Nearby theorems
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| Mirrors > Home > MPE Home > Th. List > mscl | Structured version Visualization version GIF version | ||
| Description: Closure of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 2-Oct-2015.) |
| Ref | Expression |
|---|---|
| mscl.x | β’ π = (Baseβπ) |
| mscl.d | β’ π· = (distβπ) |
| Ref | Expression |
|---|---|
| mscl | β’ ((π β MetSp β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) β β) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ovres 7581 | . . 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 MetSp 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 | msmet2 24353 | . . 3 β’ (π β MetSp β (π· βΎ (π Γ π)) β (Metβπ)) |
| 6 | metcl 24225 | . . 3 β’ (((π· βΎ (π Γ π)) β (Metβπ) 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 MetSp β§ π΄ β π β§ π΅ β π) β (π΄(π· βΎ (π Γ π))π΅) β β) |
| 8 | 2, 7 | eqeltrrd 2829 | 1 β’ ((π β MetSp 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 1534 β wcel 2099 Γ cxp 5670 βΎ cres 5674 βcfv 6542 (class class class)co 7414 βcr 11129 Basecbs 17171 distcds 17233 Metcmet 21252 MetSpcms 24211 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-sup 9457 df-inf 9458 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-n0 12495 df-z 12581 df-uz 12845 df-q 12955 df-rp 12999 df-xneg 13116 df-xadd 13117 df-xmul 13118 df-topgen 17416 df-psmet 21258 df-xmet 21259 df-met 21260 df-bl 21261 df-mopn 21262 df-top 22783 df-topon 22800 df-topsp 22822 df-bases 22836 df-xms 24213 df-ms 24214 |
| This theorem is referenced by: ngptgp 24532 nlmvscnlem2 24589 nlmvscnlem1 24590 nghmcn 24649 ipcnlem2 25159 ipcnlem1 25160 |
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