Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > psmetlecl | Structured version Visualization version GIF version | ||
| Description: Real closure of an extended metric value that is upper bounded by a real. (Contributed by Thierry Arnoux, 11-Mar-2018.) |
| Ref | Expression |
|---|---|
| psmetlecl | β’ ((π· β (PsMetβπ) β§ (π΄ β π β§ π΅ β π) β§ (πΆ β β β§ (π΄π·π΅) β€ πΆ)) β (π΄π·π΅) β β) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | psmetcl 24033 | . . . 4 β’ ((π· β (PsMetβπ) β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) β β*) | |
| 2 | 1 | 3expb 1120 | . . 3 β’ ((π· β (PsMetβπ) β§ (π΄ β π β§ π΅ β π)) β (π΄π·π΅) β β*) |
| 3 | 2 | 3adant3 1132 | . 2 β’ ((π· β (PsMetβπ) β§ (π΄ β π β§ π΅ β π) β§ (πΆ β β β§ (π΄π·π΅) β€ πΆ)) β (π΄π·π΅) β β*) |
| 4 | simp3l 1201 | . 2 β’ ((π· β (PsMetβπ) β§ (π΄ β π β§ π΅ β π) β§ (πΆ β β β§ (π΄π·π΅) β€ πΆ)) β πΆ β β) | |
| 5 | psmetge0 24038 | . . . 4 β’ ((π· β (PsMetβπ) β§ π΄ β π β§ π΅ β π) β 0 β€ (π΄π·π΅)) | |
| 6 | 5 | 3expb 1120 | . . 3 β’ ((π· β (PsMetβπ) β§ (π΄ β π β§ π΅ β π)) β 0 β€ (π΄π·π΅)) |
| 7 | 6 | 3adant3 1132 | . 2 β’ ((π· β (PsMetβπ) β§ (π΄ β π β§ π΅ β π) β§ (πΆ β β β§ (π΄π·π΅) β€ πΆ)) β 0 β€ (π΄π·π΅)) |
| 8 | simp3r 1202 | . 2 β’ ((π· β (PsMetβπ) β§ (π΄ β π β§ π΅ β π) β§ (πΆ β β β§ (π΄π·π΅) β€ πΆ)) β (π΄π·π΅) β€ πΆ) | |
| 9 | xrrege0 13157 | . 2 β’ ((((π΄π·π΅) β β* β§ πΆ β β) β§ (0 β€ (π΄π·π΅) β§ (π΄π·π΅) β€ πΆ)) β (π΄π·π΅) β β) | |
| 10 | 3, 4, 7, 8, 9 | syl22anc 837 | 1 β’ ((π· β (PsMetβπ) β§ (π΄ β π β§ π΅ β π) β§ (πΆ β β β§ (π΄π·π΅) β€ πΆ)) β (π΄π·π΅) β β) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 β wcel 2106 class class class wbr 5148 βcfv 6543 (class class class)co 7411 βcr 11111 0cc0 11112 β*cxr 11251 β€ cle 11253 PsMetcpsmet 21128 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 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 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 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-2 12279 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-psmet 21136 |
| This theorem is referenced by: blss2ps 24129 blssps 24150 |
| Copyright terms: Public domain | W3C validator |