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Mirrors > Home > MPE Home > Th. List > Mathboxes > measge0 | Structured version Visualization version GIF version |
Description: A measure is nonnegative. (Contributed by Thierry Arnoux, 9-Mar-2018.) |
Ref | Expression |
---|---|
measge0 | β’ ((π β (measuresβπ) β§ π΄ β π) β 0 β€ (πβπ΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | measvxrge0 32565 | . . 3 β’ ((π β (measuresβπ) β§ π΄ β π) β (πβπ΄) β (0[,]+β)) | |
2 | elxrge0 13303 | . . 3 β’ ((πβπ΄) β (0[,]+β) β ((πβπ΄) β β* β§ 0 β€ (πβπ΄))) | |
3 | 1, 2 | sylib 217 | . 2 β’ ((π β (measuresβπ) β§ π΄ β π) β ((πβπ΄) β β* β§ 0 β€ (πβπ΄))) |
4 | 3 | simprd 497 | 1 β’ ((π β (measuresβπ) β§ π΄ β π) β 0 β€ (πβπ΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β wcel 2107 class class class wbr 5104 βcfv 6492 (class class class)co 7350 0cc0 10985 +βcpnf 11120 β*cxr 11122 β€ cle 11124 [,]cicc 13196 measurescmeas 32555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-addrcl 11046 ax-rnegex 11056 ax-cnre 11058 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-fv 6500 df-ov 7353 df-oprab 7354 df-mpo 7355 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-icc 13200 df-esum 32388 df-meas 32556 |
This theorem is referenced by: sibfof 32701 |
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