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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 13493 | . 2 ⊢ (0[,)+∞) ⊆ ℝ | |
2 | 0xr 11306 | . . . 4 ⊢ 0 ∈ ℝ* | |
3 | 2 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 0 ∈ ℝ*) |
4 | pnfxr 11313 | . . . 4 ⊢ +∞ ∈ ℝ* | |
5 | 4 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → +∞ ∈ ℝ*) |
6 | eliccxr 13472 | . . . 4 ⊢ (𝐴 ∈ (0[,]+∞) → 𝐴 ∈ ℝ*) | |
7 | 6 | adantr 480 | . . 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 13440 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐴 ∈ (0[,]+∞)) → 0 ≤ 𝐴) | |
12 | 8, 9, 10, 11 | syl3anc 1370 | . . . 4 ⊢ (𝐴 ∈ (0[,]+∞) → 0 ≤ 𝐴) |
13 | 12 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 0 ≤ 𝐴) |
14 | pnfge 13170 | . . . . . 6 ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ +∞) | |
15 | 6, 14 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ (0[,]+∞) → 𝐴 ≤ +∞) |
16 | 15 | adantr 480 | . . . 4 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ≤ +∞) |
17 | simpr 484 | . . . 4 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ≠ +∞) | |
18 | 7, 5, 16, 17 | xrleneltd 45273 | . . 3 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 < +∞) |
19 | 3, 5, 7, 13, 18 | elicod 13434 | . 2 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ∈ (0[,)+∞)) |
20 | 1, 19 | sselid 3993 | 1 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 ≠ wne 2938 class class class wbr 5148 (class class class)co 7431 ℝcr 11152 0cc0 11153 +∞cpnf 11290 ℝ*cxr 11292 ≤ cle 11294 [,)cico 13386 [,]cicc 13387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-addrcl 11214 ax-rnegex 11224 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-ico 13390 df-icc 13391 |
This theorem is referenced by: ge0lere 45485 ovolsplit 45944 sge0tsms 46336 sge0cl 46337 sge0isum 46383 sge0xaddlem1 46389 voliunsge0lem 46428 sge0hsphoire 46545 hoidmvlelem1 46551 hoidmvlelem4 46554 hspmbl 46585 |
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