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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 12420 | . 2 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞)) | |
2 | nn0re 12355 | . . . . 5 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ) | |
3 | 2 | rexrd 11138 | . . . 4 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ*) |
4 | nn0ge0 12371 | . . . 4 ⊢ (𝐴 ∈ ℕ0 → 0 ≤ 𝐴) | |
5 | elxrge0 13302 | . . . 4 ⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴)) | |
6 | 3, 4, 5 | sylanbrc 583 | . . 3 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ (0[,]+∞)) |
7 | 0xr 11135 | . . . . 5 ⊢ 0 ∈ ℝ* | |
8 | pnfxr 11142 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
9 | 0lepnf 12981 | . . . . 5 ⊢ 0 ≤ +∞ | |
10 | ubicc2 13310 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞)) | |
11 | 7, 8, 9, 10 | mp3an 1461 | . . . 4 ⊢ +∞ ∈ (0[,]+∞) |
12 | eleq1 2825 | . . . 4 ⊢ (𝐴 = +∞ → (𝐴 ∈ (0[,]+∞) ↔ +∞ ∈ (0[,]+∞))) | |
13 | 11, 12 | mpbiri 257 | . . 3 ⊢ (𝐴 = +∞ → 𝐴 ∈ (0[,]+∞)) |
14 | 6, 13 | jaoi 855 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 = +∞) → 𝐴 ∈ (0[,]+∞)) |
15 | 1, 14 | sylbi 216 | 1 ⊢ (𝐴 ∈ ℕ0* → 𝐴 ∈ (0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 845 = wceq 1541 ∈ wcel 2106 class class class wbr 5103 (class class class)co 7349 0cc0 10984 +∞cpnf 11119 ℝ*cxr 11121 ≤ cle 11123 ℕ0cn0 12346 ℕ0*cxnn0 12418 [,]cicc 13195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7662 ax-cnex 11040 ax-resscn 11041 ax-1cn 11042 ax-icn 11043 ax-addcl 11044 ax-addrcl 11045 ax-mulcl 11046 ax-mulrcl 11047 ax-mulcom 11048 ax-addass 11049 ax-mulass 11050 ax-distr 11051 ax-i2m1 11052 ax-1ne0 11053 ax-1rid 11054 ax-rnegex 11055 ax-rrecex 11056 ax-cnre 11057 ax-pre-lttri 11058 ax-pre-lttrn 11059 ax-pre-ltadd 11060 ax-pre-mulgt0 11061 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5528 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5585 df-we 5587 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-pred 6249 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-riota 7305 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7793 df-2nd 7912 df-frecs 8179 df-wrecs 8210 df-recs 8284 df-rdg 8323 df-er 8581 df-en 8817 df-dom 8818 df-sdom 8819 df-pnf 11124 df-mnf 11125 df-xr 11126 df-ltxr 11127 df-le 11128 df-sub 11320 df-neg 11321 df-nn 12087 df-n0 12347 df-xnn0 12419 df-icc 13199 |
This theorem is referenced by: (None) |
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