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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 12598 | . 2 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞)) | |
2 | nn0re 12533 | . . . 4 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ) | |
3 | 2 | rexrd 11314 | . . 3 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ*) |
4 | pnfxr 11318 | . . . 4 ⊢ +∞ ∈ ℝ* | |
5 | eleq1 2814 | . . . 4 ⊢ (𝐴 = +∞ → (𝐴 ∈ ℝ* ↔ +∞ ∈ ℝ*)) | |
6 | 4, 5 | mpbiri 257 | . . 3 ⊢ (𝐴 = +∞ → 𝐴 ∈ ℝ*) |
7 | 3, 6 | jaoi 855 | . 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 845 = wceq 1534 ∈ wcel 2099 +∞cpnf 11295 ℝ*cxr 11297 ℕ0cn0 12524 ℕ0*cxnn0 12596 |
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 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-i2m1 11226 ax-1ne0 11227 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-pnf 11300 df-xr 11302 df-nn 12265 df-n0 12525 df-xnn0 12597 |
This theorem is referenced by: xnn0xrnemnf 12608 tayl0 26389 umgrislfupgrlem 29058 vtxdlfgrval 29422 p1evtxdeq 29450 vtxdginducedm1 29480 ewlkle 29542 upgrewlkle2 29543 upgr2pthnlp 29669 nn0xmulclb 32675 usgrcyclgt2v 34959 cusgracyclt3v 34984 aks6d1c6lem3 41870 |
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