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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 12235 | . 2 ⊢ 4 ∈ ℕ | |
2 | 1 | nnzi 12526 | 1 ⊢ 4 ∈ ℤ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 4c4 12209 ℤcz 12498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 ax-un 7671 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-i2m1 11118 ax-1ne0 11119 ax-rrecex 11122 ax-cnre 11123 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7359 df-om 7802 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-neg 11387 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-z 12499 |
This theorem is referenced by: fz0to4untppr 13543 fzo0to42pr 13658 fzo1to4tp 13659 iexpcyc 14110 sqoddm1div8 14145 4bc2eq6 14228 ef01bndlem 16065 sin01bnd 16066 cos01bnd 16067 4dvdseven 16254 flodddiv4lt 16296 6gcd4e2 16418 6lcm4e12 16491 lcmf2a3a4e12 16522 ge2nprmge4 16576 prm23lt5 16685 1259lem3 17004 ppiub 26550 bclbnd 26626 bposlem6 26635 bposlem9 26638 lgsdir2lem2 26672 m1lgs 26734 2lgsoddprmlem2 26755 chebbnd1lem2 26816 chebbnd1lem3 26817 pntlema 26942 pntlemb 26943 ex-ind-dvds 29403 hgt750lemd 33252 3lexlogpow5ineq5 40508 aks4d1p1p7 40522 aks4d1p1p5 40523 aks4d1p1 40524 flt4lem7 40975 inductionexd 42409 wallispi2lem1 44284 fmtno4prmfac 45736 31prm 45761 mod42tp1mod8 45766 8even 45877 341fppr2 45898 4fppr1 45899 9fppr8 45901 fpprel2 45905 sbgoldbo 45951 nnsum3primesle9 45958 nnsum4primeseven 45964 nnsum4primesevenALTV 45965 tgblthelfgott 45979 zlmodzxzequa 46549 zlmodzxznm 46550 zlmodzxzequap 46552 zlmodzxzldeplem3 46555 zlmodzxzldep 46557 ldepsnlinclem1 46558 ldepsnlinc 46561 |
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