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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 11568 | . 2 ⊢ 4 ∈ ℕ | |
2 | 1 | nnzi 11855 | 1 ⊢ 4 ∈ ℤ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2081 4c4 11542 ℤcz 11829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-i2m1 10451 ax-1ne0 10452 ax-rrecex 10455 ax-cnre 10456 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-ov 7019 df-om 7437 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-neg 10720 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-z 11830 |
This theorem is referenced by: fz0to4untppr 12860 fzo0to42pr 12974 fzo1to4tp 12975 iexpcyc 13419 sqoddm1div8 13454 4bc2eq6 13539 ef01bndlem 15370 sin01bnd 15371 cos01bnd 15372 4dvdseven 15557 flodddiv4lt 15599 6gcd4e2 15715 6lcm4e12 15789 lcmf2a3a4e12 15820 ge2nprmge4 15874 prm23lt5 15980 1259lem3 16295 ppiub 25462 bclbnd 25538 bposlem6 25547 bposlem9 25550 lgsdir2lem2 25584 m1lgs 25646 2lgsoddprmlem2 25667 chebbnd1lem2 25728 chebbnd1lem3 25729 pntlema 25854 pntlemb 25855 ex-ind-dvds 27932 hgt750lemd 31536 inductionexd 40009 wallispi2lem1 41918 fmtno4prmfac 43236 31prm 43262 mod42tp1mod8 43269 8even 43380 341fppr2 43401 4fppr1 43402 9fppr8 43404 fpprel2 43408 sbgoldbo 43454 nnsum3primesle9 43461 nnsum4primeseven 43467 nnsum4primesevenALTV 43468 tgblthelfgott 43482 zlmodzxzequa 44051 zlmodzxznm 44052 zlmodzxzequap 44054 zlmodzxzldeplem3 44057 zlmodzxzldep 44059 ldepsnlinclem1 44060 ldepsnlinc 44063 |
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