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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 13061 | . 2 ⊢ ((𝐵 ∈ ℝ+ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ≤ 𝐴) → 𝐴 ∈ ℝ+) | |
5 | 1, 2, 3, 4 | syl3anc 1370 | 1 ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 class class class wbr 5148 ℝcr 11152 ≤ cle 11294 ℝ+crp 13032 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-addrcl 11214 ax-rnegex 11224 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-rp 13033 |
This theorem is referenced by: rlimno1 15687 isumrpcl 15876 divlogrlim 26692 logno1 26693 chprpcl 27266 vmadivsumb 27542 vmalogdivsum2 27597 vmalogdivsum 27598 2vmadivsumlem 27599 selbergb 27608 selberg2b 27611 selberg3lem2 27617 selberg3 27618 selberg4lem1 27619 selberg4 27620 selberg3r 27628 selberg4r 27629 selberg34r 27630 pntrlog2bndlem1 27636 pntrlog2bndlem2 27637 pntrlog2bndlem3 27638 pntrlog2bndlem4 27639 pntrlog2bndlem5 27640 pntrlog2bndlem6a 27641 pntrlog2bndlem6 27642 pntrlog2bnd 27643 pntibndlem2 27650 pntlemb 27656 |
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