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Mirrors > Home > ILE Home > Th. List > oddge22np1 | Unicode version |
Description: An integer greater than one is odd iff it is one plus twice a positive integer. (Contributed by AV, 16-Aug-2021.) |
Ref | Expression |
---|---|
oddge22np1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2180 | . . . . . . . 8 | |
2 | nn0z 9032 | . . . . . . . . . . 11 | |
3 | 2 | adantl 275 | . . . . . . . . . 10 |
4 | eluz2 9288 | . . . . . . . . . . . 12 | |
5 | 2re 8754 | . . . . . . . . . . . . . . . . 17 | |
6 | 5 | a1i 9 | . . . . . . . . . . . . . . . 16 |
7 | 1red 7749 | . . . . . . . . . . . . . . . 16 | |
8 | 2nn0 8952 | . . . . . . . . . . . . . . . . . . 19 | |
9 | 8 | a1i 9 | . . . . . . . . . . . . . . . . . 18 |
10 | id 19 | . . . . . . . . . . . . . . . . . 18 | |
11 | 9, 10 | nn0mulcld 8993 | . . . . . . . . . . . . . . . . 17 |
12 | 11 | nn0red 8989 | . . . . . . . . . . . . . . . 16 |
13 | 6, 7, 12 | lesubaddd 8271 | . . . . . . . . . . . . . . 15 |
14 | 2m1e1 8802 | . . . . . . . . . . . . . . . . 17 | |
15 | 14 | breq1i 3906 | . . . . . . . . . . . . . . . 16 |
16 | nn0re 8944 | . . . . . . . . . . . . . . . . . 18 | |
17 | 2pos 8775 | . . . . . . . . . . . . . . . . . . . 20 | |
18 | 5, 17 | pm3.2i 270 | . . . . . . . . . . . . . . . . . . 19 |
19 | 18 | a1i 9 | . . . . . . . . . . . . . . . . . 18 |
20 | ledivmul 8599 | . . . . . . . . . . . . . . . . . 18 | |
21 | 7, 16, 19, 20 | syl3anc 1201 | . . . . . . . . . . . . . . . . 17 |
22 | halfgt0 8893 | . . . . . . . . . . . . . . . . . 18 | |
23 | 0red 7735 | . . . . . . . . . . . . . . . . . . 19 | |
24 | halfre 8891 | . . . . . . . . . . . . . . . . . . . 20 | |
25 | 24 | a1i 9 | . . . . . . . . . . . . . . . . . . 19 |
26 | ltletr 7821 | . . . . . . . . . . . . . . . . . . 19 | |
27 | 23, 25, 16, 26 | syl3anc 1201 | . . . . . . . . . . . . . . . . . 18 |
28 | 22, 27 | mpani 426 | . . . . . . . . . . . . . . . . 17 |
29 | 21, 28 | sylbird 169 | . . . . . . . . . . . . . . . 16 |
30 | 15, 29 | syl5bi 151 | . . . . . . . . . . . . . . 15 |
31 | 13, 30 | sylbird 169 | . . . . . . . . . . . . . 14 |
32 | 31 | com12 30 | . . . . . . . . . . . . 13 |
33 | 32 | 3ad2ant3 989 | . . . . . . . . . . . 12 |
34 | 4, 33 | sylbi 120 | . . . . . . . . . . 11 |
35 | 34 | imp 123 | . . . . . . . . . 10 |
36 | elnnz 9022 | . . . . . . . . . 10 | |
37 | 3, 35, 36 | sylanbrc 413 | . . . . . . . . 9 |
38 | 37 | ex 114 | . . . . . . . 8 |
39 | 1, 38 | syl6bir 163 | . . . . . . 7 |
40 | 39 | com13 80 | . . . . . 6 |
41 | 40 | impcom 124 | . . . . 5 |
42 | 41 | pm4.71rd 391 | . . . 4 |
43 | 42 | bicomd 140 | . . 3 |
44 | 43 | rexbidva 2411 | . 2 |
45 | nnssnn0 8938 | . . 3 | |
46 | rexss 3134 | . . 3 | |
47 | 45, 46 | mp1i 10 | . 2 |
48 | eluzge2nn0 9321 | . . 3 | |
49 | oddnn02np1 11489 | . . 3 | |
50 | 48, 49 | syl 14 | . 2 |
51 | 44, 47, 50 | 3bitr4rd 220 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3a 947 wceq 1316 wcel 1465 wrex 2394 wss 3041 class class class wbr 3899 cfv 5093 (class class class)co 5742 cr 7587 cc0 7588 c1 7589 caddc 7591 cmul 7593 clt 7768 cle 7769 cmin 7901 cdiv 8399 cn 8684 c2 8735 cn0 8935 cz 9012 cuz 9282 cdvds 11405 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-xor 1339 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-po 4188 df-iso 4189 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8304 df-ap 8311 df-div 8400 df-inn 8685 df-2 8743 df-n0 8936 df-z 9013 df-uz 9283 df-dvds 11406 |
This theorem is referenced by: (None) |
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