lem1 23089 minveclem2 23197 minveclem3b 23199 minveclem3 23200 minveclem4 23203 minveclem7 23206 dvlog2lem 24398 vacn 27549 nmcvcn 27550 smcnlem 27552 blocni 27660 minvecolem2 27731 minvecolem3 27732 minvecolem4 27736 minvecolem7 27739 metf1o 33551 mettrifi 33553 lmclim2 33554 geomcau 33555 isbnd3 33583 isbnd3b 33584 ssbnd 33587 totbndbnd 33588 equivbnd 33589 prdsbnd 33592 heibor1lem 33608 heiborlem6 33615 bfplem1 33621 bfplem2 33622 bfp 33623 rrncmslem 33631 rrnequiv 33634 rrntotbnd 33635 ioorrnopnlem 40524 |
Copyright terms: Public domain | W3C validator |