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Mirrors > Home > MPE Home > Th. List > 4z | Structured version Visualization version GIF version |
Description: 4 is an integer. (Contributed by BJ, 26-Mar-2020.) |
Ref | Expression |
---|---|
4z | ⊢ 4 ∈ ℤ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4nn 12378 | . 2 ⊢ 4 ∈ ℕ | |
2 | 1 | nnzi 12669 | 1 ⊢ 4 ∈ ℤ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2108 4c4 12352 ℤcz 12641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-un 7772 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-i2m1 11254 ax-1ne0 11255 ax-rrecex 11258 ax-cnre 11259 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-ov 7453 df-om 7906 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-neg 11525 df-nn 12296 df-2 12358 df-3 12359 df-4 12360 df-z 12642 |
This theorem is referenced by: fz0to4untppr 13689 fzo0to42pr 13805 fzo1to4tp 13806 iexpcyc 14258 sqoddm1div8 14294 4bc2eq6 14380 ef01bndlem 16234 sin01bnd 16235 cos01bnd 16236 4dvdseven 16423 flodddiv4lt 16465 6gcd4e2 16587 6lcm4e12 16665 lcmf2a3a4e12 16696 ge2nprmge4 16750 prm23lt5 16863 1259lem3 17182 ppiub 27268 bclbnd 27344 bposlem6 27353 bposlem9 27356 lgsdir2lem2 27390 m1lgs 27452 2lgsoddprmlem2 27473 chebbnd1lem2 27534 chebbnd1lem3 27535 pntlema 27660 pntlemb 27661 ex-ind-dvds 30495 hgt750lemd 34627 3lexlogpow5ineq5 42019 aks4d1p1p7 42033 aks4d1p1p5 42034 aks4d1p1 42035 flt4lem7 42616 inductionexd 44119 wallispi2lem1 45994 fmtno4prmfac 47448 31prm 47473 mod42tp1mod8 47478 8even 47589 341fppr2 47610 4fppr1 47611 9fppr8 47613 fpprel2 47617 sbgoldbo 47663 nnsum3primesle9 47670 nnsum4primeseven 47676 nnsum4primesevenALTV 47677 tgblthelfgott 47691 zlmodzxzequa 48227 zlmodzxznm 48228 zlmodzxzequap 48230 zlmodzxzldeplem3 48233 zlmodzxzldep 48235 ldepsnlinclem1 48236 ldepsnlinc 48239 |
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