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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 9251 | . 2 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | |
2 | 1 | simplbi 274 | 1 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ w3o 977 = wceq 1353 ∈ wcel 2148 ℝcr 7807 0cc0 7808 -cneg 8125 ℕcn 8915 ℤcz 9249 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-rab 2464 df-v 2739 df-un 3133 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-iota 5177 df-fv 5223 df-ov 5875 df-neg 8127 df-z 9250 |
This theorem is referenced by: zcn 9254 zrei 9255 zssre 9256 elnn0z 9262 elnnz1 9272 peano2z 9285 zaddcl 9289 ztri3or0 9291 ztri3or 9292 zletric 9293 zlelttric 9294 zltnle 9295 zleloe 9296 zletr 9298 znnsub 9300 nzadd 9301 zltp1le 9303 zleltp1 9304 znn0sub 9314 zapne 9323 zdceq 9324 zdcle 9325 zdclt 9326 zltlen 9327 nn0ge0div 9336 zextle 9340 btwnnz 9343 suprzclex 9347 msqznn 9349 peano2uz2 9356 uzind 9360 fzind 9364 fnn0ind 9365 eluzuzle 9532 uzid 9538 uzneg 9542 uz11 9546 eluzp1m1 9547 eluzp1p1 9549 eluzaddi 9550 eluzsubi 9551 uzin 9556 uz3m2nn 9569 peano2uz 9579 nn0pzuz 9583 eluz2b2 9599 uz2mulcl 9604 eqreznegel 9610 lbzbi 9612 qre 9621 elpq 9644 zltaddlt1le 10003 elfz1eq 10030 fznlem 10036 fzen 10038 uzsubsubfz 10042 fzaddel 10054 fzsuc2 10074 fzp1disj 10075 fzrev 10079 elfz1b 10085 fzneuz 10096 fzp1nel 10099 elfz0fzfz0 10121 fz0fzelfz0 10122 fznn0sub2 10123 fz0fzdiffz0 10125 elfzmlbp 10127 difelfznle 10130 elfzouz2 10156 fzonlt0 10162 fzossrbm1 10168 fzo1fzo0n0 10178 elfzo0z 10179 fzofzim 10183 eluzgtdifelfzo 10192 fzossfzop1 10207 ssfzo12bi 10220 elfzomelpfzo 10226 fzosplitprm1 10229 fzostep1 10232 flid 10279 flqbi2 10286 2tnp1ge0ge0 10296 flhalf 10297 fldiv4p1lem1div2 10300 ceiqle 10308 uzsinds 10437 zsqcl2 10592 nn0abscl 11087 zmaxcl 11226 2zsupmax 11227 2zinfmin 11244 p1modz1 11794 evennn02n 11879 evennn2n 11880 ltoddhalfle 11890 infssuzex 11942 dfgcd2 12007 algcvga 12043 isprm3 12110 dvdsnprmd 12117 sqnprm 12128 zgcdsq 12193 hashdvds 12213 fldivp1 12338 zgz 12363 4sqlem4 12382 mulgval 12918 coskpi 14140 relogexp 14164 rplogbzexp 14243 zabsle1 14271 lgsne0 14310 2sqlem2 14322 |
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