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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 22866 | . 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 ℝ*cxr 10662 ∞Metcxmet 20458 |
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-xr 10667 df-xmet 20466 |
This theorem is referenced by: xmetge0 22881 xmetlecl 22883 xmetsym 22884 xmetrtri 22892 xmetrtri2 22893 xmetgt0 22895 prdsdsf 22904 prdsxmetlem 22905 imasdsf1olem 22910 imasf1oxmet 22912 xpsdsval 22918 xblpnf 22933 bldisj 22935 blgt0 22936 xblss2 22939 blhalf 22942 xbln0 22951 blin 22958 blss 22962 xmscl 22999 prdsbl 23028 blsscls2 23041 blcld 23042 blcls 23043 comet 23050 stdbdxmet 23052 stdbdmet 23053 stdbdbl 23054 tmsxpsval2 23076 metcnpi3 23083 txmetcnp 23084 xrsmopn 23347 metdcnlem 23371 metdsf 23383 metdsge 23384 metdstri 23386 metdsle 23387 metdscnlem 23390 metnrmlem1 23394 metnrmlem3 23396 lmnn 23793 iscfil2 23796 iscau3 23808 dvlip2 24519 heicant 34808 |
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