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Mirrors > Home > MPE Home > Th. List > rpgecld | Structured version Visualization version GIF version |
Description: A number greater than or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
rpgecld.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
rpgecld.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
rpgecld.3 | ⊢ (𝜑 → 𝐵 ≤ 𝐴) |
Ref | Expression |
---|---|
rpgecld | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpgecld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
2 | rpgecld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | rpgecld.3 | . 2 ⊢ (𝜑 → 𝐵 ≤ 𝐴) | |
4 | rpgecl 12897 | . 2 ⊢ ((𝐵 ∈ ℝ+ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ≤ 𝐴) → 𝐴 ∈ ℝ+) | |
5 | 1, 2, 3, 4 | syl3anc 1371 | 1 ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 class class class wbr 5103 ℝcr 11008 ≤ cle 11148 ℝ+crp 12869 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-resscn 11066 ax-1cn 11067 ax-addrcl 11070 ax-rnegex 11080 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-rp 12870 |
This theorem is referenced by: rlimno1 15492 isumrpcl 15682 divlogrlim 25936 logno1 25937 chprpcl 26501 vmadivsumb 26777 vmalogdivsum2 26832 vmalogdivsum 26833 2vmadivsumlem 26834 selbergb 26843 selberg2b 26846 selberg3lem2 26852 selberg3 26853 selberg4lem1 26854 selberg4 26855 selberg3r 26863 selberg4r 26864 selberg34r 26865 pntrlog2bndlem1 26871 pntrlog2bndlem2 26872 pntrlog2bndlem3 26873 pntrlog2bndlem4 26874 pntrlog2bndlem5 26875 pntrlog2bndlem6a 26876 pntrlog2bndlem6 26877 pntrlog2bnd 26878 pntibndlem2 26885 pntlemb 26891 |
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