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Mirrors > Home > ILE Home > Th. List > gt0srpr | Unicode version |
Description: Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.) |
Ref | Expression |
---|---|
gt0srpr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enrer 7684 | . . . . 5 | |
2 | erdm 6519 | . . . . 5 | |
3 | 1, 2 | ax-mp 5 | . . . 4 |
4 | ltrelsr 7687 | . . . . . . 7 | |
5 | 4 | brel 4661 | . . . . . 6 |
6 | 5 | simprd 113 | . . . . 5 |
7 | df-nr 7676 | . . . . 5 | |
8 | 6, 7 | eleqtrdi 2263 | . . . 4 |
9 | ecelqsdm 6579 | . . . 4 | |
10 | 3, 8, 9 | sylancr 412 | . . 3 |
11 | opelxp 4639 | . . 3 | |
12 | 10, 11 | sylib 121 | . 2 |
13 | ltrelpr 7454 | . . . 4 | |
14 | 13 | brel 4661 | . . 3 |
15 | 14 | ancomd 265 | . 2 |
16 | df-0r 7680 | . . . . 5 | |
17 | 16 | breq1i 3994 | . . . 4 |
18 | 1pr 7503 | . . . . 5 | |
19 | ltsrprg 7696 | . . . . 5 | |
20 | 18, 18, 19 | mpanl12 434 | . . . 4 |
21 | 17, 20 | syl5bb 191 | . . 3 |
22 | ltaprg 7568 | . . . . 5 | |
23 | 18, 22 | mp3an3 1321 | . . . 4 |
24 | 23 | ancoms 266 | . . 3 |
25 | 21, 24 | bitr4d 190 | . 2 |
26 | 12, 15, 25 | pm5.21nii 699 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1348 wcel 2141 cop 3584 class class class wbr 3987 cxp 4607 cdm 4609 (class class class)co 5850 wer 6506 cec 6507 cqs 6508 cnp 7240 c1p 7241 cpp 7242 cltp 7244 cer 7245 cnr 7246 c0r 7247 cltr 7252 |
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 5853 df-oprab 5854 df-mpo 5855 df-1st 6116 df-2nd 6117 df-recs 6281 df-irdg 6346 df-1o 6392 df-2o 6393 df-oadd 6396 df-omul 6397 df-er 6509 df-ec 6511 df-qs 6515 df-ni 7253 df-pli 7254 df-mi 7255 df-lti 7256 df-plpq 7293 df-mpq 7294 df-enq 7296 df-nqqs 7297 df-plqqs 7298 df-mqqs 7299 df-1nqqs 7300 df-rq 7301 df-ltnqqs 7302 df-enq0 7373 df-nq0 7374 df-0nq0 7375 df-plq0 7376 df-mq0 7377 df-inp 7415 df-i1p 7416 df-iplp 7417 df-iltp 7419 df-enr 7675 df-nr 7676 df-ltr 7679 df-0r 7680 |
This theorem is referenced by: recexgt0sr 7722 mulgt0sr 7727 srpospr 7732 prsrpos 7734 |
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