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Mirrors > Home > ILE Home > Th. List > evennn02n | Unicode version |
Description: A nonnegative integer is even iff it is twice another nonnegative integer. (Contributed by AV, 12-Aug-2021.) |
Ref | Expression |
---|---|
evennn02n |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2220 | . . . . . . . 8 | |
2 | simpr 109 | . . . . . . . . . 10 | |
3 | 2re 8908 | . . . . . . . . . . . 12 | |
4 | 3 | a1i 9 | . . . . . . . . . . 11 |
5 | zre 9176 | . . . . . . . . . . . 12 | |
6 | 5 | adantl 275 | . . . . . . . . . . 11 |
7 | 2pos 8929 | . . . . . . . . . . . 12 | |
8 | 7 | a1i 9 | . . . . . . . . . . 11 |
9 | nn0ge0 9120 | . . . . . . . . . . . 12 | |
10 | 9 | adantr 274 | . . . . . . . . . . 11 |
11 | prodge0 8730 | . . . . . . . . . . 11 | |
12 | 4, 6, 8, 10, 11 | syl22anc 1221 | . . . . . . . . . 10 |
13 | elnn0z 9185 | . . . . . . . . . 10 | |
14 | 2, 12, 13 | sylanbrc 414 | . . . . . . . . 9 |
15 | 14 | ex 114 | . . . . . . . 8 |
16 | 1, 15 | syl6bir 163 | . . . . . . 7 |
17 | 16 | com13 80 | . . . . . 6 |
18 | 17 | impcom 124 | . . . . 5 |
19 | 18 | pm4.71rd 392 | . . . 4 |
20 | 19 | bicomd 140 | . . 3 |
21 | 20 | rexbidva 2454 | . 2 |
22 | nn0ssz 9190 | . . 3 | |
23 | rexss 3195 | . . 3 | |
24 | 22, 23 | mp1i 10 | . 2 |
25 | even2n 11777 | . . 3 | |
26 | 25 | a1i 9 | . 2 |
27 | 21, 24, 26 | 3bitr4rd 220 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 wcel 2128 wrex 2436 wss 3102 class class class wbr 3967 (class class class)co 5826 cr 7733 cc0 7734 cmul 7739 clt 7914 cle 7915 c2 8889 cn0 9095 cz 9172 cdvds 11694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-cnex 7825 ax-resscn 7826 ax-1cn 7827 ax-1re 7828 ax-icn 7829 ax-addcl 7830 ax-addrcl 7831 ax-mulcl 7832 ax-mulrcl 7833 ax-addcom 7834 ax-mulcom 7835 ax-addass 7836 ax-mulass 7837 ax-distr 7838 ax-i2m1 7839 ax-0lt1 7840 ax-1rid 7841 ax-0id 7842 ax-rnegex 7843 ax-cnre 7845 ax-pre-ltirr 7846 ax-pre-ltwlin 7847 ax-pre-lttrn 7848 ax-pre-ltadd 7850 ax-pre-mulgt0 7851 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-br 3968 df-opab 4028 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-iota 5137 df-fun 5174 df-fv 5180 df-riota 5782 df-ov 5829 df-oprab 5830 df-mpo 5831 df-pnf 7916 df-mnf 7917 df-xr 7918 df-ltxr 7919 df-le 7920 df-sub 8052 df-neg 8053 df-inn 8839 df-2 8897 df-n0 9096 df-z 9173 df-dvds 11695 |
This theorem is referenced by: (None) |
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