xrrege0 - Metamath Proof Explorer

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Theorem xrrege0 12292

Description: A nonnegative extended real that is less than a real bound is real. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xrrege0	⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ∈ ℝ)

**Proof of Theorem xrrege0**
Step	Hyp	Ref	Expression
1		ge0gtmnf 12290	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → -∞ < 𝐴)
2	1	ad2ant2r 755	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → -∞ < 𝐴)
3		simprr 791	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ≤ 𝐵)
4	2, 3	jca 509	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵))
5		xrre 12287	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ∈ ℝ)
6	4, 5	syldan 587	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2166 class class class wbr 4872 ℝcr 10250 0cc0 10251 -∞cmnf 10388 ℝ^*cxr 10389 < clt 10390 ≤ cle 10391

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-addrcl 10312 ax-rnegex 10322 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-br 4873 df-opab 4935 df-mpt 4952 df-id 5249 df-po 5262 df-so 5263 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-er 8008 df-en 8222 df-dom 8223 df-sdom 8224 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396

This theorem is referenced by: psmetlecl 22489 xmetlecl 22520 prdsxmetlem 22542 stdbdmet 22690 stdbdmopn 22692 bddnghm 22899 nmoid 22915 xrsmopn 22984 metdsre 23025 metnrmlem1a 23030 ovollecl 23648 itg2lecl 23903 probmeasb 31037 heicant 33987 mblfinlem3 33991 mblfinlem4 33992