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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 11064 | . 2 ⊢ 4 ∈ ℕ | |
2 | 1 | nnzi 11278 | 1 ⊢ 4 ∈ ℤ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 4c4 10949 ℤcz 11254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-z 11255 |
This theorem is referenced by: fz0to4untppr 12311 fzo0to42pr 12422 fzo1to4tp 12423 iexpcyc 12831 sqoddm1div8 12890 4bc2eq6 12978 ef01bndlem 14753 sin01bnd 14754 cos01bnd 14755 4dvdseven 14947 flodddiv4lt 14977 6gcd4e2 15093 6lcm4e12 15167 lcmf2a3a4e12 15198 prm23lt5 15357 1259lem3 15678 ppiub 24729 bclbnd 24805 bposlem6 24814 bposlem9 24817 lgsdir2lem2 24851 m1lgs 24913 2lgsoddprmlem2 24934 chebbnd1lem2 24959 chebbnd1lem3 24960 pntlema 25085 pntlemb 25086 4cycl4v4e 26194 4cycl4dv4e 26196 ex-ind-dvds 26710 inductionexd 37473 wallispi2lem1 38964 fmtno4prmfac 40022 31prm 40050 mod42tp1mod8 40057 8even 40160 nnsum3primesle9 40210 nnsum4primeseven 40216 nnsum4primesevenALTV 40217 tgblthelfgott 40229 tgblthelfgottOLD 40236 zlmodzxzequa 42079 zlmodzxznm 42080 zlmodzxzequap 42082 zlmodzxzldeplem3 42085 zlmodzxzldep 42087 ldepsnlinclem1 42088 ldepsnlinc 42091 |
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