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Mirrors > Home > MPE Home > Th. List > xnn0xr | Structured version Visualization version GIF version |
Description: An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.) |
Ref | Expression |
---|---|
xnn0xr | ⊢ (𝐴 ∈ ℕ0* → 𝐴 ∈ ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxnn0 12546 | . 2 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞)) | |
2 | nn0re 12481 | . . . 4 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ) | |
3 | 2 | rexrd 11264 | . . 3 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ*) |
4 | pnfxr 11268 | . . . 4 ⊢ +∞ ∈ ℝ* | |
5 | eleq1 2822 | . . . 4 ⊢ (𝐴 = +∞ → (𝐴 ∈ ℝ* ↔ +∞ ∈ ℝ*)) | |
6 | 4, 5 | mpbiri 258 | . . 3 ⊢ (𝐴 = +∞ → 𝐴 ∈ ℝ*) |
7 | 3, 6 | jaoi 856 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 = +∞) → 𝐴 ∈ ℝ*) |
8 | 1, 7 | sylbi 216 | 1 ⊢ (𝐴 ∈ ℕ0* → 𝐴 ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 846 = wceq 1542 ∈ wcel 2107 +∞cpnf 11245 ℝ*cxr 11247 ℕ0cn0 12472 ℕ0*cxnn0 12544 |
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 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-i2m1 11178 ax-1ne0 11179 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-pnf 11250 df-xr 11252 df-nn 12213 df-n0 12473 df-xnn0 12545 |
This theorem is referenced by: xnn0xrnemnf 12556 tayl0 25874 umgrislfupgrlem 28382 vtxdlfgrval 28742 p1evtxdeq 28770 vtxdginducedm1 28800 ewlkle 28862 upgrewlkle2 28863 upgr2pthnlp 28989 nn0xmulclb 31984 usgrcyclgt2v 34122 cusgracyclt3v 34147 |
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