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Mirrors > Home > ILE Home > Th. List > odd2np1lem | Unicode version |
Description: Lemma for odd2np1 11832. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
odd2np1lem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq2 2180 | . . . 4 | |
2 | 1 | rexbidv 2471 | . . 3 |
3 | eqeq2 2180 | . . . 4 | |
4 | 3 | rexbidv 2471 | . . 3 |
5 | 2, 4 | orbi12d 788 | . 2 |
6 | eqeq2 2180 | . . . . 5 | |
7 | 6 | rexbidv 2471 | . . . 4 |
8 | oveq2 5861 | . . . . . . 7 | |
9 | 8 | oveq1d 5868 | . . . . . 6 |
10 | 9 | eqeq1d 2179 | . . . . 5 |
11 | 10 | cbvrexv 2697 | . . . 4 |
12 | 7, 11 | bitrdi 195 | . . 3 |
13 | eqeq2 2180 | . . . . 5 | |
14 | 13 | rexbidv 2471 | . . . 4 |
15 | oveq1 5860 | . . . . . 6 | |
16 | 15 | eqeq1d 2179 | . . . . 5 |
17 | 16 | cbvrexv 2697 | . . . 4 |
18 | 14, 17 | bitrdi 195 | . . 3 |
19 | 12, 18 | orbi12d 788 | . 2 |
20 | eqeq2 2180 | . . . 4 | |
21 | 20 | rexbidv 2471 | . . 3 |
22 | eqeq2 2180 | . . . 4 | |
23 | 22 | rexbidv 2471 | . . 3 |
24 | 21, 23 | orbi12d 788 | . 2 |
25 | eqeq2 2180 | . . . 4 | |
26 | 25 | rexbidv 2471 | . . 3 |
27 | eqeq2 2180 | . . . 4 | |
28 | 27 | rexbidv 2471 | . . 3 |
29 | 26, 28 | orbi12d 788 | . 2 |
30 | 0z 9223 | . . . 4 | |
31 | 2cn 8949 | . . . . 5 | |
32 | 31 | mul02i 8309 | . . . 4 |
33 | oveq1 5860 | . . . . . 6 | |
34 | 33 | eqeq1d 2179 | . . . . 5 |
35 | 34 | rspcev 2834 | . . . 4 |
36 | 30, 32, 35 | mp2an 424 | . . 3 |
37 | 36 | olci 727 | . 2 |
38 | orcom 723 | . . 3 | |
39 | zcn 9217 | . . . . . . . . 9 | |
40 | mulcom 7903 | . . . . . . . . 9 | |
41 | 39, 31, 40 | sylancl 411 | . . . . . . . 8 |
42 | 41 | adantl 275 | . . . . . . 7 |
43 | 42 | eqeq1d 2179 | . . . . . 6 |
44 | eqid 2170 | . . . . . . . . 9 | |
45 | oveq2 5861 | . . . . . . . . . . . 12 | |
46 | 45 | oveq1d 5868 | . . . . . . . . . . 11 |
47 | 46 | eqeq1d 2179 | . . . . . . . . . 10 |
48 | 47 | rspcev 2834 | . . . . . . . . 9 |
49 | 44, 48 | mpan2 423 | . . . . . . . 8 |
50 | oveq1 5860 | . . . . . . . . . 10 | |
51 | 50 | eqeq2d 2182 | . . . . . . . . 9 |
52 | 51 | rexbidv 2471 | . . . . . . . 8 |
53 | 49, 52 | syl5ibcom 154 | . . . . . . 7 |
54 | 53 | adantl 275 | . . . . . 6 |
55 | 43, 54 | sylbid 149 | . . . . 5 |
56 | 55 | rexlimdva 2587 | . . . 4 |
57 | peano2z 9248 | . . . . . . . 8 | |
58 | 57 | adantl 275 | . . . . . . 7 |
59 | zcn 9217 | . . . . . . . . 9 | |
60 | mulcom 7903 | . . . . . . . . . . . . 13 | |
61 | 31, 60 | mpan2 423 | . . . . . . . . . . . 12 |
62 | 31 | mulid2i 7923 | . . . . . . . . . . . . 13 |
63 | 62 | a1i 9 | . . . . . . . . . . . 12 |
64 | 61, 63 | oveq12d 5871 | . . . . . . . . . . 11 |
65 | df-2 8937 | . . . . . . . . . . . 12 | |
66 | 65 | oveq2i 5864 | . . . . . . . . . . 11 |
67 | 64, 66 | eqtrdi 2219 | . . . . . . . . . 10 |
68 | ax-1cn 7867 | . . . . . . . . . . 11 | |
69 | adddir 7911 | . . . . . . . . . . 11 | |
70 | 68, 31, 69 | mp3an23 1324 | . . . . . . . . . 10 |
71 | mulcl 7901 | . . . . . . . . . . . 12 | |
72 | 31, 71 | mpan 422 | . . . . . . . . . . 11 |
73 | addass 7904 | . . . . . . . . . . . 12 | |
74 | 68, 68, 73 | mp3an23 1324 | . . . . . . . . . . 11 |
75 | 72, 74 | syl 14 | . . . . . . . . . 10 |
76 | 67, 70, 75 | 3eqtr4d 2213 | . . . . . . . . 9 |
77 | 59, 76 | syl 14 | . . . . . . . 8 |
78 | 77 | adantl 275 | . . . . . . 7 |
79 | oveq1 5860 | . . . . . . . . 9 | |
80 | 79 | eqeq1d 2179 | . . . . . . . 8 |
81 | 80 | rspcev 2834 | . . . . . . 7 |
82 | 58, 78, 81 | syl2anc 409 | . . . . . 6 |
83 | oveq1 5860 | . . . . . . . 8 | |
84 | 83 | eqeq2d 2182 | . . . . . . 7 |
85 | 84 | rexbidv 2471 | . . . . . 6 |
86 | 82, 85 | syl5ibcom 154 | . . . . 5 |
87 | 86 | rexlimdva 2587 | . . . 4 |
88 | 56, 87 | orim12d 781 | . . 3 |
89 | 38, 88 | syl5bi 151 | . 2 |
90 | 5, 19, 24, 29, 37, 89 | nn0ind 9326 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 703 wceq 1348 wcel 2141 wrex 2449 (class class class)co 5853 cc 7772 cc0 7774 c1 7775 caddc 7777 cmul 7779 c2 8929 cn0 9135 cz 9212 |
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 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 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-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-2 8937 df-n0 9136 df-z 9213 |
This theorem is referenced by: odd2np1 11832 |
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