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Mirrors > Home > ILE Home > Th. List > odd2np1lem | Unicode version |
Description: Lemma for odd2np1 11764. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
odd2np1lem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq2 2167 | . . . 4 | |
2 | 1 | rexbidv 2458 | . . 3 |
3 | eqeq2 2167 | . . . 4 | |
4 | 3 | rexbidv 2458 | . . 3 |
5 | 2, 4 | orbi12d 783 | . 2 |
6 | eqeq2 2167 | . . . . 5 | |
7 | 6 | rexbidv 2458 | . . . 4 |
8 | oveq2 5832 | . . . . . . 7 | |
9 | 8 | oveq1d 5839 | . . . . . 6 |
10 | 9 | eqeq1d 2166 | . . . . 5 |
11 | 10 | cbvrexv 2681 | . . . 4 |
12 | 7, 11 | bitrdi 195 | . . 3 |
13 | eqeq2 2167 | . . . . 5 | |
14 | 13 | rexbidv 2458 | . . . 4 |
15 | oveq1 5831 | . . . . . 6 | |
16 | 15 | eqeq1d 2166 | . . . . 5 |
17 | 16 | cbvrexv 2681 | . . . 4 |
18 | 14, 17 | bitrdi 195 | . . 3 |
19 | 12, 18 | orbi12d 783 | . 2 |
20 | eqeq2 2167 | . . . 4 | |
21 | 20 | rexbidv 2458 | . . 3 |
22 | eqeq2 2167 | . . . 4 | |
23 | 22 | rexbidv 2458 | . . 3 |
24 | 21, 23 | orbi12d 783 | . 2 |
25 | eqeq2 2167 | . . . 4 | |
26 | 25 | rexbidv 2458 | . . 3 |
27 | eqeq2 2167 | . . . 4 | |
28 | 27 | rexbidv 2458 | . . 3 |
29 | 26, 28 | orbi12d 783 | . 2 |
30 | 0z 9178 | . . . 4 | |
31 | 2cn 8904 | . . . . 5 | |
32 | 31 | mul02i 8265 | . . . 4 |
33 | oveq1 5831 | . . . . . 6 | |
34 | 33 | eqeq1d 2166 | . . . . 5 |
35 | 34 | rspcev 2816 | . . . 4 |
36 | 30, 32, 35 | mp2an 423 | . . 3 |
37 | 36 | olci 722 | . 2 |
38 | orcom 718 | . . 3 | |
39 | zcn 9172 | . . . . . . . . 9 | |
40 | mulcom 7861 | . . . . . . . . 9 | |
41 | 39, 31, 40 | sylancl 410 | . . . . . . . 8 |
42 | 41 | adantl 275 | . . . . . . 7 |
43 | 42 | eqeq1d 2166 | . . . . . 6 |
44 | eqid 2157 | . . . . . . . . 9 | |
45 | oveq2 5832 | . . . . . . . . . . . 12 | |
46 | 45 | oveq1d 5839 | . . . . . . . . . . 11 |
47 | 46 | eqeq1d 2166 | . . . . . . . . . 10 |
48 | 47 | rspcev 2816 | . . . . . . . . 9 |
49 | 44, 48 | mpan2 422 | . . . . . . . 8 |
50 | oveq1 5831 | . . . . . . . . . 10 | |
51 | 50 | eqeq2d 2169 | . . . . . . . . 9 |
52 | 51 | rexbidv 2458 | . . . . . . . 8 |
53 | 49, 52 | syl5ibcom 154 | . . . . . . 7 |
54 | 53 | adantl 275 | . . . . . 6 |
55 | 43, 54 | sylbid 149 | . . . . 5 |
56 | 55 | rexlimdva 2574 | . . . 4 |
57 | peano2z 9203 | . . . . . . . 8 | |
58 | 57 | adantl 275 | . . . . . . 7 |
59 | zcn 9172 | . . . . . . . . 9 | |
60 | mulcom 7861 | . . . . . . . . . . . . 13 | |
61 | 31, 60 | mpan2 422 | . . . . . . . . . . . 12 |
62 | 31 | mulid2i 7881 | . . . . . . . . . . . . 13 |
63 | 62 | a1i 9 | . . . . . . . . . . . 12 |
64 | 61, 63 | oveq12d 5842 | . . . . . . . . . . 11 |
65 | df-2 8892 | . . . . . . . . . . . 12 | |
66 | 65 | oveq2i 5835 | . . . . . . . . . . 11 |
67 | 64, 66 | eqtrdi 2206 | . . . . . . . . . 10 |
68 | ax-1cn 7825 | . . . . . . . . . . 11 | |
69 | adddir 7869 | . . . . . . . . . . 11 | |
70 | 68, 31, 69 | mp3an23 1311 | . . . . . . . . . 10 |
71 | mulcl 7859 | . . . . . . . . . . . 12 | |
72 | 31, 71 | mpan 421 | . . . . . . . . . . 11 |
73 | addass 7862 | . . . . . . . . . . . 12 | |
74 | 68, 68, 73 | mp3an23 1311 | . . . . . . . . . . 11 |
75 | 72, 74 | syl 14 | . . . . . . . . . 10 |
76 | 67, 70, 75 | 3eqtr4d 2200 | . . . . . . . . 9 |
77 | 59, 76 | syl 14 | . . . . . . . 8 |
78 | 77 | adantl 275 | . . . . . . 7 |
79 | oveq1 5831 | . . . . . . . . 9 | |
80 | 79 | eqeq1d 2166 | . . . . . . . 8 |
81 | 80 | rspcev 2816 | . . . . . . 7 |
82 | 58, 78, 81 | syl2anc 409 | . . . . . 6 |
83 | oveq1 5831 | . . . . . . . 8 | |
84 | 83 | eqeq2d 2169 | . . . . . . 7 |
85 | 84 | rexbidv 2458 | . . . . . 6 |
86 | 82, 85 | syl5ibcom 154 | . . . . 5 |
87 | 86 | rexlimdva 2574 | . . . 4 |
88 | 56, 87 | orim12d 776 | . . 3 |
89 | 38, 88 | syl5bi 151 | . 2 |
90 | 5, 19, 24, 29, 37, 89 | nn0ind 9278 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 698 wceq 1335 wcel 2128 wrex 2436 (class class class)co 5824 cc 7730 cc0 7732 c1 7733 caddc 7735 cmul 7737 c2 8884 cn0 9090 cz 9167 |
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 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-mulass 7835 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-ltadd 7848 |
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 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-iota 5135 df-fun 5172 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-inn 8834 df-2 8892 df-n0 9091 df-z 9168 |
This theorem is referenced by: odd2np1 11764 |
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