| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 24339 | . 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 1164 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ*) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 ∈ wcel 2108 × cxp 5683 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℝ*cxr 11294 ∞Metcxmet 21349 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-xr 11299 df-xmet 21357 |
| This theorem is referenced by: xmetge0 24354 xmetlecl 24356 xmetsym 24357 xmetrtri 24365 xmetrtri2 24366 xmetgt0 24368 prdsdsf 24377 prdsxmetlem 24378 imasdsf1olem 24383 imasf1oxmet 24385 xpsdsval 24391 xblpnf 24406 bldisj 24408 blgt0 24409 xblss2 24412 blhalf 24415 xbln0 24424 blin 24431 blss 24435 xmscl 24472 prdsbl 24504 blsscls2 24517 blcld 24518 blcls 24519 comet 24526 stdbdxmet 24528 stdbdmet 24529 stdbdbl 24530 tmsxpsval2 24552 metcnpi3 24559 txmetcnp 24560 xrsmopn 24834 metdcnlem 24858 metdsf 24870 metdsge 24871 metdstri 24873 metdsle 24874 metdscnlem 24877 metnrmlem1 24881 metnrmlem3 24883 lmnn 25297 iscfil2 25300 iscau3 25312 dvlip2 26034 heicant 37662 |
| Copyright terms: Public domain | W3C validator |