rpgecl - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > rpgecl		Structured version Visualization version GIF version

Theorem rpgecl 13061

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 1136	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ)
2		0red 11262	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 0 ∈ ℝ)
3		rpre 13041	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
4	3	3ad2ant1 1132	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ)
5		rpgt0 13045	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 0 < 𝐴)
6	5	3ad2ant1 1132	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 0 < 𝐴)
7		simp3 1137	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
8	2, 4, 1, 6, 7	ltletrd 11419	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 0 < 𝐵)
9		elrp 13034	. 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 2106 class class class wbr 5148 ℝcr 11152 0cc0 11153 < clt 11293 ≤ 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: divge1 13101 rpgecld 13114 logge0 26662