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Mirrors > Home > ILE Home > Th. List > ge0xaddcl | Unicode version |
Description: The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
ge0xaddcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxrge0 9766 | . 2 | |
2 | elxrge0 9766 | . 2 | |
3 | xaddcl 9648 | . . . 4 | |
4 | 3 | ad2ant2r 500 | . . 3 |
5 | xaddge0 9666 | . . . 4 | |
6 | 5 | an4s 577 | . . 3 |
7 | elxrge0 9766 | . . 3 | |
8 | 4, 6, 7 | sylanbrc 413 | . 2 |
9 | 1, 2, 8 | syl2anb 289 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 1480 class class class wbr 3929 (class class class)co 5774 cc0 7625 cpnf 7802 cxr 7804 cle 7806 cxad 9562 cicc 9679 |
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 7716 ax-resscn 7717 ax-1cn 7718 ax-1re 7719 ax-icn 7720 ax-addcl 7721 ax-addrcl 7722 ax-mulcl 7723 ax-addcom 7725 ax-addass 7727 ax-i2m1 7730 ax-0id 7733 ax-rnegex 7734 ax-pre-ltirr 7737 ax-pre-ltwlin 7738 ax-pre-lttrn 7739 ax-pre-apti 7740 ax-pre-ltadd 7741 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 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-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-pnf 7807 df-mnf 7808 df-xr 7809 df-ltxr 7810 df-le 7811 df-xadd 9565 df-icc 9683 |
This theorem is referenced by: comet 12673 bdxmet 12675 |
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