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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ge0xrre | Structured version Visualization version GIF version |
Description: A nonnegative extended real that is not +∞ is a real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
ge0xrre | ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rge0ssre 12577 | . 2 ⊢ (0[,)+∞) ⊆ ℝ | |
2 | 0xr 10410 | . . . 4 ⊢ 0 ∈ ℝ* | |
3 | 2 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 0 ∈ ℝ*) |
4 | pnfxr 10417 | . . . 4 ⊢ +∞ ∈ ℝ* | |
5 | 4 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → +∞ ∈ ℝ*) |
6 | eliccxr 12555 | . . . 4 ⊢ (𝐴 ∈ (0[,]+∞) → 𝐴 ∈ ℝ*) | |
7 | 6 | adantr 474 | . . 3 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ∈ ℝ*) |
8 | 2 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ (0[,]+∞) → 0 ∈ ℝ*) |
9 | 4 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ (0[,]+∞) → +∞ ∈ ℝ*) |
10 | id 22 | . . . . 5 ⊢ (𝐴 ∈ (0[,]+∞) → 𝐴 ∈ (0[,]+∞)) | |
11 | iccgelb 12525 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐴 ∈ (0[,]+∞)) → 0 ≤ 𝐴) | |
12 | 8, 9, 10, 11 | syl3anc 1494 | . . . 4 ⊢ (𝐴 ∈ (0[,]+∞) → 0 ≤ 𝐴) |
13 | 12 | adantr 474 | . . 3 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 0 ≤ 𝐴) |
14 | pnfge 12257 | . . . . . 6 ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ +∞) | |
15 | 6, 14 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ (0[,]+∞) → 𝐴 ≤ +∞) |
16 | 15 | adantr 474 | . . . 4 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ≤ +∞) |
17 | simpr 479 | . . . 4 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ≠ +∞) | |
18 | 7, 5, 16, 17 | xrleneltd 40334 | . . 3 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 < +∞) |
19 | 3, 5, 7, 13, 18 | elicod 12519 | . 2 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ∈ (0[,)+∞)) |
20 | 1, 19 | sseldi 3825 | 1 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2164 ≠ wne 2999 class class class wbr 4875 (class class class)co 6910 ℝcr 10258 0cc0 10259 +∞cpnf 10395 ℝ*cxr 10397 ≤ cle 10399 [,)cico 12472 [,]cicc 12473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-addrcl 10320 ax-rnegex 10330 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-po 5265 df-so 5266 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-1st 7433 df-2nd 7434 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-ico 12476 df-icc 12477 |
This theorem is referenced by: ge0lere 40552 ovolsplit 40997 sge0tsms 41386 sge0cl 41387 sge0isum 41433 sge0xaddlem1 41439 voliunsge0lem 41478 sge0hsphoire 41595 hoidmvlelem1 41601 hoidmvlelem4 41604 hspmbl 41635 |
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