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Mirrors > Home > ILE Home > Th. List > odd2np1lem | Unicode version |
Description: Lemma for odd2np1 11497. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
odd2np1lem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq2 2127 | . . . 4 | |
2 | 1 | rexbidv 2415 | . . 3 |
3 | eqeq2 2127 | . . . 4 | |
4 | 3 | rexbidv 2415 | . . 3 |
5 | 2, 4 | orbi12d 767 | . 2 |
6 | eqeq2 2127 | . . . . 5 | |
7 | 6 | rexbidv 2415 | . . . 4 |
8 | oveq2 5750 | . . . . . . 7 | |
9 | 8 | oveq1d 5757 | . . . . . 6 |
10 | 9 | eqeq1d 2126 | . . . . 5 |
11 | 10 | cbvrexv 2632 | . . . 4 |
12 | 7, 11 | syl6bb 195 | . . 3 |
13 | eqeq2 2127 | . . . . 5 | |
14 | 13 | rexbidv 2415 | . . . 4 |
15 | oveq1 5749 | . . . . . 6 | |
16 | 15 | eqeq1d 2126 | . . . . 5 |
17 | 16 | cbvrexv 2632 | . . . 4 |
18 | 14, 17 | syl6bb 195 | . . 3 |
19 | 12, 18 | orbi12d 767 | . 2 |
20 | eqeq2 2127 | . . . 4 | |
21 | 20 | rexbidv 2415 | . . 3 |
22 | eqeq2 2127 | . . . 4 | |
23 | 22 | rexbidv 2415 | . . 3 |
24 | 21, 23 | orbi12d 767 | . 2 |
25 | eqeq2 2127 | . . . 4 | |
26 | 25 | rexbidv 2415 | . . 3 |
27 | eqeq2 2127 | . . . 4 | |
28 | 27 | rexbidv 2415 | . . 3 |
29 | 26, 28 | orbi12d 767 | . 2 |
30 | 0z 9033 | . . . 4 | |
31 | 2cn 8759 | . . . . 5 | |
32 | 31 | mul02i 8120 | . . . 4 |
33 | oveq1 5749 | . . . . . 6 | |
34 | 33 | eqeq1d 2126 | . . . . 5 |
35 | 34 | rspcev 2763 | . . . 4 |
36 | 30, 32, 35 | mp2an 422 | . . 3 |
37 | 36 | olci 706 | . 2 |
38 | orcom 702 | . . 3 | |
39 | zcn 9027 | . . . . . . . . 9 | |
40 | mulcom 7717 | . . . . . . . . 9 | |
41 | 39, 31, 40 | sylancl 409 | . . . . . . . 8 |
42 | 41 | adantl 275 | . . . . . . 7 |
43 | 42 | eqeq1d 2126 | . . . . . 6 |
44 | eqid 2117 | . . . . . . . . 9 | |
45 | oveq2 5750 | . . . . . . . . . . . 12 | |
46 | 45 | oveq1d 5757 | . . . . . . . . . . 11 |
47 | 46 | eqeq1d 2126 | . . . . . . . . . 10 |
48 | 47 | rspcev 2763 | . . . . . . . . 9 |
49 | 44, 48 | mpan2 421 | . . . . . . . 8 |
50 | oveq1 5749 | . . . . . . . . . 10 | |
51 | 50 | eqeq2d 2129 | . . . . . . . . 9 |
52 | 51 | rexbidv 2415 | . . . . . . . 8 |
53 | 49, 52 | syl5ibcom 154 | . . . . . . 7 |
54 | 53 | adantl 275 | . . . . . 6 |
55 | 43, 54 | sylbid 149 | . . . . 5 |
56 | 55 | rexlimdva 2526 | . . . 4 |
57 | peano2z 9058 | . . . . . . . 8 | |
58 | 57 | adantl 275 | . . . . . . 7 |
59 | zcn 9027 | . . . . . . . . 9 | |
60 | mulcom 7717 | . . . . . . . . . . . . 13 | |
61 | 31, 60 | mpan2 421 | . . . . . . . . . . . 12 |
62 | 31 | mulid2i 7737 | . . . . . . . . . . . . 13 |
63 | 62 | a1i 9 | . . . . . . . . . . . 12 |
64 | 61, 63 | oveq12d 5760 | . . . . . . . . . . 11 |
65 | df-2 8747 | . . . . . . . . . . . 12 | |
66 | 65 | oveq2i 5753 | . . . . . . . . . . 11 |
67 | 64, 66 | syl6eq 2166 | . . . . . . . . . 10 |
68 | ax-1cn 7681 | . . . . . . . . . . 11 | |
69 | adddir 7725 | . . . . . . . . . . 11 | |
70 | 68, 31, 69 | mp3an23 1292 | . . . . . . . . . 10 |
71 | mulcl 7715 | . . . . . . . . . . . 12 | |
72 | 31, 71 | mpan 420 | . . . . . . . . . . 11 |
73 | addass 7718 | . . . . . . . . . . . 12 | |
74 | 68, 68, 73 | mp3an23 1292 | . . . . . . . . . . 11 |
75 | 72, 74 | syl 14 | . . . . . . . . . 10 |
76 | 67, 70, 75 | 3eqtr4d 2160 | . . . . . . . . 9 |
77 | 59, 76 | syl 14 | . . . . . . . 8 |
78 | 77 | adantl 275 | . . . . . . 7 |
79 | oveq1 5749 | . . . . . . . . 9 | |
80 | 79 | eqeq1d 2126 | . . . . . . . 8 |
81 | 80 | rspcev 2763 | . . . . . . 7 |
82 | 58, 78, 81 | syl2anc 408 | . . . . . 6 |
83 | oveq1 5749 | . . . . . . . 8 | |
84 | 83 | eqeq2d 2129 | . . . . . . 7 |
85 | 84 | rexbidv 2415 | . . . . . 6 |
86 | 82, 85 | syl5ibcom 154 | . . . . 5 |
87 | 86 | rexlimdva 2526 | . . . 4 |
88 | 56, 87 | orim12d 760 | . . 3 |
89 | 38, 88 | syl5bi 151 | . 2 |
90 | 5, 19, 24, 29, 37, 89 | nn0ind 9133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 682 wceq 1316 wcel 1465 wrex 2394 (class class class)co 5742 cc 7586 cc0 7588 c1 7589 caddc 7591 cmul 7593 c2 8739 cn0 8945 cz 9022 |
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-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-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 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-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-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 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-inn 8689 df-2 8747 df-n0 8946 df-z 9023 |
This theorem is referenced by: odd2np1 11497 |
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