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| Mirrors > Home > MPE Home > Th. List > xmscl | Structured version Visualization version GIF version | ||
| Description: Closure of the distance function of an extended metric space. (Contributed by Mario Carneiro, 2-Oct-2015.) |
| Ref | Expression |
|---|---|
| mscl.x | β’ π = (Baseβπ) |
| mscl.d | β’ π· = (distβπ) |
| Ref | Expression |
|---|---|
| xmscl | β’ ((π β βMetSp β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) β β*) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ovres 7575 | . . 3 β’ ((π΄ β π β§ π΅ β π) β (π΄(π· βΎ (π Γ π))π΅) = (π΄π·π΅)) | |
| 2 | 1 | 3adant1 1128 | . 2 β’ ((π β βMetSp β§ π΄ β π β§ π΅ β π) β (π΄(π· βΎ (π Γ π))π΅) = (π΄π·π΅)) |
| 3 | mscl.x | . . . 4 β’ π = (Baseβπ) | |
| 4 | mscl.d | . . . 4 β’ π· = (distβπ) | |
| 5 | 3, 4 | xmsxmet2 24185 | . . 3 β’ (π β βMetSp β (π· βΎ (π Γ π)) β (βMetβπ)) |
| 6 | xmetcl 24057 | . . 3 β’ (((π· βΎ (π Γ π)) β (βMetβπ) β§ π΄ β π β§ π΅ β π) β (π΄(π· βΎ (π Γ π))π΅) β β*) | |
| 7 | 5, 6 | syl3an1 1161 | . 2 β’ ((π β βMetSp β§ π΄ β π β§ π΅ β π) β (π΄(π· βΎ (π Γ π))π΅) β β*) |
| 8 | 2, 7 | eqeltrrd 2832 | 1 β’ ((π β βMetSp β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) β β*) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1085 = wceq 1539 β wcel 2104 Γ cxp 5673 βΎ cres 5677 βcfv 6542 (class class class)co 7411 β*cxr 11251 Basecbs 17148 distcds 17210 βMetcxmet 21129 βMetSpcxms 24043 |
| 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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-topgen 17393 df-psmet 21136 df-xmet 21137 df-bl 21139 df-mopn 21140 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-xms 24046 |
| This theorem is referenced by: (None) |
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