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Mirrors > Home > MPE Home > Th. List > xnn0xrge0 | Structured version Visualization version GIF version |
Description: An extended nonnegative integer is an extended nonnegative real. (Contributed by AV, 10-Dec-2020.) |
Ref | Expression |
---|---|
xnn0xrge0 | ⊢ (𝐴 ∈ ℕ0* → 𝐴 ∈ (0[,]+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxnn0 12191 | . 2 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞)) | |
2 | nn0re 12126 | . . . . 5 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ) | |
3 | 2 | rexrd 10910 | . . . 4 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ*) |
4 | nn0ge0 12142 | . . . 4 ⊢ (𝐴 ∈ ℕ0 → 0 ≤ 𝐴) | |
5 | elxrge0 13072 | . . . 4 ⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴)) | |
6 | 3, 4, 5 | sylanbrc 586 | . . 3 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ (0[,]+∞)) |
7 | 0xr 10907 | . . . . 5 ⊢ 0 ∈ ℝ* | |
8 | pnfxr 10914 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
9 | 0lepnf 12751 | . . . . 5 ⊢ 0 ≤ +∞ | |
10 | ubicc2 13080 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞)) | |
11 | 7, 8, 9, 10 | mp3an 1463 | . . . 4 ⊢ +∞ ∈ (0[,]+∞) |
12 | eleq1 2827 | . . . 4 ⊢ (𝐴 = +∞ → (𝐴 ∈ (0[,]+∞) ↔ +∞ ∈ (0[,]+∞))) | |
13 | 11, 12 | mpbiri 261 | . . 3 ⊢ (𝐴 = +∞ → 𝐴 ∈ (0[,]+∞)) |
14 | 6, 13 | jaoi 857 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 = +∞) → 𝐴 ∈ (0[,]+∞)) |
15 | 1, 14 | sylbi 220 | 1 ⊢ (𝐴 ∈ ℕ0* → 𝐴 ∈ (0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 847 = wceq 1543 ∈ wcel 2112 class class class wbr 5069 (class class class)co 7234 0cc0 10756 +∞cpnf 10891 ℝ*cxr 10893 ≤ cle 10895 ℕ0cn0 12117 ℕ0*cxnn0 12189 [,]cicc 12965 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 ax-un 7544 ax-cnex 10812 ax-resscn 10813 ax-1cn 10814 ax-icn 10815 ax-addcl 10816 ax-addrcl 10817 ax-mulcl 10818 ax-mulrcl 10819 ax-mulcom 10820 ax-addass 10821 ax-mulass 10822 ax-distr 10823 ax-i2m1 10824 ax-1ne0 10825 ax-1rid 10826 ax-rnegex 10827 ax-rrecex 10828 ax-cnre 10829 ax-pre-lttri 10830 ax-pre-lttrn 10831 ax-pre-ltadd 10832 ax-pre-mulgt0 10833 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3711 df-csb 3828 df-dif 3885 df-un 3887 df-in 3889 df-ss 3899 df-pss 3901 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4836 df-iun 4922 df-br 5070 df-opab 5132 df-mpt 5152 df-tr 5178 df-id 5471 df-eprel 5477 df-po 5485 df-so 5486 df-fr 5526 df-we 5528 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-res 5580 df-ima 5581 df-pred 6178 df-ord 6236 df-on 6237 df-lim 6238 df-suc 6239 df-iota 6358 df-fun 6402 df-fn 6403 df-f 6404 df-f1 6405 df-fo 6406 df-f1o 6407 df-fv 6408 df-riota 7191 df-ov 7237 df-oprab 7238 df-mpo 7239 df-om 7666 df-wrecs 8070 df-recs 8131 df-rdg 8169 df-er 8414 df-en 8650 df-dom 8651 df-sdom 8652 df-pnf 10896 df-mnf 10897 df-xr 10898 df-ltxr 10899 df-le 10900 df-sub 11091 df-neg 11092 df-nn 11858 df-n0 12118 df-xnn0 12190 df-icc 12969 |
This theorem is referenced by: (None) |
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