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Mirrors > Home > ILE Home > Th. List > rpge0 | GIF version |
Description: A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.) |
Ref | Expression |
---|---|
rpge0 | ⊢ (𝐴 ∈ ℝ+ → 0 ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpre 9441 | . 2 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
2 | rpgt0 9446 | . 2 ⊢ (𝐴 ∈ ℝ+ → 0 < 𝐴) | |
3 | 0re 7759 | . . 3 ⊢ 0 ∈ ℝ | |
4 | ltle 7844 | . . 3 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐴 → 0 ≤ 𝐴)) | |
5 | 3, 4 | mpan 420 | . 2 ⊢ (𝐴 ∈ ℝ → (0 < 𝐴 → 0 ≤ 𝐴)) |
6 | 1, 2, 5 | sylc 62 | 1 ⊢ (𝐴 ∈ ℝ+ → 0 ≤ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 class class class wbr 3924 ℝcr 7612 0cc0 7613 < clt 7793 ≤ cle 7794 ℝ+crp 9434 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1re 7707 ax-addrcl 7710 ax-rnegex 7722 ax-pre-ltirr 7725 ax-pre-lttrn 7727 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-cnv 4542 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-rp 9435 |
This theorem is referenced by: rprege0 9449 rpge0d 9480 rpsqrtcl 10806 ef01bndlem 11452 bdmet 12660 |
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