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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 13063 | . 2 ⊢ ((𝐵 ∈ ℝ+ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ≤ 𝐴) → 𝐴 ∈ ℝ+) | |
| 5 | 1, 2, 3, 4 | syl3anc 1373 | 1 ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 class class class wbr 5143 ℝcr 11154 ≤ cle 11296 ℝ+crp 13034 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-addrcl 11216 ax-rnegex 11226 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-rp 13035 |
| This theorem is referenced by: rlimno1 15690 isumrpcl 15879 divlogrlim 26677 logno1 26678 chprpcl 27251 vmadivsumb 27527 vmalogdivsum2 27582 vmalogdivsum 27583 2vmadivsumlem 27584 selbergb 27593 selberg2b 27596 selberg3lem2 27602 selberg3 27603 selberg4lem1 27604 selberg4 27605 selberg3r 27613 selberg4r 27614 selberg34r 27615 pntrlog2bndlem1 27621 pntrlog2bndlem2 27622 pntrlog2bndlem3 27623 pntrlog2bndlem4 27624 pntrlog2bndlem5 27625 pntrlog2bndlem6a 27626 pntrlog2bndlem6 27627 pntrlog2bnd 27628 pntibndlem2 27635 pntlemb 27641 |
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