![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 13085 | . 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 2108 class class class wbr 5166 ℝcr 11183 ≤ cle 11325 ℝ+crp 13057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-resscn 11241 ax-1cn 11242 ax-addrcl 11245 ax-rnegex 11255 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-rp 13058 |
This theorem is referenced by: rlimno1 15702 isumrpcl 15891 divlogrlim 26695 logno1 26696 chprpcl 27269 vmadivsumb 27545 vmalogdivsum2 27600 vmalogdivsum 27601 2vmadivsumlem 27602 selbergb 27611 selberg2b 27614 selberg3lem2 27620 selberg3 27621 selberg4lem1 27622 selberg4 27623 selberg3r 27631 selberg4r 27632 selberg34r 27633 pntrlog2bndlem1 27639 pntrlog2bndlem2 27640 pntrlog2bndlem3 27641 pntrlog2bndlem4 27642 pntrlog2bndlem5 27643 pntrlog2bndlem6a 27644 pntrlog2bndlem6 27645 pntrlog2bnd 27646 pntibndlem2 27653 pntlemb 27659 |
Copyright terms: Public domain | W3C validator |