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Mirrors > Home > ILE Home > Th. List > addlelt | Unicode version |
Description: If the sum of a real number and a positive real number is less than or equal to a third real number, the first real number is less than the third real number. (Contributed by AV, 1-Jul-2021.) |
Ref | Expression |
---|---|
addlelt |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpgt0 9453 | . . . 4 | |
2 | 1 | 3ad2ant3 1004 | . . 3 |
3 | rpre 9448 | . . . . 5 | |
4 | 3 | 3ad2ant3 1004 | . . . 4 |
5 | simp1 981 | . . . 4 | |
6 | 4, 5 | ltaddposd 8291 | . . 3 |
7 | 2, 6 | mpbid 146 | . 2 |
8 | simpl 108 | . . . . 5 | |
9 | 3 | adantl 275 | . . . . 5 |
10 | 8, 9 | readdcld 7795 | . . . 4 |
11 | 10 | 3adant2 1000 | . . 3 |
12 | simp2 982 | . . 3 | |
13 | ltletr 7853 | . . 3 | |
14 | 5, 11, 12, 13 | syl3anc 1216 | . 2 |
15 | 7, 14 | mpand 425 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wcel 1480 class class class wbr 3929 (class class class)co 5774 cr 7619 cc0 7620 caddc 7623 clt 7800 cle 7801 crp 9441 |
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 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-i2m1 7725 ax-0id 7728 ax-rnegex 7729 ax-pre-ltwlin 7733 ax-pre-ltadd 7736 |
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 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-rp 9442 |
This theorem is referenced by: zltaddlt1le 9789 |
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