![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nn0ge0 | Structured version Visualization version GIF version |
Description: A nonnegative integer is greater than or equal to zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
nn0ge0 | ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 12416 | . . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | nngt0 12185 | . . . 4 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
3 | id 22 | . . . . 5 ⊢ (𝑁 = 0 → 𝑁 = 0) | |
4 | 3 | eqcomd 2743 | . . . 4 ⊢ (𝑁 = 0 → 0 = 𝑁) |
5 | 2, 4 | orim12i 908 | . . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (0 < 𝑁 ∨ 0 = 𝑁)) |
6 | 1, 5 | sylbi 216 | . 2 ⊢ (𝑁 ∈ ℕ0 → (0 < 𝑁 ∨ 0 = 𝑁)) |
7 | 0re 11158 | . . 3 ⊢ 0 ∈ ℝ | |
8 | nn0re 12423 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
9 | leloe 11242 | . . 3 ⊢ ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁))) | |
10 | 7, 8, 9 | sylancr 588 | . 2 ⊢ (𝑁 ∈ ℕ0 → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁))) |
11 | 6, 10 | mpbird 257 | 1 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∨ wo 846 = wceq 1542 ∈ wcel 2107 class class class wbr 5106 ℝcr 11051 0cc0 11052 < clt 11190 ≤ cle 11191 ℕcn 12154 ℕ0cn0 12414 |
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 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-n0 12415 |
This theorem is referenced by: nn0nlt0 12440 nn0ge0i 12441 nn0le0eq0 12442 nn0p1gt0 12443 0mnnnnn0 12446 nn0addge1 12460 nn0addge2 12461 nn0negleid 12466 nn0ge0d 12477 nn0ge0div 12573 xnn0ge0 13055 xnn0xadd0 13167 nn0rp0 13373 xnn0xrge0 13424 0elfz 13539 fz0fzelfz0 13548 fz0fzdiffz0 13551 fzctr 13554 difelfzle 13555 fzoun 13610 nn0p1elfzo 13616 elfzodifsumelfzo 13639 fvinim0ffz 13692 subfzo0 13695 adddivflid 13724 modmuladdnn0 13821 addmodid 13825 modifeq2int 13839 modfzo0difsn 13849 nn0sq11 14038 zzlesq 14111 bernneq 14133 bernneq3 14135 faclbnd 14191 faclbnd6 14200 facubnd 14201 bcval5 14219 hashneq0 14265 fi1uzind 14397 ccat0 14465 ccat2s1fvw 14527 repswswrd 14673 nn0sqeq1 15162 rprisefaccl 15907 dvdseq 16197 evennn02n 16233 nn0ehalf 16261 nn0oddm1d2 16268 bitsinv1 16323 smuval2 16363 gcdn0gt0 16399 nn0gcdid0 16402 absmulgcd 16431 algcvgblem 16454 algcvga 16456 lcmgcdnn 16488 lcmfun 16522 lcmfass 16523 2mulprm 16570 nonsq 16635 hashgcdlem 16661 odzdvds 16668 pcfaclem 16771 prmirredlem 20896 prmirred 20898 coe1sclmul 21656 coe1sclmul2 21658 fvmptnn04ifb 22203 mdegle0 25445 plypf1 25576 dgrlt 25630 fta1 25671 taylfval 25721 logbgcd1irr 26147 eldmgm 26374 basellem3 26435 bcmono 26628 lgsdinn0 26696 2sq2 26784 2sqnn0 26789 2sqreulem1 26797 dchrisumlem1 26840 dchrisumlem2 26841 wwlksnextwrd 28845 wwlksnextfun 28846 wwlksnextinj 28847 wwlksnextproplem2 28858 wwlksnextproplem3 28859 wrdt2ind 31810 xrsmulgzz 31872 hashf2 32686 hasheuni 32687 reprinfz1 33238 0nn0m1nnn0 33706 faclimlem1 34319 rrntotbnd 36298 factwoffsmonot 40618 gcdnn0id 40818 pell14qrgt0 41185 pell1qrgaplem 41199 monotoddzzfi 41269 jm2.17a 41287 jm2.22 41322 rmxdiophlem 41342 rexanuz2nf 43735 wallispilem3 44315 stirlinglem7 44328 elfz2z 45554 fz0addge0 45558 elfzlble 45559 2ffzoeq 45567 iccpartigtl 45622 sqrtpwpw2p 45737 flsqrt 45792 nn0e 45896 nn0sumltlt 46433 nn0eo 46621 fllog2 46661 dignn0fr 46694 dignnld 46696 dig1 46701 itcovalt2lem2lem1 46766 |
Copyright terms: Public domain | W3C validator |