Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 11825 | . . . . 5 | |
2 | odd2np1 11825 | . . . . 5 | |
3 | 1, 2 | bi2anan9 601 | . . . 4 |
4 | reeanv 2639 | . . . . 5 | |
5 | 2z 9233 | . . . . . . . . 9 | |
6 | zaddcl 9245 | . . . . . . . . . 10 | |
7 | 6 | peano2zd 9330 | . . . . . . . . 9 |
8 | dvdsmul1 11768 | . . . . . . . . 9 | |
9 | 5, 7, 8 | sylancr 412 | . . . . . . . 8 |
10 | zcn 9210 | . . . . . . . . 9 | |
11 | zcn 9210 | . . . . . . . . 9 | |
12 | addcl 7892 | . . . . . . . . . . . . 13 | |
13 | 2cn 8942 | . . . . . . . . . . . . . 14 | |
14 | ax-1cn 7860 | . . . . . . . . . . . . . 14 | |
15 | adddi 7899 | . . . . . . . . . . . . . 14 | |
16 | 13, 14, 15 | mp3an13 1323 | . . . . . . . . . . . . 13 |
17 | 12, 16 | syl 14 | . . . . . . . . . . . 12 |
18 | adddi 7899 | . . . . . . . . . . . . . 14 | |
19 | 13, 18 | mp3an1 1319 | . . . . . . . . . . . . 13 |
20 | 19 | oveq1d 5866 | . . . . . . . . . . . 12 |
21 | 17, 20 | eqtrd 2203 | . . . . . . . . . . 11 |
22 | 2t1e2 9024 | . . . . . . . . . . . . 13 | |
23 | df-2 8930 | . . . . . . . . . . . . 13 | |
24 | 22, 23 | eqtri 2191 | . . . . . . . . . . . 12 |
25 | 24 | oveq2i 5862 | . . . . . . . . . . 11 |
26 | 21, 25 | eqtrdi 2219 | . . . . . . . . . 10 |
27 | mulcl 7894 | . . . . . . . . . . . 12 | |
28 | 13, 27 | mpan 422 | . . . . . . . . . . 11 |
29 | mulcl 7894 | . . . . . . . . . . . 12 | |
30 | 13, 29 | mpan 422 | . . . . . . . . . . 11 |
31 | add4 8073 | . . . . . . . . . . . 12 | |
32 | 14, 14, 31 | mpanr12 437 | . . . . . . . . . . 11 |
33 | 28, 30, 32 | syl2an 287 | . . . . . . . . . 10 |
34 | 26, 33 | eqtrd 2203 | . . . . . . . . 9 |
35 | 10, 11, 34 | syl2an 287 | . . . . . . . 8 |
36 | 9, 35 | breqtrd 4013 | . . . . . . 7 |
37 | oveq12 5860 | . . . . . . . 8 | |
38 | 37 | breq2d 3999 | . . . . . . 7 |
39 | 36, 38 | syl5ibcom 154 | . . . . . 6 |
40 | 39 | rexlimivv 2593 | . . . . 5 |
41 | 4, 40 | sylbir 134 | . . . 4 |
42 | 3, 41 | syl6bi 162 | . . 3 |
43 | 42 | imp 123 | . 2 |
44 | 43 | an4s 583 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wceq 1348 wcel 2141 wrex 2449 class class class wbr 3987 (class class class)co 5851 cc 7765 c1 7768 caddc 7770 cmul 7772 c2 8922 cz 9205 cdvds 11742 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-mulrcl 7866 ax-addcom 7867 ax-mulcom 7868 ax-addass 7869 ax-mulass 7870 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-1rid 7874 ax-0id 7875 ax-rnegex 7876 ax-precex 7877 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-apti 7882 ax-pre-ltadd 7883 ax-pre-mulgt0 7884 ax-pre-mulext 7885 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-xor 1371 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-id 4276 df-po 4279 df-iso 4280 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-reap 8487 df-ap 8494 df-div 8583 df-inn 8872 df-2 8930 df-n0 9129 df-z 9206 df-dvds 11743 |
This theorem is referenced by: pythagtriplem11 12221 |
Copyright terms: Public domain | W3C validator |