xrrege0 - Metamath Proof Explorer

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Theorem xrrege0 13141

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 13139	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → -∞ < 𝐴)
2	1	ad2ant2r 747	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → -∞ < 𝐴)
3		simprr 772	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ≤ 𝐵)
4	2, 3	jca 511	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵))
5		xrre 13136	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ∈ ℝ)
6	4, 5	syldan 591	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2109 class class class wbr 5110 ℝcr 11074 0cc0 11075 -∞cmnf 11213 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-addrcl 11136 ax-rnegex 11146 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-po 5549 df-so 5550 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221

This theorem is referenced by: psmetlecl 24210 xmetlecl 24241 prdsxmetlem 24263 stdbdmet 24411 stdbdmopn 24413 bddnghm 24621 nmoid 24637 xrsmopn 24708 metdsre 24749 metnrmlem1a 24754 ovollecl 25391 itg2lecl 25646 probmeasb 34428 heicant 37656 mblfinlem3 37660 mblfinlem4 37661