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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 13473 | . 2 ⊢ (0[,)+∞) ⊆ ℝ | |
2 | 0xr 11299 | . . . 4 ⊢ 0 ∈ ℝ* | |
3 | 2 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 0 ∈ ℝ*) |
4 | pnfxr 11306 | . . . 4 ⊢ +∞ ∈ ℝ* | |
5 | 4 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → +∞ ∈ ℝ*) |
6 | eliccxr 13452 | . . . 4 ⊢ (𝐴 ∈ (0[,]+∞) → 𝐴 ∈ ℝ*) | |
7 | 6 | adantr 479 | . . 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 13420 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐴 ∈ (0[,]+∞)) → 0 ≤ 𝐴) | |
12 | 8, 9, 10, 11 | syl3anc 1368 | . . . 4 ⊢ (𝐴 ∈ (0[,]+∞) → 0 ≤ 𝐴) |
13 | 12 | adantr 479 | . . 3 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 0 ≤ 𝐴) |
14 | pnfge 13150 | . . . . . 6 ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ +∞) | |
15 | 6, 14 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ (0[,]+∞) → 𝐴 ≤ +∞) |
16 | 15 | adantr 479 | . . . 4 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ≤ +∞) |
17 | simpr 483 | . . . 4 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ≠ +∞) | |
18 | 7, 5, 16, 17 | xrleneltd 44734 | . . 3 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 < +∞) |
19 | 3, 5, 7, 13, 18 | elicod 13414 | . 2 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ∈ (0[,)+∞)) |
20 | 1, 19 | sselid 3980 | 1 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 ≠ wne 2937 class class class wbr 5152 (class class class)co 7426 ℝcr 11145 0cc0 11146 +∞cpnf 11283 ℝ*cxr 11285 ≤ cle 11287 [,)cico 13366 [,]cicc 13367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-addrcl 11207 ax-rnegex 11217 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-ico 13370 df-icc 13371 |
This theorem is referenced by: ge0lere 44946 ovolsplit 45405 sge0tsms 45797 sge0cl 45798 sge0isum 45844 sge0xaddlem1 45850 voliunsge0lem 45889 sge0hsphoire 46006 hoidmvlelem1 46012 hoidmvlelem4 46015 hspmbl 46046 |
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