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Mirrors > Home > ILE Home > Th. List > addgt0sr | Unicode version |
Description: The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.) |
Ref | Expression |
---|---|
addgt0sr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . . 4 | |
2 | ltrelsr 7693 | . . . . . . 7 | |
3 | 2 | brel 4661 | . . . . . 6 |
4 | 3 | simprd 113 | . . . . 5 |
5 | 2 | brel 4661 | . . . . . 6 |
6 | 5 | simprd 113 | . . . . 5 |
7 | 0r 7705 | . . . . . 6 | |
8 | ltasrg 7725 | . . . . . 6 | |
9 | 7, 8 | mp3an1 1319 | . . . . 5 |
10 | 4, 6, 9 | syl2anr 288 | . . . 4 |
11 | 1, 10 | mpbid 146 | . . 3 |
12 | 6 | adantr 274 | . . . 4 |
13 | 0idsr 7722 | . . . . 5 | |
14 | 13 | breq1d 3997 | . . . 4 |
15 | 12, 14 | syl 14 | . . 3 |
16 | 11, 15 | mpbid 146 | . 2 |
17 | ltsosr 7719 | . . 3 | |
18 | 17, 2 | sotri 5004 | . 2 |
19 | 16, 18 | syldan 280 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 2141 class class class wbr 3987 (class class class)co 5851 cnr 7252 c0r 7253 cplr 7256 cltr 7258 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-eprel 4272 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-irdg 6347 df-1o 6393 df-2o 6394 df-oadd 6397 df-omul 6398 df-er 6511 df-ec 6513 df-qs 6517 df-ni 7259 df-pli 7260 df-mi 7261 df-lti 7262 df-plpq 7299 df-mpq 7300 df-enq 7302 df-nqqs 7303 df-plqqs 7304 df-mqqs 7305 df-1nqqs 7306 df-rq 7307 df-ltnqqs 7308 df-enq0 7379 df-nq0 7380 df-0nq0 7381 df-plq0 7382 df-mq0 7383 df-inp 7421 df-i1p 7422 df-iplp 7423 df-iltp 7425 df-enr 7681 df-nr 7682 df-plr 7683 df-ltr 7685 df-0r 7686 |
This theorem is referenced by: (None) |
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