| 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 13496 | . 2 ⊢ (0[,)+∞) ⊆ ℝ | |
| 2 | 0xr 11308 | . . . 4 ⊢ 0 ∈ ℝ* | |
| 3 | 2 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 0 ∈ ℝ*) |
| 4 | pnfxr 11315 | . . . 4 ⊢ +∞ ∈ ℝ* | |
| 5 | 4 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → +∞ ∈ ℝ*) |
| 6 | eliccxr 13475 | . . . 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 13443 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐴 ∈ (0[,]+∞)) → 0 ≤ 𝐴) | |
| 12 | 8, 9, 10, 11 | syl3anc 1373 | . . . 4 ⊢ (𝐴 ∈ (0[,]+∞) → 0 ≤ 𝐴) |
| 13 | 12 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 0 ≤ 𝐴) |
| 14 | pnfge 13172 | . . . . . 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 45334 | . . 3 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 < +∞) |
| 19 | 3, 5, 7, 13, 18 | elicod 13437 | . 2 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ∈ (0[,)+∞)) |
| 20 | 1, 19 | sselid 3981 | 1 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ∈ ℝ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 0cc0 11155 +∞cpnf 11292 ℝ*cxr 11294 ≤ cle 11296 [,)cico 13389 [,]cicc 13390 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-addrcl 11216 ax-rnegex 11226 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-ico 13393 df-icc 13394 |
| This theorem is referenced by: ge0lere 45545 ovolsplit 46003 sge0tsms 46395 sge0cl 46396 sge0isum 46442 sge0xaddlem1 46448 voliunsge0lem 46487 sge0hsphoire 46604 hoidmvlelem1 46610 hoidmvlelem4 46613 hspmbl 46644 |
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