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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 14097 | . . . 4 β’ ((π· β (PsMetβπ) β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) β β*) | |
| 2 | 1 | 3expb 1205 | . . 3 β’ ((π· β (PsMetβπ) β§ (π΄ β π β§ π΅ β π)) β (π΄π·π΅) β β*) |
| 3 | 2 | 3adant3 1018 | . 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 1026 | . 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 14102 | . . . 4 β’ ((π· β (PsMetβπ) β§ π΄ β π β§ π΅ β π) β 0 β€ (π΄π·π΅)) | |
| 6 | 5 | 3expb 1205 | . . 3 β’ ((π· β (PsMetβπ) β§ (π΄ β π β§ π΅ β π)) β 0 β€ (π΄π·π΅)) |
| 7 | 6 | 3adant3 1018 | . 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 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 (π΄π·π΅) β€ πΆ) | |
| 9 | xrrege0 9838 | . 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 1249 | 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 979 β wcel 2158 class class class wbr 4015 βcfv 5228 (class class class)co 5888 βcr 7823 0cc0 7824 β*cxr 8004 β€ cle 8006 PsMetcpsmet 13696 |
| 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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulrcl 7923 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-precex 7934 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-ltadd 7940 ax-pre-mulgt0 7941 |
| This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-po 4308 df-iso 4309 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-map 6663 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-2 8991 df-xadd 9786 df-psmet 13704 |
| This theorem is referenced by: blss2ps 14177 blssps 14198 |
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