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Metamath Proof Explorer

Theorem odd2np1

Description: An integer is odd iff it is one plus twice another integer. (Contributed by Scott Fenton, 3-Apr-2014) (Revised by Mario Carneiro, 19-Apr-2014)

Ref Expression
Assertion odd2np1 
|- ( N e. ZZ -> ( -. 2 || N <-> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) )

Proof

Step Hyp Ref Expression
1 2z 
 |-  2 e. ZZ
2 divides 
 |-  ( ( 2 e. ZZ /\ N e. ZZ ) -> ( 2 || N <-> E. k e. ZZ ( k x. 2 ) = N ) )
3 1 2 mpan 
 |-  ( N e. ZZ -> ( 2 || N <-> E. k e. ZZ ( k x. 2 ) = N ) )
4 3 notbid 
 |-  ( N e. ZZ -> ( -. 2 || N <-> -. E. k e. ZZ ( k x. 2 ) = N ) )
5 elznn0 
 |-  ( N e. ZZ <-> ( N e. RR /\ ( N e. NN0 \/ -u N e. NN0 ) ) )
6 odd2np1lem 
 |-  ( N e. NN0 -> ( E. n e. ZZ ( ( 2 x. n ) + 1 ) = N \/ E. k e. ZZ ( k x. 2 ) = N ) )
7 6 adantl 
 |-  ( ( N e. RR /\ N e. NN0 ) -> ( E. n e. ZZ ( ( 2 x. n ) + 1 ) = N \/ E. k e. ZZ ( k x. 2 ) = N ) )
8 peano2z 
 |-  ( x e. ZZ -> ( x + 1 ) e. ZZ )
9 znegcl 
 |-  ( ( x + 1 ) e. ZZ -> -u ( x + 1 ) e. ZZ )
10 8 9 syl 
 |-  ( x e. ZZ -> -u ( x + 1 ) e. ZZ )
11 10 ad2antlr 
 |-  ( ( ( N e. RR /\ x e. ZZ ) /\ ( ( 2 x. x ) + 1 ) = -u N ) -> -u ( x + 1 ) e. ZZ )
12 zcn 
 |-  ( x e. ZZ -> x e. CC )
13 2cn 
 |-  2 e. CC
14 mulcl 
 |-  ( ( 2 e. CC /\ x e. CC ) -> ( 2 x. x ) e. CC )
15 13 14 mpan 
 |-  ( x e. CC -> ( 2 x. x ) e. CC )
16 peano2cn 
 |-  ( ( 2 x. x ) e. CC -> ( ( 2 x. x ) + 1 ) e. CC )
17 15 16 syl 
 |-  ( x e. CC -> ( ( 2 x. x ) + 1 ) e. CC )
18 12 17 syl 
 |-  ( x e. ZZ -> ( ( 2 x. x ) + 1 ) e. CC )
19 18 adantl 
 |-  ( ( N e. RR /\ x e. ZZ ) -> ( ( 2 x. x ) + 1 ) e. CC )
20 simpl 
 |-  ( ( N e. RR /\ x e. ZZ ) -> N e. RR )
21 20 recnd 
 |-  ( ( N e. RR /\ x e. ZZ ) -> N e. CC )
22 negcon2 
 |-  ( ( ( ( 2 x. x ) + 1 ) e. CC /\ N e. CC ) -> ( ( ( 2 x. x ) + 1 ) = -u N <-> N = -u ( ( 2 x. x ) + 1 ) ) )
23 19 21 22 syl2anc 
 |-  ( ( N e. RR /\ x e. ZZ ) -> ( ( ( 2 x. x ) + 1 ) = -u N <-> N = -u ( ( 2 x. x ) + 1 ) ) )
24 eqcom 
 |-  ( N = -u ( ( 2 x. x ) + 1 ) <-> -u ( ( 2 x. x ) + 1 ) = N )
25 13 12 14 sylancr 
 |-  ( x e. ZZ -> ( 2 x. x ) e. CC )
26 ax-1cn 
 |-  1 e. CC
27 13 26 mulcli 
 |-  ( 2 x. 1 ) e. CC
28 addsubass 
 |-  ( ( ( 2 x. x ) e. CC /\ ( 2 x. 1 ) e. CC /\ 1 e. CC ) -> ( ( ( 2 x. x ) + ( 2 x. 1 ) ) - 1 ) = ( ( 2 x. x ) + ( ( 2 x. 1 ) - 1 ) ) )
29 27 26 28 mp3an23 
 |-  ( ( 2 x. x ) e. CC -> ( ( ( 2 x. x ) + ( 2 x. 1 ) ) - 1 ) = ( ( 2 x. x ) + ( ( 2 x. 1 ) - 1 ) ) )
30 25 29 syl 
 |-  ( x e. ZZ -> ( ( ( 2 x. x ) + ( 2 x. 1 ) ) - 1 ) = ( ( 2 x. x ) + ( ( 2 x. 1 ) - 1 ) ) )
31 2t1e2 
 |-  ( 2 x. 1 ) = 2
32 31 oveq1i 
 |-  ( ( 2 x. 1 ) - 1 ) = ( 2 - 1 )
33 2m1e1 
 |-  ( 2 - 1 ) = 1
34 32 33 eqtri 
 |-  ( ( 2 x. 1 ) - 1 ) = 1
35 34 oveq2i 
 |-  ( ( 2 x. x ) + ( ( 2 x. 1 ) - 1 ) ) = ( ( 2 x. x ) + 1 )
36 30 35 eqtr2di 
 |-  ( x e. ZZ -> ( ( 2 x. x ) + 1 ) = ( ( ( 2 x. x ) + ( 2 x. 1 ) ) - 1 ) )
37 adddi 
 |-  ( ( 2 e. CC /\ x e. CC /\ 1 e. CC ) -> ( 2 x. ( x + 1 ) ) = ( ( 2 x. x ) + ( 2 x. 1 ) ) )
38 13 26 37 mp3an13 
 |-  ( x e. CC -> ( 2 x. ( x + 1 ) ) = ( ( 2 x. x ) + ( 2 x. 1 ) ) )
39 12 38 syl 
 |-  ( x e. ZZ -> ( 2 x. ( x + 1 ) ) = ( ( 2 x. x ) + ( 2 x. 1 ) ) )
40 39 oveq1d 
 |-  ( x e. ZZ -> ( ( 2 x. ( x + 1 ) ) - 1 ) = ( ( ( 2 x. x ) + ( 2 x. 1 ) ) - 1 ) )
41 36 40 eqtr4d 
 |-  ( x e. ZZ -> ( ( 2 x. x ) + 1 ) = ( ( 2 x. ( x + 1 ) ) - 1 ) )
42 41 negeqd 
 |-  ( x e. ZZ -> -u ( ( 2 x. x ) + 1 ) = -u ( ( 2 x. ( x + 1 ) ) - 1 ) )
43 8 zcnd 
 |-  ( x e. ZZ -> ( x + 1 ) e. CC )
44 mulneg2 
 |-  ( ( 2 e. CC /\ ( x + 1 ) e. CC ) -> ( 2 x. -u ( x + 1 ) ) = -u ( 2 x. ( x + 1 ) ) )
45 13 43 44 sylancr 
 |-  ( x e. ZZ -> ( 2 x. -u ( x + 1 ) ) = -u ( 2 x. ( x + 1 ) ) )
46 45 oveq1d 
 |-  ( x e. ZZ -> ( ( 2 x. -u ( x + 1 ) ) + 1 ) = ( -u ( 2 x. ( x + 1 ) ) + 1 ) )
47 mulcl 
 |-  ( ( 2 e. CC /\ ( x + 1 ) e. CC ) -> ( 2 x. ( x + 1 ) ) e. CC )
48 13 43 47 sylancr 
 |-  ( x e. ZZ -> ( 2 x. ( x + 1 ) ) e. CC )
49 negsubdi 
 |-  ( ( ( 2 x. ( x + 1 ) ) e. CC /\ 1 e. CC ) -> -u ( ( 2 x. ( x + 1 ) ) - 1 ) = ( -u ( 2 x. ( x + 1 ) ) + 1 ) )
50 48 26 49 sylancl 
 |-  ( x e. ZZ -> -u ( ( 2 x. ( x + 1 ) ) - 1 ) = ( -u ( 2 x. ( x + 1 ) ) + 1 ) )
51 46 50 eqtr4d 
 |-  ( x e. ZZ -> ( ( 2 x. -u ( x + 1 ) ) + 1 ) = -u ( ( 2 x. ( x + 1 ) ) - 1 ) )
52 42 51 eqtr4d 
 |-  ( x e. ZZ -> -u ( ( 2 x. x ) + 1 ) = ( ( 2 x. -u ( x + 1 ) ) + 1 ) )
53 52 adantl 
 |-  ( ( N e. RR /\ x e. ZZ ) -> -u ( ( 2 x. x ) + 1 ) = ( ( 2 x. -u ( x + 1 ) ) + 1 ) )
54 53 eqeq1d 
 |-  ( ( N e. RR /\ x e. ZZ ) -> ( -u ( ( 2 x. x ) + 1 ) = N <-> ( ( 2 x. -u ( x + 1 ) ) + 1 ) = N ) )
55 24 54 bitrid 
 |-  ( ( N e. RR /\ x e. ZZ ) -> ( N = -u ( ( 2 x. x ) + 1 ) <-> ( ( 2 x. -u ( x + 1 ) ) + 1 ) = N ) )
56 23 55 bitrd 
 |-  ( ( N e. RR /\ x e. ZZ ) -> ( ( ( 2 x. x ) + 1 ) = -u N <-> ( ( 2 x. -u ( x + 1 ) ) + 1 ) = N ) )
57 56 biimpa 
 |-  ( ( ( N e. RR /\ x e. ZZ ) /\ ( ( 2 x. x ) + 1 ) = -u N ) -> ( ( 2 x. -u ( x + 1 ) ) + 1 ) = N )
58 oveq2 
 |-  ( n = -u ( x + 1 ) -> ( 2 x. n ) = ( 2 x. -u ( x + 1 ) ) )
59 58 oveq1d 
 |-  ( n = -u ( x + 1 ) -> ( ( 2 x. n ) + 1 ) = ( ( 2 x. -u ( x + 1 ) ) + 1 ) )
60 59 eqeq1d 
 |-  ( n = -u ( x + 1 ) -> ( ( ( 2 x. n ) + 1 ) = N <-> ( ( 2 x. -u ( x + 1 ) ) + 1 ) = N ) )
61 60 rspcev 
 |-  ( ( -u ( x + 1 ) e. ZZ /\ ( ( 2 x. -u ( x + 1 ) ) + 1 ) = N ) -> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N )
62 11 57 61 syl2anc 
 |-  ( ( ( N e. RR /\ x e. ZZ ) /\ ( ( 2 x. x ) + 1 ) = -u N ) -> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N )
63 62 rexlimdva2 
 |-  ( N e. RR -> ( E. x e. ZZ ( ( 2 x. x ) + 1 ) = -u N -> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) )
64 znegcl 
 |-  ( y e. ZZ -> -u y e. ZZ )
65 64 ad2antlr 
 |-  ( ( ( N e. RR /\ y e. ZZ ) /\ ( y x. 2 ) = -u N ) -> -u y e. ZZ )
66 zcn 
 |-  ( y e. ZZ -> y e. CC )
67 mulcl 
 |-  ( ( y e. CC /\ 2 e. CC ) -> ( y x. 2 ) e. CC )
68 66 13 67 sylancl 
 |-  ( y e. ZZ -> ( y x. 2 ) e. CC )
69 recn 
 |-  ( N e. RR -> N e. CC )
70 negcon2 
 |-  ( ( ( y x. 2 ) e. CC /\ N e. CC ) -> ( ( y x. 2 ) = -u N <-> N = -u ( y x. 2 ) ) )
71 68 69 70 syl2anr 
 |-  ( ( N e. RR /\ y e. ZZ ) -> ( ( y x. 2 ) = -u N <-> N = -u ( y x. 2 ) ) )
72 eqcom 
 |-  ( N = -u ( y x. 2 ) <-> -u ( y x. 2 ) = N )
73 mulneg1 
 |-  ( ( y e. CC /\ 2 e. CC ) -> ( -u y x. 2 ) = -u ( y x. 2 ) )
74 66 13 73 sylancl 
 |-  ( y e. ZZ -> ( -u y x. 2 ) = -u ( y x. 2 ) )
75 74 adantl 
 |-  ( ( N e. RR /\ y e. ZZ ) -> ( -u y x. 2 ) = -u ( y x. 2 ) )
76 75 eqeq1d 
 |-  ( ( N e. RR /\ y e. ZZ ) -> ( ( -u y x. 2 ) = N <-> -u ( y x. 2 ) = N ) )
77 72 76 bitr4id 
 |-  ( ( N e. RR /\ y e. ZZ ) -> ( N = -u ( y x. 2 ) <-> ( -u y x. 2 ) = N ) )
78 71 77 bitrd 
 |-  ( ( N e. RR /\ y e. ZZ ) -> ( ( y x. 2 ) = -u N <-> ( -u y x. 2 ) = N ) )
79 78 biimpa 
 |-  ( ( ( N e. RR /\ y e. ZZ ) /\ ( y x. 2 ) = -u N ) -> ( -u y x. 2 ) = N )
80 oveq1 
 |-  ( k = -u y -> ( k x. 2 ) = ( -u y x. 2 ) )
81 80 eqeq1d 
 |-  ( k = -u y -> ( ( k x. 2 ) = N <-> ( -u y x. 2 ) = N ) )
82 81 rspcev 
 |-  ( ( -u y e. ZZ /\ ( -u y x. 2 ) = N ) -> E. k e. ZZ ( k x. 2 ) = N )
83 65 79 82 syl2anc 
 |-  ( ( ( N e. RR /\ y e. ZZ ) /\ ( y x. 2 ) = -u N ) -> E. k e. ZZ ( k x. 2 ) = N )
84 83 rexlimdva2 
 |-  ( N e. RR -> ( E. y e. ZZ ( y x. 2 ) = -u N -> E. k e. ZZ ( k x. 2 ) = N ) )
85 63 84 orim12d 
 |-  ( N e. RR -> ( ( E. x e. ZZ ( ( 2 x. x ) + 1 ) = -u N \/ E. y e. ZZ ( y x. 2 ) = -u N ) -> ( E. n e. ZZ ( ( 2 x. n ) + 1 ) = N \/ E. k e. ZZ ( k x. 2 ) = N ) ) )
86 odd2np1lem 
 |-  ( -u N e. NN0 -> ( E. x e. ZZ ( ( 2 x. x ) + 1 ) = -u N \/ E. y e. ZZ ( y x. 2 ) = -u N ) )
87 85 86 impel 
 |-  ( ( N e. RR /\ -u N e. NN0 ) -> ( E. n e. ZZ ( ( 2 x. n ) + 1 ) = N \/ E. k e. ZZ ( k x. 2 ) = N ) )
88 7 87 jaodan 
 |-  ( ( N e. RR /\ ( N e. NN0 \/ -u N e. NN0 ) ) -> ( E. n e. ZZ ( ( 2 x. n ) + 1 ) = N \/ E. k e. ZZ ( k x. 2 ) = N ) )
89 5 88 sylbi 
 |-  ( N e. ZZ -> ( E. n e. ZZ ( ( 2 x. n ) + 1 ) = N \/ E. k e. ZZ ( k x. 2 ) = N ) )
90 halfnz 
 |-  -. ( 1 / 2 ) e. ZZ
91 reeanv 
 |-  ( E. n e. ZZ E. k e. ZZ ( ( ( 2 x. n ) + 1 ) = N /\ ( k x. 2 ) = N ) <-> ( E. n e. ZZ ( ( 2 x. n ) + 1 ) = N /\ E. k e. ZZ ( k x. 2 ) = N ) )
92 eqtr3 
 |-  ( ( ( ( 2 x. n ) + 1 ) = N /\ ( k x. 2 ) = N ) -> ( ( 2 x. n ) + 1 ) = ( k x. 2 ) )
93 zcn 
 |-  ( k e. ZZ -> k e. CC )
94 mulcom 
 |-  ( ( k e. CC /\ 2 e. CC ) -> ( k x. 2 ) = ( 2 x. k ) )
95 93 13 94 sylancl 
 |-  ( k e. ZZ -> ( k x. 2 ) = ( 2 x. k ) )
96 95 eqeq2d 
 |-  ( k e. ZZ -> ( ( ( 2 x. n ) + 1 ) = ( k x. 2 ) <-> ( ( 2 x. n ) + 1 ) = ( 2 x. k ) ) )
97 96 adantl 
 |-  ( ( n e. ZZ /\ k e. ZZ ) -> ( ( ( 2 x. n ) + 1 ) = ( k x. 2 ) <-> ( ( 2 x. n ) + 1 ) = ( 2 x. k ) ) )
98 mulcl 
 |-  ( ( 2 e. CC /\ k e. CC ) -> ( 2 x. k ) e. CC )
99 13 93 98 sylancr 
 |-  ( k e. ZZ -> ( 2 x. k ) e. CC )
100 zcn 
 |-  ( n e. ZZ -> n e. CC )
101 mulcl 
 |-  ( ( 2 e. CC /\ n e. CC ) -> ( 2 x. n ) e. CC )
102 13 100 101 sylancr 
 |-  ( n e. ZZ -> ( 2 x. n ) e. CC )
103 subadd 
 |-  ( ( ( 2 x. k ) e. CC /\ ( 2 x. n ) e. CC /\ 1 e. CC ) -> ( ( ( 2 x. k ) - ( 2 x. n ) ) = 1 <-> ( ( 2 x. n ) + 1 ) = ( 2 x. k ) ) )
104 26 103 mp3an3 
 |-  ( ( ( 2 x. k ) e. CC /\ ( 2 x. n ) e. CC ) -> ( ( ( 2 x. k ) - ( 2 x. n ) ) = 1 <-> ( ( 2 x. n ) + 1 ) = ( 2 x. k ) ) )
105 99 102 104 syl2anr 
 |-  ( ( n e. ZZ /\ k e. ZZ ) -> ( ( ( 2 x. k ) - ( 2 x. n ) ) = 1 <-> ( ( 2 x. n ) + 1 ) = ( 2 x. k ) ) )
106 subcl 
 |-  ( ( k e. CC /\ n e. CC ) -> ( k - n ) e. CC )
107 2cnne0 
 |-  ( 2 e. CC /\ 2 =/= 0 )
108 eqcom 
 |-  ( ( k - n ) = ( 1 / 2 ) <-> ( 1 / 2 ) = ( k - n ) )
109 divmul 
 |-  ( ( 1 e. CC /\ ( k - n ) e. CC /\ ( 2 e. CC /\ 2 =/= 0 ) ) -> ( ( 1 / 2 ) = ( k - n ) <-> ( 2 x. ( k - n ) ) = 1 ) )
110 108 109 bitrid 
 |-  ( ( 1 e. CC /\ ( k - n ) e. CC /\ ( 2 e. CC /\ 2 =/= 0 ) ) -> ( ( k - n ) = ( 1 / 2 ) <-> ( 2 x. ( k - n ) ) = 1 ) )
111 26 107 110 mp3an13 
 |-  ( ( k - n ) e. CC -> ( ( k - n ) = ( 1 / 2 ) <-> ( 2 x. ( k - n ) ) = 1 ) )
112 106 111 syl 
 |-  ( ( k e. CC /\ n e. CC ) -> ( ( k - n ) = ( 1 / 2 ) <-> ( 2 x. ( k - n ) ) = 1 ) )
113 112 ancoms 
 |-  ( ( n e. CC /\ k e. CC ) -> ( ( k - n ) = ( 1 / 2 ) <-> ( 2 x. ( k - n ) ) = 1 ) )
114 subdi 
 |-  ( ( 2 e. CC /\ k e. CC /\ n e. CC ) -> ( 2 x. ( k - n ) ) = ( ( 2 x. k ) - ( 2 x. n ) ) )
115 13 114 mp3an1 
 |-  ( ( k e. CC /\ n e. CC ) -> ( 2 x. ( k - n ) ) = ( ( 2 x. k ) - ( 2 x. n ) ) )
116 115 ancoms 
 |-  ( ( n e. CC /\ k e. CC ) -> ( 2 x. ( k - n ) ) = ( ( 2 x. k ) - ( 2 x. n ) ) )
117 116 eqeq1d 
 |-  ( ( n e. CC /\ k e. CC ) -> ( ( 2 x. ( k - n ) ) = 1 <-> ( ( 2 x. k ) - ( 2 x. n ) ) = 1 ) )
118 113 117 bitrd 
 |-  ( ( n e. CC /\ k e. CC ) -> ( ( k - n ) = ( 1 / 2 ) <-> ( ( 2 x. k ) - ( 2 x. n ) ) = 1 ) )
119 100 93 118 syl2an 
 |-  ( ( n e. ZZ /\ k e. ZZ ) -> ( ( k - n ) = ( 1 / 2 ) <-> ( ( 2 x. k ) - ( 2 x. n ) ) = 1 ) )
120 zsubcl 
 |-  ( ( k e. ZZ /\ n e. ZZ ) -> ( k - n ) e. ZZ )
121 eleq1 
 |-  ( ( k - n ) = ( 1 / 2 ) -> ( ( k - n ) e. ZZ <-> ( 1 / 2 ) e. ZZ ) )
122 120 121 syl5ibcom 
 |-  ( ( k e. ZZ /\ n e. ZZ ) -> ( ( k - n ) = ( 1 / 2 ) -> ( 1 / 2 ) e. ZZ ) )
123 122 ancoms 
 |-  ( ( n e. ZZ /\ k e. ZZ ) -> ( ( k - n ) = ( 1 / 2 ) -> ( 1 / 2 ) e. ZZ ) )
124 119 123 sylbird 
 |-  ( ( n e. ZZ /\ k e. ZZ ) -> ( ( ( 2 x. k ) - ( 2 x. n ) ) = 1 -> ( 1 / 2 ) e. ZZ ) )
125 105 124 sylbird 
 |-  ( ( n e. ZZ /\ k e. ZZ ) -> ( ( ( 2 x. n ) + 1 ) = ( 2 x. k ) -> ( 1 / 2 ) e. ZZ ) )
126 97 125 sylbid 
 |-  ( ( n e. ZZ /\ k e. ZZ ) -> ( ( ( 2 x. n ) + 1 ) = ( k x. 2 ) -> ( 1 / 2 ) e. ZZ ) )
127 92 126 syl5 
 |-  ( ( n e. ZZ /\ k e. ZZ ) -> ( ( ( ( 2 x. n ) + 1 ) = N /\ ( k x. 2 ) = N ) -> ( 1 / 2 ) e. ZZ ) )
128 127 rexlimivv 
 |-  ( E. n e. ZZ E. k e. ZZ ( ( ( 2 x. n ) + 1 ) = N /\ ( k x. 2 ) = N ) -> ( 1 / 2 ) e. ZZ )
129 91 128 sylbir 
 |-  ( ( E. n e. ZZ ( ( 2 x. n ) + 1 ) = N /\ E. k e. ZZ ( k x. 2 ) = N ) -> ( 1 / 2 ) e. ZZ )
130 90 129 mto 
 |-  -. ( E. n e. ZZ ( ( 2 x. n ) + 1 ) = N /\ E. k e. ZZ ( k x. 2 ) = N )
131 pm5.17 
 |-  ( ( ( E. n e. ZZ ( ( 2 x. n ) + 1 ) = N \/ E. k e. ZZ ( k x. 2 ) = N ) /\ -. ( E. n e. ZZ ( ( 2 x. n ) + 1 ) = N /\ E. k e. ZZ ( k x. 2 ) = N ) ) <-> ( E. n e. ZZ ( ( 2 x. n ) + 1 ) = N <-> -. E. k e. ZZ ( k x. 2 ) = N ) )
132 bicom 
 |-  ( ( E. n e. ZZ ( ( 2 x. n ) + 1 ) = N <-> -. E. k e. ZZ ( k x. 2 ) = N ) <-> ( -. E. k e. ZZ ( k x. 2 ) = N <-> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) )
133 131 132 bitri 
 |-  ( ( ( E. n e. ZZ ( ( 2 x. n ) + 1 ) = N \/ E. k e. ZZ ( k x. 2 ) = N ) /\ -. ( E. n e. ZZ ( ( 2 x. n ) + 1 ) = N /\ E. k e. ZZ ( k x. 2 ) = N ) ) <-> ( -. E. k e. ZZ ( k x. 2 ) = N <-> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) )
134 89 130 133 sylanblc 
 |-  ( N e. ZZ -> ( -. E. k e. ZZ ( k x. 2 ) = N <-> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) )
135 4 134 bitrd 
 |-  ( N e. ZZ -> ( -. 2 || N <-> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) )