rpgecl - Metamath Proof Explorer

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Theorem rpgecl 12981

Description: A number greater than or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)

**Assertion**
Ref	Expression
rpgecl	⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ⁺)

**Proof of Theorem rpgecl**
Step	Hyp	Ref	Expression
1		simp2 1137	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ)
2		0red 11177	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 0 ∈ ℝ)
3		rpre 12960	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
4	3	3ad2ant1 1133	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ)
5		rpgt0 12964	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 0 < 𝐴)
6	5	3ad2ant1 1133	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 0 < 𝐴)
7		simp3 1138	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
8	2, 4, 1, 6, 7	ltletrd 11334	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 0 < 𝐵)
9		elrp 12953	. 2 ⊢ (𝐵 ∈ ℝ⁺ ↔ (𝐵 ∈ ℝ ∧ 0 < 𝐵))
10	1, 8, 9	sylanbrc 583	1 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ⁺)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1086 ∈ wcel 2109 class class class wbr 5107 ℝcr 11067 0cc0 11068 < clt 11208 ≤ cle 11209 ℝ⁺crp 12951

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-resscn 11125 ax-1cn 11126 ax-addrcl 11129 ax-rnegex 11139 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-rp 12952

This theorem is referenced by: divge1 13021 rpgecld 13034 logge0 26514