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| Mirrors > Home > ILE Home > Th. List > psmetlecl | 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 14229 | . . . 4 β’ ((π· β (PsMetβπ) β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) β β*) | |
| 2 | 1 | 3expb 1206 | . . 3 β’ ((π· β (PsMetβπ) β§ (π΄ β π β§ π΅ β π)) β (π΄π·π΅) β β*) |
| 3 | 2 | 3adant3 1019 | . 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 1027 | . 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 14234 | . . . 4 β’ ((π· β (PsMetβπ) β§ π΄ β π β§ π΅ β π) β 0 β€ (π΄π·π΅)) | |
| 6 | 5 | 3expb 1206 | . . 3 β’ ((π· β (PsMetβπ) β§ (π΄ β π β§ π΅ β π)) β 0 β€ (π΄π·π΅)) |
| 7 | 6 | 3adant3 1019 | . 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 1028 | . 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 9844 | . 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 1250 | 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 set class |
| Syntax hints: β wi 4 β§ wa 104 β§ w3a 980 β wcel 2160 class class class wbr 4018 βcfv 5231 (class class class)co 5891 βcr 7829 0cc0 7830 β*cxr 8010 β€ cle 8012 PsMetcpsmet 13815 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7921 ax-resscn 7922 ax-1cn 7923 ax-1re 7924 ax-icn 7925 ax-addcl 7926 ax-addrcl 7927 ax-mulcl 7928 ax-mulrcl 7929 ax-addcom 7930 ax-mulcom 7931 ax-addass 7932 ax-mulass 7933 ax-distr 7934 ax-i2m1 7935 ax-0lt1 7936 ax-1rid 7937 ax-0id 7938 ax-rnegex 7939 ax-precex 7940 ax-cnre 7941 ax-pre-ltirr 7942 ax-pre-ltwlin 7943 ax-pre-lttrn 7944 ax-pre-ltadd 7946 ax-pre-mulgt0 7947 |
| This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-po 4311 df-iso 4312 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-map 6668 df-pnf 8013 df-mnf 8014 df-xr 8015 df-ltxr 8016 df-le 8017 df-sub 8149 df-neg 8150 df-2 8997 df-xadd 9792 df-psmet 13823 |
| This theorem is referenced by: blss2ps 14309 blssps 14330 |
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