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Mirrors > Home > ILE Home > Th. List > addgtge0d | Unicode version |
Description: Addition of positive and nonnegative numbers is positive. (Contributed by Asger C. Ipsen, 12-May-2021.) |
Ref | Expression |
---|---|
leidd.1 | |
ltnegd.2 | |
addgtge0d.3 | |
addgtge0d.4 |
Ref | Expression |
---|---|
addgtge0d |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leidd.1 | . 2 | |
2 | ltnegd.2 | . 2 | |
3 | addgtge0d.3 | . 2 | |
4 | addgtge0d.4 | . 2 | |
5 | addgtge0 8215 | . 2 | |
6 | 1, 2, 3, 4, 5 | syl22anc 1217 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 1480 class class class wbr 3929 (class class class)co 5774 cr 7622 cc0 7623 caddc 7626 clt 7803 cle 7804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-addcom 7723 ax-addass 7725 ax-i2m1 7728 ax-0id 7731 ax-rnegex 7732 ax-pre-ltwlin 7736 ax-pre-ltadd 7739 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-iota 5088 df-fv 5131 df-ov 5777 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 |
This theorem is referenced by: cosordlem 12933 |
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