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Mirrors > Home > MPE Home > Th. List > abvge0 | Structured version Visualization version GIF version |
Description: The absolute value of a number is greater than or equal to zero. (Contributed by Mario Carneiro, 8-Sep-2014.) |
Ref | Expression |
---|---|
abvf.a | β’ π΄ = (AbsValβπ ) |
abvf.b | β’ π΅ = (Baseβπ ) |
Ref | Expression |
---|---|
abvge0 | β’ ((πΉ β π΄ β§ π β π΅) β 0 β€ (πΉβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abvf.a | . . . 4 β’ π΄ = (AbsValβπ ) | |
2 | abvf.b | . . . 4 β’ π΅ = (Baseβπ ) | |
3 | 1, 2 | abvfge0 20709 | . . 3 β’ (πΉ β π΄ β πΉ:π΅βΆ(0[,)+β)) |
4 | 3 | ffvelcdmda 7099 | . 2 β’ ((πΉ β π΄ β§ π β π΅) β (πΉβπ) β (0[,)+β)) |
5 | elrege0 13471 | . . 3 β’ ((πΉβπ) β (0[,)+β) β ((πΉβπ) β β β§ 0 β€ (πΉβπ))) | |
6 | 5 | simprbi 495 | . 2 β’ ((πΉβπ) β (0[,)+β) β 0 β€ (πΉβπ)) |
7 | 4, 6 | syl 17 | 1 β’ ((πΉ β π΄ β§ π β π΅) β 0 β€ (πΉβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 class class class wbr 5152 βcfv 6553 (class class class)co 7426 βcr 11145 0cc0 11146 +βcpnf 11283 β€ cle 11287 [,)cico 13366 Basecbs 17187 AbsValcabv 20703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-addrcl 11207 ax-rnegex 11217 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-ico 13370 df-abv 20704 |
This theorem is referenced by: abvgt0 20715 abvneg 20721 abvcxp 27568 ostth2lem2 27587 |
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