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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 12570 | . 2 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞)) | |
2 | nn0re 12505 | . . . 4 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ) | |
3 | 2 | rexrd 11288 | . . 3 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ*) |
4 | pnfxr 11292 | . . . 4 ⊢ +∞ ∈ ℝ* | |
5 | eleq1 2816 | . . . 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 1534 ∈ wcel 2099 +∞cpnf 11269 ℝ*cxr 11271 ℕ0cn0 12496 ℕ0*cxnn0 12568 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-i2m1 11200 ax-1ne0 11201 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-pnf 11274 df-xr 11276 df-nn 12237 df-n0 12497 df-xnn0 12569 |
This theorem is referenced by: xnn0xrnemnf 12580 tayl0 26289 umgrislfupgrlem 28928 vtxdlfgrval 29292 p1evtxdeq 29320 vtxdginducedm1 29350 ewlkle 29412 upgrewlkle2 29413 upgr2pthnlp 29539 nn0xmulclb 32535 usgrcyclgt2v 34731 cusgracyclt3v 34756 aks6d1c6lem3 41628 |
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