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| Mirrors > Home > MPE Home > Th. List > xmetlecl | Structured version Visualization version GIF version | ||
| Description: Real closure of an extended metric value that is upper bounded by a real. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| Ref | Expression |
|---|---|
| xmetlecl | β’ ((π· β (βMetβπ) 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 | xmetcl 23736 | . . . 4 β’ ((π· β (βMetβπ) β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) β β*) | |
| 2 | 1 | 3expb 1120 | . . 3 β’ ((π· β (βMetβπ) β§ (π΄ β π β§ π΅ β π)) β (π΄π·π΅) β β*) |
| 3 | 2 | 3adant3 1132 | . 2 β’ ((π· β (βMetβπ) 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 (βMetβπ) β§ (π΄ β π β§ π΅ β π) β§ (πΆ β β β§ (π΄π·π΅) β€ πΆ)) β πΆ β β) | |
| 5 | xmetge0 23749 | . . . 4 β’ ((π· β (βMetβπ) β§ π΄ β π β§ π΅ β π) β 0 β€ (π΄π·π΅)) | |
| 6 | 5 | 3expb 1120 | . . 3 β’ ((π· β (βMetβπ) β§ (π΄ β π β§ π΅ β π)) β 0 β€ (π΄π·π΅)) |
| 7 | 6 | 3adant3 1132 | . 2 β’ ((π· β (βMetβπ) 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 (βMetβπ) 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 13118 | . 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 (βMetβπ) 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 5125 βcfv 6516 (class class class)co 7377 βcr 11074 0cc0 11075 β*cxr 11212 β€ cle 11214 βMetcxmet 20833 |
| 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 2702 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 |
| 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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3364 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-id 5551 df-po 5565 df-so 5566 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-1st 7941 df-2nd 7942 df-er 8670 df-map 8789 df-en 8906 df-dom 8907 df-sdom 8908 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11411 df-neg 11412 df-div 11837 df-2 12240 df-rp 12940 df-xneg 13057 df-xadd 13058 df-xmul 13059 df-xmet 20841 |
| This theorem is referenced by: blss2 23809 blss 23830 xmeter 23838 metdcnlem 24251 |
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