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Mirrors > Home > ILE Home > Th. List > odd2np1lem | Unicode version |
Description: Lemma for odd2np1 10272. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
odd2np1lem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq2 2090 | . . . 4 | |
2 | 1 | rexbidv 2369 | . . 3 |
3 | eqeq2 2090 | . . . 4 | |
4 | 3 | rexbidv 2369 | . . 3 |
5 | 2, 4 | orbi12d 739 | . 2 |
6 | eqeq2 2090 | . . . . 5 | |
7 | 6 | rexbidv 2369 | . . . 4 |
8 | oveq2 5540 | . . . . . . 7 | |
9 | 8 | oveq1d 5547 | . . . . . 6 |
10 | 9 | eqeq1d 2089 | . . . . 5 |
11 | 10 | cbvrexv 2578 | . . . 4 |
12 | 7, 11 | syl6bb 194 | . . 3 |
13 | eqeq2 2090 | . . . . 5 | |
14 | 13 | rexbidv 2369 | . . . 4 |
15 | oveq1 5539 | . . . . . 6 | |
16 | 15 | eqeq1d 2089 | . . . . 5 |
17 | 16 | cbvrexv 2578 | . . . 4 |
18 | 14, 17 | syl6bb 194 | . . 3 |
19 | 12, 18 | orbi12d 739 | . 2 |
20 | eqeq2 2090 | . . . 4 | |
21 | 20 | rexbidv 2369 | . . 3 |
22 | eqeq2 2090 | . . . 4 | |
23 | 22 | rexbidv 2369 | . . 3 |
24 | 21, 23 | orbi12d 739 | . 2 |
25 | eqeq2 2090 | . . . 4 | |
26 | 25 | rexbidv 2369 | . . 3 |
27 | eqeq2 2090 | . . . 4 | |
28 | 27 | rexbidv 2369 | . . 3 |
29 | 26, 28 | orbi12d 739 | . 2 |
30 | 0z 8362 | . . . 4 | |
31 | 2cn 8110 | . . . . 5 | |
32 | 31 | mul02i 7494 | . . . 4 |
33 | oveq1 5539 | . . . . . 6 | |
34 | 33 | eqeq1d 2089 | . . . . 5 |
35 | 34 | rspcev 2701 | . . . 4 |
36 | 30, 32, 35 | mp2an 416 | . . 3 |
37 | 36 | olci 683 | . 2 |
38 | orcom 679 | . . 3 | |
39 | zcn 8356 | . . . . . . . . 9 | |
40 | mulcom 7102 | . . . . . . . . 9 | |
41 | 39, 31, 40 | sylancl 404 | . . . . . . . 8 |
42 | 41 | adantl 271 | . . . . . . 7 |
43 | 42 | eqeq1d 2089 | . . . . . 6 |
44 | eqid 2081 | . . . . . . . . 9 | |
45 | oveq2 5540 | . . . . . . . . . . . 12 | |
46 | 45 | oveq1d 5547 | . . . . . . . . . . 11 |
47 | 46 | eqeq1d 2089 | . . . . . . . . . 10 |
48 | 47 | rspcev 2701 | . . . . . . . . 9 |
49 | 44, 48 | mpan2 415 | . . . . . . . 8 |
50 | oveq1 5539 | . . . . . . . . . 10 | |
51 | 50 | eqeq2d 2092 | . . . . . . . . 9 |
52 | 51 | rexbidv 2369 | . . . . . . . 8 |
53 | 49, 52 | syl5ibcom 153 | . . . . . . 7 |
54 | 53 | adantl 271 | . . . . . 6 |
55 | 43, 54 | sylbid 148 | . . . . 5 |
56 | 55 | rexlimdva 2477 | . . . 4 |
57 | peano2z 8387 | . . . . . . . 8 | |
58 | 57 | adantl 271 | . . . . . . 7 |
59 | zcn 8356 | . . . . . . . . 9 | |
60 | mulcom 7102 | . . . . . . . . . . . . 13 | |
61 | 31, 60 | mpan2 415 | . . . . . . . . . . . 12 |
62 | 31 | mulid2i 7122 | . . . . . . . . . . . . 13 |
63 | 62 | a1i 9 | . . . . . . . . . . . 12 |
64 | 61, 63 | oveq12d 5550 | . . . . . . . . . . 11 |
65 | df-2 8098 | . . . . . . . . . . . 12 | |
66 | 65 | oveq2i 5543 | . . . . . . . . . . 11 |
67 | 64, 66 | syl6eq 2129 | . . . . . . . . . 10 |
68 | ax-1cn 7069 | . . . . . . . . . . 11 | |
69 | adddir 7110 | . . . . . . . . . . 11 | |
70 | 68, 31, 69 | mp3an23 1260 | . . . . . . . . . 10 |
71 | mulcl 7100 | . . . . . . . . . . . 12 | |
72 | 31, 71 | mpan 414 | . . . . . . . . . . 11 |
73 | addass 7103 | . . . . . . . . . . . 12 | |
74 | 68, 68, 73 | mp3an23 1260 | . . . . . . . . . . 11 |
75 | 72, 74 | syl 14 | . . . . . . . . . 10 |
76 | 67, 70, 75 | 3eqtr4d 2123 | . . . . . . . . 9 |
77 | 59, 76 | syl 14 | . . . . . . . 8 |
78 | 77 | adantl 271 | . . . . . . 7 |
79 | oveq1 5539 | . . . . . . . . 9 | |
80 | 79 | eqeq1d 2089 | . . . . . . . 8 |
81 | 80 | rspcev 2701 | . . . . . . 7 |
82 | 58, 78, 81 | syl2anc 403 | . . . . . 6 |
83 | oveq1 5539 | . . . . . . . 8 | |
84 | 83 | eqeq2d 2092 | . . . . . . 7 |
85 | 84 | rexbidv 2369 | . . . . . 6 |
86 | 82, 85 | syl5ibcom 153 | . . . . 5 |
87 | 86 | rexlimdva 2477 | . . . 4 |
88 | 56, 87 | orim12d 732 | . . 3 |
89 | 38, 88 | syl5bi 150 | . 2 |
90 | 5, 19, 24, 29, 37, 89 | nn0ind 8461 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wo 661 wceq 1284 wcel 1433 wrex 2349 (class class class)co 5532 cc 6979 cc0 6981 c1 6982 caddc 6984 cmul 6986 c2 8089 cn0 8288 cz 8351 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-inn 8040 df-2 8098 df-n0 8289 df-z 8352 |
This theorem is referenced by: odd2np1 10272 |
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