Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > measle0 | Structured version Visualization version GIF version |
Description: If the measure of a given set is bounded by zero, it is zero. (Contributed by Thierry Arnoux, 20-Oct-2017.) |
Ref | Expression |
---|---|
measle0 | β’ ((π β (measuresβπ) β§ π΄ β π β§ (πβπ΄) β€ 0) β (πβπ΄) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 1138 | . 2 β’ ((π β (measuresβπ) β§ π΄ β π β§ (πβπ΄) β€ 0) β (πβπ΄) β€ 0) | |
2 | measvxrge0 32577 | . . . . 5 β’ ((π β (measuresβπ) β§ π΄ β π) β (πβπ΄) β (0[,]+β)) | |
3 | elxrge0 13302 | . . . . 5 β’ ((πβπ΄) β (0[,]+β) β ((πβπ΄) β β* β§ 0 β€ (πβπ΄))) | |
4 | 2, 3 | sylib 217 | . . . 4 β’ ((π β (measuresβπ) β§ π΄ β π) β ((πβπ΄) β β* β§ 0 β€ (πβπ΄))) |
5 | 4 | 3adant3 1132 | . . 3 β’ ((π β (measuresβπ) β§ π΄ β π β§ (πβπ΄) β€ 0) β ((πβπ΄) β β* β§ 0 β€ (πβπ΄))) |
6 | 5 | simprd 496 | . 2 β’ ((π β (measuresβπ) β§ π΄ β π β§ (πβπ΄) β€ 0) β 0 β€ (πβπ΄)) |
7 | 5 | simpld 495 | . . 3 β’ ((π β (measuresβπ) β§ π΄ β π β§ (πβπ΄) β€ 0) β (πβπ΄) β β*) |
8 | 0xr 11135 | . . 3 β’ 0 β β* | |
9 | xrletri3 13001 | . . 3 β’ (((πβπ΄) β β* β§ 0 β β*) β ((πβπ΄) = 0 β ((πβπ΄) β€ 0 β§ 0 β€ (πβπ΄)))) | |
10 | 7, 8, 9 | sylancl 586 | . 2 β’ ((π β (measuresβπ) β§ π΄ β π β§ (πβπ΄) β€ 0) β ((πβπ΄) = 0 β ((πβπ΄) β€ 0 β§ 0 β€ (πβπ΄)))) |
11 | 1, 6, 10 | mpbir2and 711 | 1 β’ ((π β (measuresβπ) β§ π΄ β π β§ (πβπ΄) β€ 0) β (πβπ΄) = 0) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 class class class wbr 5103 βcfv 6491 (class class class)co 7349 0cc0 10984 +βcpnf 11119 β*cxr 11121 β€ cle 11123 [,]cicc 13195 measurescmeas 32567 |
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 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7662 ax-cnex 11040 ax-resscn 11041 ax-1cn 11042 ax-addrcl 11045 ax-rnegex 11055 ax-cnre 11057 ax-pre-lttri 11058 ax-pre-lttrn 11059 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5528 df-po 5542 df-so 5543 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-ov 7352 df-oprab 7353 df-mpo 7354 df-er 8581 df-en 8817 df-dom 8818 df-sdom 8819 df-pnf 11124 df-mnf 11125 df-xr 11126 df-ltxr 11127 df-le 11128 df-icc 13199 df-esum 32400 df-meas 32568 |
This theorem is referenced by: aean 32616 |
Copyright terms: Public domain | W3C validator |