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Mirrors > Home > ILE Home > Th. List > opoe | Unicode version |
Description: The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
opoe |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | odd2np1 11570 | . . . . 5 | |
2 | odd2np1 11570 | . . . . 5 | |
3 | 1, 2 | bi2anan9 595 | . . . 4 |
4 | reeanv 2600 | . . . . 5 | |
5 | 2z 9082 | . . . . . . . . 9 | |
6 | zaddcl 9094 | . . . . . . . . . 10 | |
7 | 6 | peano2zd 9176 | . . . . . . . . 9 |
8 | dvdsmul1 11515 | . . . . . . . . 9 | |
9 | 5, 7, 8 | sylancr 410 | . . . . . . . 8 |
10 | zcn 9059 | . . . . . . . . 9 | |
11 | zcn 9059 | . . . . . . . . 9 | |
12 | addcl 7745 | . . . . . . . . . . . . 13 | |
13 | 2cn 8791 | . . . . . . . . . . . . . 14 | |
14 | ax-1cn 7713 | . . . . . . . . . . . . . 14 | |
15 | adddi 7752 | . . . . . . . . . . . . . 14 | |
16 | 13, 14, 15 | mp3an13 1306 | . . . . . . . . . . . . 13 |
17 | 12, 16 | syl 14 | . . . . . . . . . . . 12 |
18 | adddi 7752 | . . . . . . . . . . . . . 14 | |
19 | 13, 18 | mp3an1 1302 | . . . . . . . . . . . . 13 |
20 | 19 | oveq1d 5789 | . . . . . . . . . . . 12 |
21 | 17, 20 | eqtrd 2172 | . . . . . . . . . . 11 |
22 | 2t1e2 8873 | . . . . . . . . . . . . 13 | |
23 | df-2 8779 | . . . . . . . . . . . . 13 | |
24 | 22, 23 | eqtri 2160 | . . . . . . . . . . . 12 |
25 | 24 | oveq2i 5785 | . . . . . . . . . . 11 |
26 | 21, 25 | syl6eq 2188 | . . . . . . . . . 10 |
27 | mulcl 7747 | . . . . . . . . . . . 12 | |
28 | 13, 27 | mpan 420 | . . . . . . . . . . 11 |
29 | mulcl 7747 | . . . . . . . . . . . 12 | |
30 | 13, 29 | mpan 420 | . . . . . . . . . . 11 |
31 | add4 7923 | . . . . . . . . . . . 12 | |
32 | 14, 14, 31 | mpanr12 435 | . . . . . . . . . . 11 |
33 | 28, 30, 32 | syl2an 287 | . . . . . . . . . 10 |
34 | 26, 33 | eqtrd 2172 | . . . . . . . . 9 |
35 | 10, 11, 34 | syl2an 287 | . . . . . . . 8 |
36 | 9, 35 | breqtrd 3954 | . . . . . . 7 |
37 | oveq12 5783 | . . . . . . . 8 | |
38 | 37 | breq2d 3941 | . . . . . . 7 |
39 | 36, 38 | syl5ibcom 154 | . . . . . 6 |
40 | 39 | rexlimivv 2555 | . . . . 5 |
41 | 4, 40 | sylbir 134 | . . . 4 |
42 | 3, 41 | syl6bi 162 | . . 3 |
43 | 42 | imp 123 | . 2 |
44 | 43 | an4s 577 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wceq 1331 wcel 1480 wrex 2417 class class class wbr 3929 (class class class)co 5774 cc 7618 c1 7621 caddc 7623 cmul 7625 c2 8771 cz 9054 cdvds 11493 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-xor 1354 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-n0 8978 df-z 9055 df-dvds 11494 |
This theorem is referenced by: (None) |
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