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Theorem rpgecl 12967

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 1138	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ)
2		0red 11142	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 0 ∈ ℝ)
3		rpre 12946	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
4	3	3ad2ant1 1134	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ)
5		rpgt0 12950	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 0 < 𝐴)
6	5	3ad2ant1 1134	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 0 < 𝐴)
7		simp3 1139	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
8	2, 4, 1, 6, 7	ltletrd 11301	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 0 < 𝐵)
9		elrp 12939	. 2 ⊢ (𝐵 ∈ ℝ⁺ ↔ (𝐵 ∈ ℝ ∧ 0 < 𝐵))
10	1, 8, 9	sylanbrc 584	1 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ⁺)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1087 ∈ wcel 2114 class class class wbr 5086 ℝcr 11032 0cc0 11033 < clt 11174 ≤ cle 11175 ℝ⁺crp 12937

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-resscn 11090 ax-1cn 11091 ax-addrcl 11094 ax-rnegex 11104 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5521 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-er 8638 df-en 8889 df-dom 8890 df-sdom 8891 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-rp 12938

This theorem is referenced by: divge1 13007 rpgecld 13020 logge0 26586