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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 22867 | . 2 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ) | |
2 | fovrn 7307 | . 2 ⊢ ((𝐷:(𝑋 × 𝑋)⟶ℝ ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ) | |
3 | 1, 2 | syl3an1 1155 | 1 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 ∈ wcel 2105 × cxp 5546 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 Metcmet 20459 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-met 20467 |
This theorem is referenced by: mettri2 22878 metrtri 22894 prdsmet 22907 imasf1omet 22913 blpnf 22934 bl2in 22937 mscl 22998 metss2lem 23048 methaus 23057 nmf2 23129 metdsre 23388 iscmet3lem1 23821 minveclem2 23956 minveclem3b 23958 minveclem3 23959 minveclem4 23962 minveclem7 23965 dvlog2lem 25162 vacn 28398 nmcvcn 28399 smcnlem 28401 blocni 28509 minvecolem2 28579 minvecolem3 28580 minvecolem4 28584 minvecolem7 28587 metf1o 34911 mettrifi 34913 lmclim2 34914 geomcau 34915 isbnd3 34943 isbnd3b 34944 ssbnd 34947 totbndbnd 34948 equivbnd 34949 prdsbnd 34952 heibor1lem 34968 heiborlem6 34975 bfplem1 34981 bfplem2 34982 bfp 34983 rrncmslem 34991 rrnequiv 34994 rrntotbnd 34995 ioorrnopnlem 42466 |
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