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Mirrors > Home > MPE Home > Th. List > xmetcl | 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 |
---|---|
xmetcl | ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xmetf 24355 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) | |
2 | fovcdm 7603 | . 2 ⊢ ((𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ*) | |
3 | 1, 2 | syl3an1 1162 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 ∈ wcel 2106 × cxp 5687 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℝ*cxr 11292 ∞Metcxmet 21367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 df-xr 11297 df-xmet 21375 |
This theorem is referenced by: xmetge0 24370 xmetlecl 24372 xmetsym 24373 xmetrtri 24381 xmetrtri2 24382 xmetgt0 24384 prdsdsf 24393 prdsxmetlem 24394 imasdsf1olem 24399 imasf1oxmet 24401 xpsdsval 24407 xblpnf 24422 bldisj 24424 blgt0 24425 xblss2 24428 blhalf 24431 xbln0 24440 blin 24447 blss 24451 xmscl 24488 prdsbl 24520 blsscls2 24533 blcld 24534 blcls 24535 comet 24542 stdbdxmet 24544 stdbdmet 24545 stdbdbl 24546 tmsxpsval2 24568 metcnpi3 24575 txmetcnp 24576 xrsmopn 24848 metdcnlem 24872 metdsf 24884 metdsge 24885 metdstri 24887 metdsle 24888 metdscnlem 24891 metnrmlem1 24895 metnrmlem3 24897 lmnn 25311 iscfil2 25314 iscau3 25326 dvlip2 26049 heicant 37642 |
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