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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 12411 | . 2 ⊢ ((𝐵 ∈ ℝ+ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ≤ 𝐴) → 𝐴 ∈ ℝ+) | |
5 | 1, 2, 3, 4 | syl3anc 1367 | 1 ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 class class class wbr 5059 ℝcr 10530 ≤ cle 10670 ℝ+crp 12383 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-addrcl 10592 ax-rnegex 10602 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-rp 12384 |
This theorem is referenced by: rlimno1 15004 isumrpcl 15192 divlogrlim 25212 logno1 25213 chprpcl 25777 vmadivsumb 26053 vmalogdivsum2 26108 vmalogdivsum 26109 2vmadivsumlem 26110 selbergb 26119 selberg2b 26122 selberg3lem2 26128 selberg3 26129 selberg4lem1 26130 selberg4 26131 selberg3r 26139 selberg4r 26140 selberg34r 26141 pntrlog2bndlem1 26147 pntrlog2bndlem2 26148 pntrlog2bndlem3 26149 pntrlog2bndlem4 26150 pntrlog2bndlem5 26151 pntrlog2bndlem6a 26152 pntrlog2bndlem6 26153 pntrlog2bnd 26154 pntibndlem2 26161 pntlemb 26167 |
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