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Theorem rpgecl 12933

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 11133	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 0 ∈ ℝ)
3		rpre 12912	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
4	3	3ad2ant1 1133	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ)
5		rpgt0 12916	. . . 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 11291	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 0 < 𝐵)
9		elrp 12905	. 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 2113 class class class wbr 5096 ℝcr 11023 0cc0 11024 < clt 11164 ≤ cle 11165 ℝ⁺crp 12903

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-resscn 11081 ax-1cn 11082 ax-addrcl 11085 ax-rnegex 11095 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-br 5097 df-opab 5159 df-mpt 5178 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-er 8633 df-en 8882 df-dom 8883 df-sdom 8884 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-rp 12904

This theorem is referenced by: divge1 12973 rpgecld 12986 logge0 26568