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Mirrors > Home > MPE Home > Th. List > odd2np1lem | Structured version Visualization version Unicode version |
Description: Lemma for odd2np1 15065. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
odd2np1lem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq2 2633 | . . . 4 | |
2 | 1 | rexbidv 3052 | . . 3 |
3 | eqeq2 2633 | . . . 4 | |
4 | 3 | rexbidv 3052 | . . 3 |
5 | 2, 4 | orbi12d 746 | . 2 |
6 | eqeq2 2633 | . . . . 5 | |
7 | 6 | rexbidv 3052 | . . . 4 |
8 | oveq2 6658 | . . . . . . 7 | |
9 | 8 | oveq1d 6665 | . . . . . 6 |
10 | 9 | eqeq1d 2624 | . . . . 5 |
11 | 10 | cbvrexv 3172 | . . . 4 |
12 | 7, 11 | syl6bb 276 | . . 3 |
13 | eqeq2 2633 | . . . . 5 | |
14 | 13 | rexbidv 3052 | . . . 4 |
15 | oveq1 6657 | . . . . . 6 | |
16 | 15 | eqeq1d 2624 | . . . . 5 |
17 | 16 | cbvrexv 3172 | . . . 4 |
18 | 14, 17 | syl6bb 276 | . . 3 |
19 | 12, 18 | orbi12d 746 | . 2 |
20 | eqeq2 2633 | . . . 4 | |
21 | 20 | rexbidv 3052 | . . 3 |
22 | eqeq2 2633 | . . . 4 | |
23 | 22 | rexbidv 3052 | . . 3 |
24 | 21, 23 | orbi12d 746 | . 2 |
25 | eqeq2 2633 | . . . 4 | |
26 | 25 | rexbidv 3052 | . . 3 |
27 | eqeq2 2633 | . . . 4 | |
28 | 27 | rexbidv 3052 | . . 3 |
29 | 26, 28 | orbi12d 746 | . 2 |
30 | 0z 11388 | . . . 4 | |
31 | 2cn 11091 | . . . . 5 | |
32 | 31 | mul02i 10225 | . . . 4 |
33 | oveq1 6657 | . . . . . 6 | |
34 | 33 | eqeq1d 2624 | . . . . 5 |
35 | 34 | rspcev 3309 | . . . 4 |
36 | 30, 32, 35 | mp2an 708 | . . 3 |
37 | 36 | olci 406 | . 2 |
38 | orcom 402 | . . 3 | |
39 | zcn 11382 | . . . . . . . . 9 | |
40 | mulcom 10022 | . . . . . . . . 9 | |
41 | 39, 31, 40 | sylancl 694 | . . . . . . . 8 |
42 | 41 | adantl 482 | . . . . . . 7 |
43 | 42 | eqeq1d 2624 | . . . . . 6 |
44 | eqid 2622 | . . . . . . . . 9 | |
45 | oveq2 6658 | . . . . . . . . . . . 12 | |
46 | 45 | oveq1d 6665 | . . . . . . . . . . 11 |
47 | 46 | eqeq1d 2624 | . . . . . . . . . 10 |
48 | 47 | rspcev 3309 | . . . . . . . . 9 |
49 | 44, 48 | mpan2 707 | . . . . . . . 8 |
50 | oveq1 6657 | . . . . . . . . . 10 | |
51 | 50 | eqeq2d 2632 | . . . . . . . . 9 |
52 | 51 | rexbidv 3052 | . . . . . . . 8 |
53 | 49, 52 | syl5ibcom 235 | . . . . . . 7 |
54 | 53 | adantl 482 | . . . . . 6 |
55 | 43, 54 | sylbid 230 | . . . . 5 |
56 | 55 | rexlimdva 3031 | . . . 4 |
57 | peano2z 11418 | . . . . . . . 8 | |
58 | 57 | adantl 482 | . . . . . . 7 |
59 | zcn 11382 | . . . . . . . . 9 | |
60 | mulcom 10022 | . . . . . . . . . . . . 13 | |
61 | 31, 60 | mpan2 707 | . . . . . . . . . . . 12 |
62 | 31 | mulid2i 10043 | . . . . . . . . . . . . 13 |
63 | 62 | a1i 11 | . . . . . . . . . . . 12 |
64 | 61, 63 | oveq12d 6668 | . . . . . . . . . . 11 |
65 | df-2 11079 | . . . . . . . . . . . 12 | |
66 | 65 | oveq2i 6661 | . . . . . . . . . . 11 |
67 | 64, 66 | syl6eq 2672 | . . . . . . . . . 10 |
68 | ax-1cn 9994 | . . . . . . . . . . 11 | |
69 | adddir 10031 | . . . . . . . . . . 11 | |
70 | 68, 31, 69 | mp3an23 1416 | . . . . . . . . . 10 |
71 | mulcl 10020 | . . . . . . . . . . . 12 | |
72 | 31, 71 | mpan 706 | . . . . . . . . . . 11 |
73 | addass 10023 | . . . . . . . . . . . 12 | |
74 | 68, 68, 73 | mp3an23 1416 | . . . . . . . . . . 11 |
75 | 72, 74 | syl 17 | . . . . . . . . . 10 |
76 | 67, 70, 75 | 3eqtr4d 2666 | . . . . . . . . 9 |
77 | 59, 76 | syl 17 | . . . . . . . 8 |
78 | 77 | adantl 482 | . . . . . . 7 |
79 | oveq1 6657 | . . . . . . . . 9 | |
80 | 79 | eqeq1d 2624 | . . . . . . . 8 |
81 | 80 | rspcev 3309 | . . . . . . 7 |
82 | 58, 78, 81 | syl2anc 693 | . . . . . 6 |
83 | oveq1 6657 | . . . . . . . 8 | |
84 | 83 | eqeq2d 2632 | . . . . . . 7 |
85 | 84 | rexbidv 3052 | . . . . . 6 |
86 | 82, 85 | syl5ibcom 235 | . . . . 5 |
87 | 86 | rexlimdva 3031 | . . . 4 |
88 | 56, 87 | orim12d 883 | . . 3 |
89 | 38, 88 | syl5bi 232 | . 2 |
90 | 5, 19, 24, 29, 37, 89 | nn0ind 11472 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wo 383 wa 384 wceq 1483 wcel 1990 wrex 2913 (class class class)co 6650 cc 9934 cc0 9936 c1 9937 caddc 9939 cmul 9941 c2 11070 cn0 11292 cz 11377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 |
This theorem is referenced by: odd2np1 15065 |
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