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Mirrors > Home > MPE Home > Th. List > metcl | Structured version Visualization version GIF 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 | ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metf 22182 | . 2 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ) | |
2 | fovrn 6846 | . 2 ⊢ ((𝐷:(𝑋 × 𝑋)⟶ℝ ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ) | |
3 | 1, 2 | syl3an1 1399 | 1 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1054 ∈ wcel 2030 × cxp 5141 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 ℝcr 9973 Metcme 19780 |
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-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 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-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-map 7901 df-met 19788 |
This theorem is referenced by: mettri2 22193 metrtri 22209 prdsmet 22222 imasf1omet 22228 blpnf 22249 bl2in 22252 mscl 22313 metss2lem 22363 methaus 22372 nmf2 22444 metdsre 22703 iscmet3lem1 23135 minveclem2 23243 minveclem3b 23245 minveclem3 23246 minveclem4 23249 minveclem7 23252 dvlog2lem 24443 vacn 27677 nmcvcn 27678 smcnlem 27680 blocni 27788 minvecolem2 27859 minvecolem3 27860 minvecolem4 27864 minvecolem7 27867 metf1o 33681 mettrifi 33683 lmclim2 33684 geomcau 33685 isbnd3 33713 isbnd3b 33714 ssbnd 33717 totbndbnd 33718 equivbnd 33719 prdsbnd 33722 heibor1lem 33738 heiborlem6 33745 bfplem1 33751 bfplem2 33752 bfp 33753 rrncmslem 33761 rrnequiv 33764 rrntotbnd 33765 ioorrnopnlem 40842 |
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