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Mirrors > Home > ILE Home > Th. List > rpgt0d | GIF version |
Description: A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
rpred.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
Ref | Expression |
---|---|
rpgt0d | ⊢ (𝜑 → 0 < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpred.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
2 | rpgt0 9421 | . 2 ⊢ (𝐴 ∈ ℝ+ → 0 < 𝐴) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → 0 < 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1465 class class class wbr 3899 0cc0 7588 < clt 7768 ℝ+crp 9409 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-rab 2402 df-v 2662 df-un 3045 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-rp 9410 |
This theorem is referenced by: rpregt0d 9458 ltmulgt11d 9487 ltmulgt12d 9488 gt0divd 9489 ge0divd 9490 lediv12ad 9511 expgt0 10294 nnesq 10379 bccl2 10482 resqrexlemp1rp 10746 resqrexlemover 10750 resqrexlemnm 10758 resqrexlemgt0 10760 resqrexlemglsq 10762 sqrtgt0d 10899 reccn2ap 11050 fsumlt 11201 eirraplem 11410 prmind2 11728 sqrt2irrlem 11766 ssblex 12527 mulc1cncf 12672 cncfmptc 12678 mulcncflem 12686 cnplimclemle 12733 pilem3 12791 trilpolemeq1 13160 taupi 13166 |
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