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Theorem odd2np1lem 13921
Description: Lemma for odd2np1 13922. (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 2482 . . . 4  |-  ( j  =  0  ->  (
( ( 2  x.  n )  +  1 )  =  j  <->  ( (
2  x.  n )  +  1 )  =  0 ) )
21rexbidv 2978 . . 3  |-  ( j  =  0  ->  ( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  j  <->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  0 ) )
3 eqeq2 2482 . . . 4  |-  ( j  =  0  ->  (
( k  x.  2 )  =  j  <->  ( k  x.  2 )  =  0 ) )
43rexbidv 2978 . . 3  |-  ( j  =  0  ->  ( E. k  e.  ZZ  ( k  x.  2 )  =  j  <->  E. k  e.  ZZ  ( k  x.  2 )  =  0 ) )
52, 4orbi12d 709 . 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 2482 . . . . 5  |-  ( j  =  m  ->  (
( ( 2  x.  n )  +  1 )  =  j  <->  ( (
2  x.  n )  +  1 )  =  m ) )
76rexbidv 2978 . . . 4  |-  ( j  =  m  ->  ( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  j  <->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  m ) )
8 oveq2 6303 . . . . . . 7  |-  ( n  =  x  ->  (
2  x.  n )  =  ( 2  x.  x ) )
98oveq1d 6310 . . . . . 6  |-  ( n  =  x  ->  (
( 2  x.  n
)  +  1 )  =  ( ( 2  x.  x )  +  1 ) )
109eqeq1d 2469 . . . . 5  |-  ( n  =  x  ->  (
( ( 2  x.  n )  +  1 )  =  m  <->  ( (
2  x.  x )  +  1 )  =  m ) )
1110cbvrexv 3094 . . . 4  |-  ( E. n  e.  ZZ  (
( 2  x.  n
)  +  1 )  =  m  <->  E. x  e.  ZZ  ( ( 2  x.  x )  +  1 )  =  m )
127, 11syl6bb 261 . . 3  |-  ( j  =  m  ->  ( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  j  <->  E. x  e.  ZZ  ( ( 2  x.  x )  +  1 )  =  m ) )
13 eqeq2 2482 . . . . 5  |-  ( j  =  m  ->  (
( k  x.  2 )  =  j  <->  ( k  x.  2 )  =  m ) )
1413rexbidv 2978 . . . 4  |-  ( j  =  m  ->  ( E. k  e.  ZZ  ( k  x.  2 )  =  j  <->  E. k  e.  ZZ  ( k  x.  2 )  =  m ) )
15 oveq1 6302 . . . . . 6  |-  ( k  =  y  ->  (
k  x.  2 )  =  ( y  x.  2 ) )
1615eqeq1d 2469 . . . . 5  |-  ( k  =  y  ->  (
( k  x.  2 )  =  m  <->  ( y  x.  2 )  =  m ) )
1716cbvrexv 3094 . . . 4  |-  ( E. k  e.  ZZ  (
k  x.  2 )  =  m  <->  E. y  e.  ZZ  ( y  x.  2 )  =  m )
1814, 17syl6bb 261 . . 3  |-  ( j  =  m  ->  ( E. k  e.  ZZ  ( k  x.  2 )  =  j  <->  E. y  e.  ZZ  ( y  x.  2 )  =  m ) )
1912, 18orbi12d 709 . 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 2482 . . . 4  |-  ( j  =  ( m  + 
1 )  ->  (
( ( 2  x.  n )  +  1 )  =  j  <->  ( (
2  x.  n )  +  1 )  =  ( m  +  1 ) ) )
2120rexbidv 2978 . . 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 2482 . . . 4  |-  ( j  =  ( m  + 
1 )  ->  (
( k  x.  2 )  =  j  <->  ( k  x.  2 )  =  ( m  +  1 ) ) )
2322rexbidv 2978 . . 3  |-  ( j  =  ( m  + 
1 )  ->  ( E. k  e.  ZZ  ( k  x.  2 )  =  j  <->  E. k  e.  ZZ  ( k  x.  2 )  =  ( m  +  1 ) ) )
2421, 23orbi12d 709 . 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 2482 . . . 4  |-  ( j  =  N  ->  (
( ( 2  x.  n )  +  1 )  =  j  <->  ( (
2  x.  n )  +  1 )  =  N ) )
2625rexbidv 2978 . . 3  |-  ( j  =  N  ->  ( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  j  <->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N ) )
27 eqeq2 2482 . . . 4  |-  ( j  =  N  ->  (
( k  x.  2 )  =  j  <->  ( k  x.  2 )  =  N ) )
2827rexbidv 2978 . . 3  |-  ( j  =  N  ->  ( E. k  e.  ZZ  ( k  x.  2 )  =  j  <->  E. k  e.  ZZ  ( k  x.  2 )  =  N ) )
2926, 28orbi12d 709 . 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 10887 . . . 4  |-  0  e.  ZZ
31 2cn 10618 . . . . 5  |-  2  e.  CC
3231mul02i 9780 . . . 4  |-  ( 0  x.  2 )  =  0
33 oveq1 6302 . . . . . 6  |-  ( k  =  0  ->  (
k  x.  2 )  =  ( 0  x.  2 ) )
3433eqeq1d 2469 . . . . 5  |-  ( k  =  0  ->  (
( k  x.  2 )  =  0  <->  (
0  x.  2 )  =  0 ) )
3534rspcev 3219 . . . 4  |-  ( ( 0  e.  ZZ  /\  ( 0  x.  2 )  =  0 )  ->  E. k  e.  ZZ  ( k  x.  2 )  =  0 )
3630, 32, 35mp2an 672 . . 3  |-  E. k  e.  ZZ  ( k  x.  2 )  =  0
3736olci 391 . 2  |-  ( E. n  e.  ZZ  (
( 2  x.  n
)  +  1 )  =  0  \/  E. k  e.  ZZ  (
k  x.  2 )  =  0 )
38 orcom 387 . . 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 10881 . . . . . . . . 9  |-  ( y  e.  ZZ  ->  y  e.  CC )
40 mulcom 9590 . . . . . . . . 9  |-  ( ( y  e.  CC  /\  2  e.  CC )  ->  ( y  x.  2 )  =  ( 2  x.  y ) )
4139, 31, 40sylancl 662 . . . . . . . 8  |-  ( y  e.  ZZ  ->  (
y  x.  2 )  =  ( 2  x.  y ) )
4241adantl 466 . . . . . . 7  |-  ( ( m  e.  NN0  /\  y  e.  ZZ )  ->  ( y  x.  2 )  =  ( 2  x.  y ) )
4342eqeq1d 2469 . . . . . 6  |-  ( ( m  e.  NN0  /\  y  e.  ZZ )  ->  ( ( y  x.  2 )  =  m  <-> 
( 2  x.  y
)  =  m ) )
44 eqid 2467 . . . . . . . . 9  |-  ( ( 2  x.  y )  +  1 )  =  ( ( 2  x.  y )  +  1 )
45 oveq2 6303 . . . . . . . . . . . 12  |-  ( n  =  y  ->  (
2  x.  n )  =  ( 2  x.  y ) )
4645oveq1d 6310 . . . . . . . . . . 11  |-  ( n  =  y  ->  (
( 2  x.  n
)  +  1 )  =  ( ( 2  x.  y )  +  1 ) )
4746eqeq1d 2469 . . . . . . . . . 10  |-  ( n  =  y  ->  (
( ( 2  x.  n )  +  1 )  =  ( ( 2  x.  y )  +  1 )  <->  ( (
2  x.  y )  +  1 )  =  ( ( 2  x.  y )  +  1 ) ) )
4847rspcev 3219 . . . . . . . . 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 671 . . . . . . . 8  |-  ( y  e.  ZZ  ->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  ( ( 2  x.  y
)  +  1 ) )
50 oveq1 6302 . . . . . . . . . 10  |-  ( ( 2  x.  y )  =  m  ->  (
( 2  x.  y
)  +  1 )  =  ( m  + 
1 ) )
5150eqeq2d 2481 . . . . . . . . 9  |-  ( ( 2  x.  y )  =  m  ->  (
( ( 2  x.  n )  +  1 )  =  ( ( 2  x.  y )  +  1 )  <->  ( (
2  x.  n )  +  1 )  =  ( m  +  1 ) ) )
5251rexbidv 2978 . . . . . . . 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 220 . . . . . . 7  |-  ( y  e.  ZZ  ->  (
( 2  x.  y
)  =  m  ->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  ( m  +  1 ) ) )
5453adantl 466 . . . . . 6  |-  ( ( m  e.  NN0  /\  y  e.  ZZ )  ->  ( ( 2  x.  y )  =  m  ->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  ( m  +  1 ) ) )
5543, 54sylbid 215 . . . . 5  |-  ( ( m  e.  NN0  /\  y  e.  ZZ )  ->  ( ( y  x.  2 )  =  m  ->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  ( m  +  1 ) ) )
5655rexlimdva 2959 . . . 4  |-  ( m  e.  NN0  ->  ( E. y  e.  ZZ  (
y  x.  2 )  =  m  ->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  ( m  +  1 ) ) )
57 peano2z 10916 . . . . . . . 8  |-  ( x  e.  ZZ  ->  (
x  +  1 )  e.  ZZ )
5857adantl 466 . . . . . . 7  |-  ( ( m  e.  NN0  /\  x  e.  ZZ )  ->  ( x  +  1 )  e.  ZZ )
59 zcn 10881 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  x  e.  CC )
60 mulcom 9590 . . . . . . . . . . . . 13  |-  ( ( x  e.  CC  /\  2  e.  CC )  ->  ( x  x.  2 )  =  ( 2  x.  x ) )
6131, 60mpan2 671 . . . . . . . . . . . 12  |-  ( x  e.  CC  ->  (
x  x.  2 )  =  ( 2  x.  x ) )
6231mulid2i 9611 . . . . . . . . . . . . 13  |-  ( 1  x.  2 )  =  2
6362a1i 11 . . . . . . . . . . . 12  |-  ( x  e.  CC  ->  (
1  x.  2 )  =  2 )
6461, 63oveq12d 6313 . . . . . . . . . . 11  |-  ( x  e.  CC  ->  (
( x  x.  2 )  +  ( 1  x.  2 ) )  =  ( ( 2  x.  x )  +  2 ) )
65 df-2 10606 . . . . . . . . . . . 12  |-  2  =  ( 1  +  1 )
6665oveq2i 6306 . . . . . . . . . . 11  |-  ( ( 2  x.  x )  +  2 )  =  ( ( 2  x.  x )  +  ( 1  +  1 ) )
6764, 66syl6eq 2524 . . . . . . . . . 10  |-  ( x  e.  CC  ->  (
( x  x.  2 )  +  ( 1  x.  2 ) )  =  ( ( 2  x.  x )  +  ( 1  +  1 ) ) )
68 ax-1cn 9562 . . . . . . . . . . 11  |-  1  e.  CC
69 adddir 9599 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  1  e.  CC  /\  2  e.  CC )  ->  (
( x  +  1 )  x.  2 )  =  ( ( x  x.  2 )  +  ( 1  x.  2 ) ) )
7068, 31, 69mp3an23 1316 . . . . . . . . . 10  |-  ( x  e.  CC  ->  (
( x  +  1 )  x.  2 )  =  ( ( x  x.  2 )  +  ( 1  x.  2 ) ) )
71 mulcl 9588 . . . . . . . . . . . 12  |-  ( ( 2  e.  CC  /\  x  e.  CC )  ->  ( 2  x.  x
)  e.  CC )
7231, 71mpan 670 . . . . . . . . . . 11  |-  ( x  e.  CC  ->  (
2  x.  x )  e.  CC )
73 addass 9591 . . . . . . . . . . . 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 1316 . . . . . . . . . . 11  |-  ( ( 2  x.  x )  e.  CC  ->  (
( ( 2  x.  x )  +  1 )  +  1 )  =  ( ( 2  x.  x )  +  ( 1  +  1 ) ) )
7572, 74syl 16 . . . . . . . . . 10  |-  ( x  e.  CC  ->  (
( ( 2  x.  x )  +  1 )  +  1 )  =  ( ( 2  x.  x )  +  ( 1  +  1 ) ) )
7667, 70, 753eqtr4d 2518 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
( x  +  1 )  x.  2 )  =  ( ( ( 2  x.  x )  +  1 )  +  1 ) )
7759, 76syl 16 . . . . . . . 8  |-  ( x  e.  ZZ  ->  (
( x  +  1 )  x.  2 )  =  ( ( ( 2  x.  x )  +  1 )  +  1 ) )
7877adantl 466 . . . . . . 7  |-  ( ( m  e.  NN0  /\  x  e.  ZZ )  ->  ( ( x  + 
1 )  x.  2 )  =  ( ( ( 2  x.  x
)  +  1 )  +  1 ) )
79 oveq1 6302 . . . . . . . . 9  |-  ( k  =  ( x  + 
1 )  ->  (
k  x.  2 )  =  ( ( x  +  1 )  x.  2 ) )
8079eqeq1d 2469 . . . . . . . 8  |-  ( k  =  ( x  + 
1 )  ->  (
( k  x.  2 )  =  ( ( ( 2  x.  x
)  +  1 )  +  1 )  <->  ( (
x  +  1 )  x.  2 )  =  ( ( ( 2  x.  x )  +  1 )  +  1 ) ) )
8180rspcev 3219 . . . . . . 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 661 . . . . . 6  |-  ( ( m  e.  NN0  /\  x  e.  ZZ )  ->  E. k  e.  ZZ  ( k  x.  2 )  =  ( ( ( 2  x.  x
)  +  1 )  +  1 ) )
83 oveq1 6302 . . . . . . . 8  |-  ( ( ( 2  x.  x
)  +  1 )  =  m  ->  (
( ( 2  x.  x )  +  1 )  +  1 )  =  ( m  + 
1 ) )
8483eqeq2d 2481 . . . . . . 7  |-  ( ( ( 2  x.  x
)  +  1 )  =  m  ->  (
( k  x.  2 )  =  ( ( ( 2  x.  x
)  +  1 )  +  1 )  <->  ( k  x.  2 )  =  ( m  +  1 ) ) )
8584rexbidv 2978 . . . . . 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 220 . . . . 5  |-  ( ( m  e.  NN0  /\  x  e.  ZZ )  ->  ( ( ( 2  x.  x )  +  1 )  =  m  ->  E. k  e.  ZZ  ( k  x.  2 )  =  ( m  +  1 ) ) )
8786rexlimdva 2959 . . . 4  |-  ( m  e.  NN0  ->  ( E. x  e.  ZZ  (
( 2  x.  x
)  +  1 )  =  m  ->  E. k  e.  ZZ  ( k  x.  2 )  =  ( m  +  1 ) ) )
8856, 87orim12d 836 . . 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 217 . 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 10969 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 368    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2818  (class class class)co 6295   CCcc 9502   0cc0 9504   1c1 9505    + caddc 9507    x. cmul 9509   2c2 10597   NN0cn0 10807   ZZcz 10876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877
This theorem is referenced by:  odd2np1  13922
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