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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 20663 | . . 3 β’ (πΉ β π΄ β πΉ:π΅βΆ(0[,)+β)) |
4 | 3 | ffvelcdmda 7079 | . 2 β’ ((πΉ β π΄ β§ π β π΅) β (πΉβπ) β (0[,)+β)) |
5 | elrege0 13434 | . . 3 β’ ((πΉβπ) β (0[,)+β) β ((πΉβπ) β β β§ 0 β€ (πΉβπ))) | |
6 | 5 | simprbi 496 | . 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 395 = wceq 1533 β wcel 2098 class class class wbr 5141 βcfv 6536 (class class class)co 7404 βcr 11108 0cc0 11109 +βcpnf 11246 β€ cle 11250 [,)cico 13329 Basecbs 17151 AbsValcabv 20657 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-addrcl 11170 ax-rnegex 11180 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-ico 13333 df-abv 20658 |
This theorem is referenced by: abvgt0 20669 abvneg 20675 abvcxp 27499 ostth2lem2 27518 |
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