xrrege0 - Metamath Proof Explorer

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Theorem xrrege0 13124

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 13122	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → -∞ < 𝐴)
2	1	ad2ant2r 753	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → -∞ < 𝐴)
3		simprr 778	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ≤ 𝐵)
4	2, 3	jca 516	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵))
5		xrre 13119	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ∈ ℝ)
6	4, 5	syldan 597	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2119 class class class wbr 5079 ℝcr 11035 0cc0 11036 -∞cmnf 11175 ℝ^*cxr 11176 < clt 11177 ≤ cle 11178

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-addrcl 11097 ax-rnegex 11107 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5161 df-id 5520 df-po 5533 df-so 5534 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-er 8640 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183

This theorem is referenced by: psmetlecl 24305 xmetlecl 24336 prdsxmetlem 24358 stdbdmet 24506 stdbdmopn 24508 bddnghm 24716 nmoid 24732 xrsmopn 24803 metdsre 24844 metnrmlem1a 24849 ovollecl 25475 itg2lecl 25730 probmeasb 34621 heicant 38029 mblfinlem3 38033 mblfinlem4 38034