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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 13374 | . 2 ⊢ (0[,)+∞) ⊆ ℝ | |
2 | 0xr 11203 | . . . 4 ⊢ 0 ∈ ℝ* | |
3 | 2 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 0 ∈ ℝ*) |
4 | pnfxr 11210 | . . . 4 ⊢ +∞ ∈ ℝ* | |
5 | 4 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → +∞ ∈ ℝ*) |
6 | eliccxr 13353 | . . . 4 ⊢ (𝐴 ∈ (0[,]+∞) → 𝐴 ∈ ℝ*) | |
7 | 6 | adantr 482 | . . 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 13321 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐴 ∈ (0[,]+∞)) → 0 ≤ 𝐴) | |
12 | 8, 9, 10, 11 | syl3anc 1372 | . . . 4 ⊢ (𝐴 ∈ (0[,]+∞) → 0 ≤ 𝐴) |
13 | 12 | adantr 482 | . . 3 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 0 ≤ 𝐴) |
14 | pnfge 13052 | . . . . . 6 ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ +∞) | |
15 | 6, 14 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ (0[,]+∞) → 𝐴 ≤ +∞) |
16 | 15 | adantr 482 | . . . 4 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ≤ +∞) |
17 | simpr 486 | . . . 4 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ≠ +∞) | |
18 | 7, 5, 16, 17 | xrleneltd 43564 | . . 3 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 < +∞) |
19 | 3, 5, 7, 13, 18 | elicod 13315 | . 2 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ∈ (0[,)+∞)) |
20 | 1, 19 | sselid 3943 | 1 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 ≠ wne 2944 class class class wbr 5106 (class class class)co 7358 ℝcr 11051 0cc0 11052 +∞cpnf 11187 ℝ*cxr 11189 ≤ cle 11191 [,)cico 13267 [,]cicc 13268 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-addrcl 11113 ax-rnegex 11123 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-ico 13271 df-icc 13272 |
This theorem is referenced by: ge0lere 43777 ovolsplit 44236 sge0tsms 44628 sge0cl 44629 sge0isum 44675 sge0xaddlem1 44681 voliunsge0lem 44720 sge0hsphoire 44837 hoidmvlelem1 44843 hoidmvlelem4 44846 hspmbl 44877 |
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