MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  odd2np1lem Structured version   Unicode version

Theorem odd2np1lem 14307
Description: Lemma for odd2np1 14308. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
odd2np1lem  |-  ( N  e.  NN0  ->  ( E. n  e.  ZZ  (
( 2  x.  n
)  +  1 )  =  N  \/  E. k  e.  ZZ  (
k  x.  2 )  =  N ) )
Distinct variable groups:    k, N    n, N

Proof of Theorem odd2np1lem
Dummy variables  j  m  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq2 2439 . . . 4  |-  ( j  =  0  ->  (
( ( 2  x.  n )  +  1 )  =  j  <->  ( (
2  x.  n )  +  1 )  =  0 ) )
21rexbidv 2878 . . 3  |-  ( j  =  0  ->  ( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  j  <->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  0 ) )
3 eqeq2 2439 . . . 4  |-  ( j  =  0  ->  (
( k  x.  2 )  =  j  <->  ( k  x.  2 )  =  0 ) )
43rexbidv 2878 . . 3  |-  ( j  =  0  ->  ( E. k  e.  ZZ  ( k  x.  2 )  =  j  <->  E. k  e.  ZZ  ( k  x.  2 )  =  0 ) )
52, 4orbi12d 714 . 2  |-  ( j  =  0  ->  (
( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  j  \/  E. k  e.  ZZ  ( k  x.  2 )  =  j )  <->  ( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  0  \/  E. k  e.  ZZ  ( k  x.  2 )  =  0 ) ) )
6 eqeq2 2439 . . . . 5  |-  ( j  =  m  ->  (
( ( 2  x.  n )  +  1 )  =  j  <->  ( (
2  x.  n )  +  1 )  =  m ) )
76rexbidv 2878 . . . 4  |-  ( j  =  m  ->  ( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  j  <->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  m ) )
8 oveq2 6257 . . . . . . 7  |-  ( n  =  x  ->  (
2  x.  n )  =  ( 2  x.  x ) )
98oveq1d 6264 . . . . . 6  |-  ( n  =  x  ->  (
( 2  x.  n
)  +  1 )  =  ( ( 2  x.  x )  +  1 ) )
109eqeq1d 2430 . . . . 5  |-  ( n  =  x  ->  (
( ( 2  x.  n )  +  1 )  =  m  <->  ( (
2  x.  x )  +  1 )  =  m ) )
1110cbvrexv 2997 . . . 4  |-  ( E. n  e.  ZZ  (
( 2  x.  n
)  +  1 )  =  m  <->  E. x  e.  ZZ  ( ( 2  x.  x )  +  1 )  =  m )
127, 11syl6bb 264 . . 3  |-  ( j  =  m  ->  ( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  j  <->  E. x  e.  ZZ  ( ( 2  x.  x )  +  1 )  =  m ) )
13 eqeq2 2439 . . . . 5  |-  ( j  =  m  ->  (
( k  x.  2 )  =  j  <->  ( k  x.  2 )  =  m ) )
1413rexbidv 2878 . . . 4  |-  ( j  =  m  ->  ( E. k  e.  ZZ  ( k  x.  2 )  =  j  <->  E. k  e.  ZZ  ( k  x.  2 )  =  m ) )
15 oveq1 6256 . . . . . 6  |-  ( k  =  y  ->  (
k  x.  2 )  =  ( y  x.  2 ) )
1615eqeq1d 2430 . . . . 5  |-  ( k  =  y  ->  (
( k  x.  2 )  =  m  <->  ( y  x.  2 )  =  m ) )
1716cbvrexv 2997 . . . 4  |-  ( E. k  e.  ZZ  (
k  x.  2 )  =  m  <->  E. y  e.  ZZ  ( y  x.  2 )  =  m )
1814, 17syl6bb 264 . . 3  |-  ( j  =  m  ->  ( E. k  e.  ZZ  ( k  x.  2 )  =  j  <->  E. y  e.  ZZ  ( y  x.  2 )  =  m ) )
1912, 18orbi12d 714 . 2  |-  ( j  =  m  ->  (
( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  j  \/  E. k  e.  ZZ  ( k  x.  2 )  =  j )  <->  ( E. x  e.  ZZ  ( ( 2  x.  x )  +  1 )  =  m  \/  E. y  e.  ZZ  ( y  x.  2 )  =  m ) ) )
20 eqeq2 2439 . . . 4  |-  ( j  =  ( m  + 
1 )  ->  (
( ( 2  x.  n )  +  1 )  =  j  <->  ( (
2  x.  n )  +  1 )  =  ( m  +  1 ) ) )
2120rexbidv 2878 . . 3  |-  ( j  =  ( m  + 
1 )  ->  ( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  j  <->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  ( m  +  1 ) ) )
22 eqeq2 2439 . . . 4  |-  ( j  =  ( m  + 
1 )  ->  (
( k  x.  2 )  =  j  <->  ( k  x.  2 )  =  ( m  +  1 ) ) )
2322rexbidv 2878 . . 3  |-  ( j  =  ( m  + 
1 )  ->  ( E. k  e.  ZZ  ( k  x.  2 )  =  j  <->  E. k  e.  ZZ  ( k  x.  2 )  =  ( m  +  1 ) ) )
2421, 23orbi12d 714 . 2  |-  ( j  =  ( m  + 
1 )  ->  (
( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  j  \/  E. k  e.  ZZ  ( k  x.  2 )  =  j )  <->  ( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  ( m  +  1 )  \/  E. k  e.  ZZ  ( k  x.  2 )  =  ( m  +  1 ) ) ) )
25 eqeq2 2439 . . . 4  |-  ( j  =  N  ->  (
( ( 2  x.  n )  +  1 )  =  j  <->  ( (
2  x.  n )  +  1 )  =  N ) )
2625rexbidv 2878 . . 3  |-  ( j  =  N  ->  ( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  j  <->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N ) )
27 eqeq2 2439 . . . 4  |-  ( j  =  N  ->  (
( k  x.  2 )  =  j  <->  ( k  x.  2 )  =  N ) )
2827rexbidv 2878 . . 3  |-  ( j  =  N  ->  ( E. k  e.  ZZ  ( k  x.  2 )  =  j  <->  E. k  e.  ZZ  ( k  x.  2 )  =  N ) )
2926, 28orbi12d 714 . 2  |-  ( j  =  N  ->  (
( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  j  \/  E. k  e.  ZZ  ( k  x.  2 )  =  j )  <->  ( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N  \/  E. k  e.  ZZ  ( k  x.  2 )  =  N ) ) )
30 0z 10899 . . . 4  |-  0  e.  ZZ
31 2cn 10631 . . . . 5  |-  2  e.  CC
3231mul02i 9773 . . . 4  |-  ( 0  x.  2 )  =  0
33 oveq1 6256 . . . . . 6  |-  ( k  =  0  ->  (
k  x.  2 )  =  ( 0  x.  2 ) )
3433eqeq1d 2430 . . . . 5  |-  ( k  =  0  ->  (
( k  x.  2 )  =  0  <->  (
0  x.  2 )  =  0 ) )
3534rspcev 3125 . . . 4  |-  ( ( 0  e.  ZZ  /\  ( 0  x.  2 )  =  0 )  ->  E. k  e.  ZZ  ( k  x.  2 )  =  0 )
3630, 32, 35mp2an 676 . . 3  |-  E. k  e.  ZZ  ( k  x.  2 )  =  0
3736olci 392 . 2  |-  ( E. n  e.  ZZ  (
( 2  x.  n
)  +  1 )  =  0  \/  E. k  e.  ZZ  (
k  x.  2 )  =  0 )
38 orcom 388 . . 3  |-  ( ( E. x  e.  ZZ  ( ( 2  x.  x )  +  1 )  =  m  \/ 
E. y  e.  ZZ  ( y  x.  2 )  =  m )  <-> 
( E. y  e.  ZZ  ( y  x.  2 )  =  m  \/  E. x  e.  ZZ  ( ( 2  x.  x )  +  1 )  =  m ) )
39 zcn 10893 . . . . . . . . 9  |-  ( y  e.  ZZ  ->  y  e.  CC )
40 mulcom 9576 . . . . . . . . 9  |-  ( ( y  e.  CC  /\  2  e.  CC )  ->  ( y  x.  2 )  =  ( 2  x.  y ) )
4139, 31, 40sylancl 666 . . . . . . . 8  |-  ( y  e.  ZZ  ->  (
y  x.  2 )  =  ( 2  x.  y ) )
4241adantl 467 . . . . . . 7  |-  ( ( m  e.  NN0  /\  y  e.  ZZ )  ->  ( y  x.  2 )  =  ( 2  x.  y ) )
4342eqeq1d 2430 . . . . . 6  |-  ( ( m  e.  NN0  /\  y  e.  ZZ )  ->  ( ( y  x.  2 )  =  m  <-> 
( 2  x.  y
)  =  m ) )
44 eqid 2428 . . . . . . . . 9  |-  ( ( 2  x.  y )  +  1 )  =  ( ( 2  x.  y )  +  1 )
45 oveq2 6257 . . . . . . . . . . . 12  |-  ( n  =  y  ->  (
2  x.  n )  =  ( 2  x.  y ) )
4645oveq1d 6264 . . . . . . . . . . 11  |-  ( n  =  y  ->  (
( 2  x.  n
)  +  1 )  =  ( ( 2  x.  y )  +  1 ) )
4746eqeq1d 2430 . . . . . . . . . 10  |-  ( n  =  y  ->  (
( ( 2  x.  n )  +  1 )  =  ( ( 2  x.  y )  +  1 )  <->  ( (
2  x.  y )  +  1 )  =  ( ( 2  x.  y )  +  1 ) ) )
4847rspcev 3125 . . . . . . . . 9  |-  ( ( y  e.  ZZ  /\  ( ( 2  x.  y )  +  1 )  =  ( ( 2  x.  y )  +  1 ) )  ->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  ( ( 2  x.  y )  +  1 ) )
4944, 48mpan2 675 . . . . . . . 8  |-  ( y  e.  ZZ  ->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  ( ( 2  x.  y
)  +  1 ) )
50 oveq1 6256 . . . . . . . . . 10  |-  ( ( 2  x.  y )  =  m  ->  (
( 2  x.  y
)  +  1 )  =  ( m  + 
1 ) )
5150eqeq2d 2438 . . . . . . . . 9  |-  ( ( 2  x.  y )  =  m  ->  (
( ( 2  x.  n )  +  1 )  =  ( ( 2  x.  y )  +  1 )  <->  ( (
2  x.  n )  +  1 )  =  ( m  +  1 ) ) )
5251rexbidv 2878 . . . . . . . 8  |-  ( ( 2  x.  y )  =  m  ->  ( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  ( ( 2  x.  y )  +  1 )  <->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  ( m  +  1 ) ) )
5349, 52syl5ibcom 223 . . . . . . 7  |-  ( y  e.  ZZ  ->  (
( 2  x.  y
)  =  m  ->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  ( m  +  1 ) ) )
5453adantl 467 . . . . . 6  |-  ( ( m  e.  NN0  /\  y  e.  ZZ )  ->  ( ( 2  x.  y )  =  m  ->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  ( m  +  1 ) ) )
5543, 54sylbid 218 . . . . 5  |-  ( ( m  e.  NN0  /\  y  e.  ZZ )  ->  ( ( y  x.  2 )  =  m  ->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  ( m  +  1 ) ) )
5655rexlimdva 2856 . . . 4  |-  ( m  e.  NN0  ->  ( E. y  e.  ZZ  (
y  x.  2 )  =  m  ->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  ( m  +  1 ) ) )
57 peano2z 10929 . . . . . . . 8  |-  ( x  e.  ZZ  ->  (
x  +  1 )  e.  ZZ )
5857adantl 467 . . . . . . 7  |-  ( ( m  e.  NN0  /\  x  e.  ZZ )  ->  ( x  +  1 )  e.  ZZ )
59 zcn 10893 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  x  e.  CC )
60 mulcom 9576 . . . . . . . . . . . . 13  |-  ( ( x  e.  CC  /\  2  e.  CC )  ->  ( x  x.  2 )  =  ( 2  x.  x ) )
6131, 60mpan2 675 . . . . . . . . . . . 12  |-  ( x  e.  CC  ->  (
x  x.  2 )  =  ( 2  x.  x ) )
6231mulid2i 9597 . . . . . . . . . . . . 13  |-  ( 1  x.  2 )  =  2
6362a1i 11 . . . . . . . . . . . 12  |-  ( x  e.  CC  ->  (
1  x.  2 )  =  2 )
6461, 63oveq12d 6267 . . . . . . . . . . 11  |-  ( x  e.  CC  ->  (
( x  x.  2 )  +  ( 1  x.  2 ) )  =  ( ( 2  x.  x )  +  2 ) )
65 df-2 10619 . . . . . . . . . . . 12  |-  2  =  ( 1  +  1 )
6665oveq2i 6260 . . . . . . . . . . 11  |-  ( ( 2  x.  x )  +  2 )  =  ( ( 2  x.  x )  +  ( 1  +  1 ) )
6764, 66syl6eq 2478 . . . . . . . . . 10  |-  ( x  e.  CC  ->  (
( x  x.  2 )  +  ( 1  x.  2 ) )  =  ( ( 2  x.  x )  +  ( 1  +  1 ) ) )
68 ax-1cn 9548 . . . . . . . . . . 11  |-  1  e.  CC
69 adddir 9585 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  1  e.  CC  /\  2  e.  CC )  ->  (
( x  +  1 )  x.  2 )  =  ( ( x  x.  2 )  +  ( 1  x.  2 ) ) )
7068, 31, 69mp3an23 1352 . . . . . . . . . 10  |-  ( x  e.  CC  ->  (
( x  +  1 )  x.  2 )  =  ( ( x  x.  2 )  +  ( 1  x.  2 ) ) )
71 mulcl 9574 . . . . . . . . . . . 12  |-  ( ( 2  e.  CC  /\  x  e.  CC )  ->  ( 2  x.  x
)  e.  CC )
7231, 71mpan 674 . . . . . . . . . . 11  |-  ( x  e.  CC  ->  (
2  x.  x )  e.  CC )
73 addass 9577 . . . . . . . . . . . 12  |-  ( ( ( 2  x.  x
)  e.  CC  /\  1  e.  CC  /\  1  e.  CC )  ->  (
( ( 2  x.  x )  +  1 )  +  1 )  =  ( ( 2  x.  x )  +  ( 1  +  1 ) ) )
7468, 68, 73mp3an23 1352 . . . . . . . . . . 11  |-  ( ( 2  x.  x )  e.  CC  ->  (
( ( 2  x.  x )  +  1 )  +  1 )  =  ( ( 2  x.  x )  +  ( 1  +  1 ) ) )
7572, 74syl 17 . . . . . . . . . 10  |-  ( x  e.  CC  ->  (
( ( 2  x.  x )  +  1 )  +  1 )  =  ( ( 2  x.  x )  +  ( 1  +  1 ) ) )
7667, 70, 753eqtr4d 2472 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
( x  +  1 )  x.  2 )  =  ( ( ( 2  x.  x )  +  1 )  +  1 ) )
7759, 76syl 17 . . . . . . . 8  |-  ( x  e.  ZZ  ->  (
( x  +  1 )  x.  2 )  =  ( ( ( 2  x.  x )  +  1 )  +  1 ) )
7877adantl 467 . . . . . . 7  |-  ( ( m  e.  NN0  /\  x  e.  ZZ )  ->  ( ( x  + 
1 )  x.  2 )  =  ( ( ( 2  x.  x
)  +  1 )  +  1 ) )
79 oveq1 6256 . . . . . . . . 9  |-  ( k  =  ( x  + 
1 )  ->  (
k  x.  2 )  =  ( ( x  +  1 )  x.  2 ) )
8079eqeq1d 2430 . . . . . . . 8  |-  ( k  =  ( x  + 
1 )  ->  (
( k  x.  2 )  =  ( ( ( 2  x.  x
)  +  1 )  +  1 )  <->  ( (
x  +  1 )  x.  2 )  =  ( ( ( 2  x.  x )  +  1 )  +  1 ) ) )
8180rspcev 3125 . . . . . . 7  |-  ( ( ( x  +  1 )  e.  ZZ  /\  ( ( x  + 
1 )  x.  2 )  =  ( ( ( 2  x.  x
)  +  1 )  +  1 ) )  ->  E. k  e.  ZZ  ( k  x.  2 )  =  ( ( ( 2  x.  x
)  +  1 )  +  1 ) )
8258, 78, 81syl2anc 665 . . . . . 6  |-  ( ( m  e.  NN0  /\  x  e.  ZZ )  ->  E. k  e.  ZZ  ( k  x.  2 )  =  ( ( ( 2  x.  x
)  +  1 )  +  1 ) )
83 oveq1 6256 . . . . . . . 8  |-  ( ( ( 2  x.  x
)  +  1 )  =  m  ->  (
( ( 2  x.  x )  +  1 )  +  1 )  =  ( m  + 
1 ) )
8483eqeq2d 2438 . . . . . . 7  |-  ( ( ( 2  x.  x
)  +  1 )  =  m  ->  (
( k  x.  2 )  =  ( ( ( 2  x.  x
)  +  1 )  +  1 )  <->  ( k  x.  2 )  =  ( m  +  1 ) ) )
8584rexbidv 2878 . . . . . 6  |-  ( ( ( 2  x.  x
)  +  1 )  =  m  ->  ( E. k  e.  ZZ  ( k  x.  2 )  =  ( ( ( 2  x.  x
)  +  1 )  +  1 )  <->  E. k  e.  ZZ  ( k  x.  2 )  =  ( m  +  1 ) ) )
8682, 85syl5ibcom 223 . . . . 5  |-  ( ( m  e.  NN0  /\  x  e.  ZZ )  ->  ( ( ( 2  x.  x )  +  1 )  =  m  ->  E. k  e.  ZZ  ( k  x.  2 )  =  ( m  +  1 ) ) )
8786rexlimdva 2856 . . . 4  |-  ( m  e.  NN0  ->  ( E. x  e.  ZZ  (
( 2  x.  x
)  +  1 )  =  m  ->  E. k  e.  ZZ  ( k  x.  2 )  =  ( m  +  1 ) ) )
8856, 87orim12d 846 . . 3  |-  ( m  e.  NN0  ->  ( ( E. y  e.  ZZ  ( y  x.  2 )  =  m  \/ 
E. x  e.  ZZ  ( ( 2  x.  x )  +  1 )  =  m )  ->  ( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  ( m  +  1 )  \/  E. k  e.  ZZ  ( k  x.  2 )  =  ( m  +  1 ) ) ) )
8938, 88syl5bi 220 . 2  |-  ( m  e.  NN0  ->  ( ( E. x  e.  ZZ  ( ( 2  x.  x )  +  1 )  =  m  \/ 
E. y  e.  ZZ  ( y  x.  2 )  =  m )  ->  ( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  ( m  +  1 )  \/  E. k  e.  ZZ  ( k  x.  2 )  =  ( m  +  1 ) ) ) )
905, 19, 24, 29, 37, 89nn0ind 10981 1  |-  ( N  e.  NN0  ->  ( E. n  e.  ZZ  (
( 2  x.  n
)  +  1 )  =  N  \/  E. k  e.  ZZ  (
k  x.  2 )  =  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872   E.wrex 2715  (class class class)co 6249   CCcc 9488   0cc0 9490   1c1 9491    + caddc 9493    x. cmul 9495   2c2 10610   NN0cn0 10820   ZZcz 10888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-nn 10561  df-2 10619  df-n0 10821  df-z 10889
This theorem is referenced by:  odd2np1  14308
  Copyright terms: Public domain W3C validator