Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > zre | GIF version |
Description: An integer is a real. (Contributed by NM, 8-Jan-2002.) |
Ref | Expression |
---|---|
zre | ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elz 9184 | . 2 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | |
2 | 1 | simplbi 272 | 1 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ w3o 966 = wceq 1342 ∈ wcel 2135 ℝcr 7743 0cc0 7744 -cneg 8061 ℕcn 8848 ℤcz 9182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 df-rab 2451 df-v 2723 df-un 3115 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-iota 5147 df-fv 5190 df-ov 5839 df-neg 8063 df-z 9183 |
This theorem is referenced by: zcn 9187 zrei 9188 zssre 9189 elnn0z 9195 elnnz1 9205 peano2z 9218 zaddcl 9222 ztri3or0 9224 ztri3or 9225 zletric 9226 zlelttric 9227 zltnle 9228 zleloe 9229 zletr 9231 znnsub 9233 nzadd 9234 zltp1le 9236 zleltp1 9237 znn0sub 9247 zapne 9256 zdceq 9257 zdcle 9258 zdclt 9259 zltlen 9260 nn0ge0div 9269 zextle 9273 btwnnz 9276 suprzclex 9280 msqznn 9282 peano2uz2 9289 uzind 9293 fzind 9297 fnn0ind 9298 eluzuzle 9465 uzid 9471 uzneg 9475 uz11 9479 eluzp1m1 9480 eluzp1p1 9482 eluzaddi 9483 eluzsubi 9484 uzin 9489 uz3m2nn 9502 peano2uz 9512 nn0pzuz 9516 eluz2b2 9532 uz2mulcl 9537 eqreznegel 9543 lbzbi 9545 qre 9554 elpq 9577 zltaddlt1le 9934 elfz1eq 9960 fznlem 9966 fzen 9968 uzsubsubfz 9972 fzaddel 9984 fzsuc2 10004 fzp1disj 10005 fzrev 10009 elfz1b 10015 fzneuz 10026 fzp1nel 10029 elfz0fzfz0 10051 fz0fzelfz0 10052 fznn0sub2 10053 fz0fzdiffz0 10055 elfzmlbp 10057 difelfznle 10060 elfzouz2 10086 fzonlt0 10092 fzossrbm1 10098 fzo1fzo0n0 10108 elfzo0z 10109 fzofzim 10113 eluzgtdifelfzo 10122 fzossfzop1 10137 ssfzo12bi 10150 elfzomelpfzo 10156 fzosplitprm1 10159 fzostep1 10162 flid 10209 flqbi2 10216 2tnp1ge0ge0 10226 flhalf 10227 fldiv4p1lem1div2 10230 ceiqle 10238 uzsinds 10367 zsqcl2 10522 nn0abscl 11013 zmaxcl 11152 2zsupmax 11153 2zinfmin 11170 p1modz1 11720 evennn02n 11804 evennn2n 11805 ltoddhalfle 11815 infssuzex 11867 dfgcd2 11932 algcvga 11962 isprm3 12029 dvdsnprmd 12036 sqnprm 12047 zgcdsq 12110 hashdvds 12130 fldivp1 12255 coskpi 13310 relogexp 13334 rplogbzexp 13413 |
Copyright terms: Public domain | W3C validator |