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Theorem odd2np1lem 12602
Description: Lemma for odd2np1 12603. (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 2305 . . . 4  |-  ( j  =  0  ->  (
( ( 2  x.  n )  +  1 )  =  j  <->  ( (
2  x.  n )  +  1 )  =  0 ) )
21rexbidv 2577 . . 3  |-  ( j  =  0  ->  ( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  j  <->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  0 ) )
3 eqeq2 2305 . . . 4  |-  ( j  =  0  ->  (
( k  x.  2 )  =  j  <->  ( k  x.  2 )  =  0 ) )
43rexbidv 2577 . . 3  |-  ( j  =  0  ->  ( E. k  e.  ZZ  ( k  x.  2 )  =  j  <->  E. k  e.  ZZ  ( k  x.  2 )  =  0 ) )
52, 4orbi12d 690 . 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 2305 . . . . 5  |-  ( j  =  m  ->  (
( ( 2  x.  n )  +  1 )  =  j  <->  ( (
2  x.  n )  +  1 )  =  m ) )
76rexbidv 2577 . . . 4  |-  ( j  =  m  ->  ( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  j  <->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  m ) )
8 oveq2 5882 . . . . . . 7  |-  ( n  =  x  ->  (
2  x.  n )  =  ( 2  x.  x ) )
98oveq1d 5889 . . . . . 6  |-  ( n  =  x  ->  (
( 2  x.  n
)  +  1 )  =  ( ( 2  x.  x )  +  1 ) )
109eqeq1d 2304 . . . . 5  |-  ( n  =  x  ->  (
( ( 2  x.  n )  +  1 )  =  m  <->  ( (
2  x.  x )  +  1 )  =  m ) )
1110cbvrexv 2778 . . . 4  |-  ( E. n  e.  ZZ  (
( 2  x.  n
)  +  1 )  =  m  <->  E. x  e.  ZZ  ( ( 2  x.  x )  +  1 )  =  m )
127, 11syl6bb 252 . . 3  |-  ( j  =  m  ->  ( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  j  <->  E. x  e.  ZZ  ( ( 2  x.  x )  +  1 )  =  m ) )
13 eqeq2 2305 . . . . 5  |-  ( j  =  m  ->  (
( k  x.  2 )  =  j  <->  ( k  x.  2 )  =  m ) )
1413rexbidv 2577 . . . 4  |-  ( j  =  m  ->  ( E. k  e.  ZZ  ( k  x.  2 )  =  j  <->  E. k  e.  ZZ  ( k  x.  2 )  =  m ) )
15 oveq1 5881 . . . . . 6  |-  ( k  =  y  ->  (
k  x.  2 )  =  ( y  x.  2 ) )
1615eqeq1d 2304 . . . . 5  |-  ( k  =  y  ->  (
( k  x.  2 )  =  m  <->  ( y  x.  2 )  =  m ) )
1716cbvrexv 2778 . . . 4  |-  ( E. k  e.  ZZ  (
k  x.  2 )  =  m  <->  E. y  e.  ZZ  ( y  x.  2 )  =  m )
1814, 17syl6bb 252 . . 3  |-  ( j  =  m  ->  ( E. k  e.  ZZ  ( k  x.  2 )  =  j  <->  E. y  e.  ZZ  ( y  x.  2 )  =  m ) )
1912, 18orbi12d 690 . 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 2305 . . . 4  |-  ( j  =  ( m  + 
1 )  ->  (
( ( 2  x.  n )  +  1 )  =  j  <->  ( (
2  x.  n )  +  1 )  =  ( m  +  1 ) ) )
2120rexbidv 2577 . . 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 2305 . . . 4  |-  ( j  =  ( m  + 
1 )  ->  (
( k  x.  2 )  =  j  <->  ( k  x.  2 )  =  ( m  +  1 ) ) )
2322rexbidv 2577 . . 3  |-  ( j  =  ( m  + 
1 )  ->  ( E. k  e.  ZZ  ( k  x.  2 )  =  j  <->  E. k  e.  ZZ  ( k  x.  2 )  =  ( m  +  1 ) ) )
2421, 23orbi12d 690 . 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 2305 . . . 4  |-  ( j  =  N  ->  (
( ( 2  x.  n )  +  1 )  =  j  <->  ( (
2  x.  n )  +  1 )  =  N ) )
2625rexbidv 2577 . . 3  |-  ( j  =  N  ->  ( E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  j  <->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N ) )
27 eqeq2 2305 . . . 4  |-  ( j  =  N  ->  (
( k  x.  2 )  =  j  <->  ( k  x.  2 )  =  N ) )
2827rexbidv 2577 . . 3  |-  ( j  =  N  ->  ( E. k  e.  ZZ  ( k  x.  2 )  =  j  <->  E. k  e.  ZZ  ( k  x.  2 )  =  N ) )
2926, 28orbi12d 690 . 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 10051 . . . 4  |-  0  e.  ZZ
31 2cn 9832 . . . . 5  |-  2  e.  CC
3231mul02i 9017 . . . 4  |-  ( 0  x.  2 )  =  0
33 oveq1 5881 . . . . . 6  |-  ( k  =  0  ->  (
k  x.  2 )  =  ( 0  x.  2 ) )
3433eqeq1d 2304 . . . . 5  |-  ( k  =  0  ->  (
( k  x.  2 )  =  0  <->  (
0  x.  2 )  =  0 ) )
3534rspcev 2897 . . . 4  |-  ( ( 0  e.  ZZ  /\  ( 0  x.  2 )  =  0 )  ->  E. k  e.  ZZ  ( k  x.  2 )  =  0 )
3630, 32, 35mp2an 653 . . 3  |-  E. k  e.  ZZ  ( k  x.  2 )  =  0
3736olci 380 . 2  |-  ( E. n  e.  ZZ  (
( 2  x.  n
)  +  1 )  =  0  \/  E. k  e.  ZZ  (
k  x.  2 )  =  0 )
38 orcom 376 . . 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 10045 . . . . . . . . 9  |-  ( y  e.  ZZ  ->  y  e.  CC )
40 mulcom 8839 . . . . . . . . 9  |-  ( ( y  e.  CC  /\  2  e.  CC )  ->  ( y  x.  2 )  =  ( 2  x.  y ) )
4139, 31, 40sylancl 643 . . . . . . . 8  |-  ( y  e.  ZZ  ->  (
y  x.  2 )  =  ( 2  x.  y ) )
4241adantl 452 . . . . . . 7  |-  ( ( m  e.  NN0  /\  y  e.  ZZ )  ->  ( y  x.  2 )  =  ( 2  x.  y ) )
4342eqeq1d 2304 . . . . . 6  |-  ( ( m  e.  NN0  /\  y  e.  ZZ )  ->  ( ( y  x.  2 )  =  m  <-> 
( 2  x.  y
)  =  m ) )
44 eqid 2296 . . . . . . . . 9  |-  ( ( 2  x.  y )  +  1 )  =  ( ( 2  x.  y )  +  1 )
45 oveq2 5882 . . . . . . . . . . . 12  |-  ( n  =  y  ->  (
2  x.  n )  =  ( 2  x.  y ) )
4645oveq1d 5889 . . . . . . . . . . 11  |-  ( n  =  y  ->  (
( 2  x.  n
)  +  1 )  =  ( ( 2  x.  y )  +  1 ) )
4746eqeq1d 2304 . . . . . . . . . 10  |-  ( n  =  y  ->  (
( ( 2  x.  n )  +  1 )  =  ( ( 2  x.  y )  +  1 )  <->  ( (
2  x.  y )  +  1 )  =  ( ( 2  x.  y )  +  1 ) ) )
4847rspcev 2897 . . . . . . . . 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 652 . . . . . . . 8  |-  ( y  e.  ZZ  ->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  ( ( 2  x.  y
)  +  1 ) )
50 oveq1 5881 . . . . . . . . . 10  |-  ( ( 2  x.  y )  =  m  ->  (
( 2  x.  y
)  +  1 )  =  ( m  + 
1 ) )
5150eqeq2d 2307 . . . . . . . . 9  |-  ( ( 2  x.  y )  =  m  ->  (
( ( 2  x.  n )  +  1 )  =  ( ( 2  x.  y )  +  1 )  <->  ( (
2  x.  n )  +  1 )  =  ( m  +  1 ) ) )
5251rexbidv 2577 . . . . . . . 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 211 . . . . . . 7  |-  ( y  e.  ZZ  ->  (
( 2  x.  y
)  =  m  ->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  ( m  +  1 ) ) )
5453adantl 452 . . . . . 6  |-  ( ( m  e.  NN0  /\  y  e.  ZZ )  ->  ( ( 2  x.  y )  =  m  ->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  ( m  +  1 ) ) )
5543, 54sylbid 206 . . . . 5  |-  ( ( m  e.  NN0  /\  y  e.  ZZ )  ->  ( ( y  x.  2 )  =  m  ->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  ( m  +  1 ) ) )
5655rexlimdva 2680 . . . 4  |-  ( m  e.  NN0  ->  ( E. y  e.  ZZ  (
y  x.  2 )  =  m  ->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  ( m  +  1 ) ) )
57 peano2z 10076 . . . . . . . 8  |-  ( x  e.  ZZ  ->  (
x  +  1 )  e.  ZZ )
5857adantl 452 . . . . . . 7  |-  ( ( m  e.  NN0  /\  x  e.  ZZ )  ->  ( x  +  1 )  e.  ZZ )
59 zcn 10045 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  x  e.  CC )
60 mulcom 8839 . . . . . . . . . . . . 13  |-  ( ( x  e.  CC  /\  2  e.  CC )  ->  ( x  x.  2 )  =  ( 2  x.  x ) )
6131, 60mpan2 652 . . . . . . . . . . . 12  |-  ( x  e.  CC  ->  (
x  x.  2 )  =  ( 2  x.  x ) )
6231mulid2i 8856 . . . . . . . . . . . . 13  |-  ( 1  x.  2 )  =  2
6362a1i 10 . . . . . . . . . . . 12  |-  ( x  e.  CC  ->  (
1  x.  2 )  =  2 )
6461, 63oveq12d 5892 . . . . . . . . . . 11  |-  ( x  e.  CC  ->  (
( x  x.  2 )  +  ( 1  x.  2 ) )  =  ( ( 2  x.  x )  +  2 ) )
65 df-2 9820 . . . . . . . . . . . 12  |-  2  =  ( 1  +  1 )
6665oveq2i 5885 . . . . . . . . . . 11  |-  ( ( 2  x.  x )  +  2 )  =  ( ( 2  x.  x )  +  ( 1  +  1 ) )
6764, 66syl6eq 2344 . . . . . . . . . 10  |-  ( x  e.  CC  ->  (
( x  x.  2 )  +  ( 1  x.  2 ) )  =  ( ( 2  x.  x )  +  ( 1  +  1 ) ) )
68 ax-1cn 8811 . . . . . . . . . . 11  |-  1  e.  CC
69 adddir 8846 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  1  e.  CC  /\  2  e.  CC )  ->  (
( x  +  1 )  x.  2 )  =  ( ( x  x.  2 )  +  ( 1  x.  2 ) ) )
7068, 31, 69mp3an23 1269 . . . . . . . . . 10  |-  ( x  e.  CC  ->  (
( x  +  1 )  x.  2 )  =  ( ( x  x.  2 )  +  ( 1  x.  2 ) ) )
71 mulcl 8837 . . . . . . . . . . . 12  |-  ( ( 2  e.  CC  /\  x  e.  CC )  ->  ( 2  x.  x
)  e.  CC )
7231, 71mpan 651 . . . . . . . . . . 11  |-  ( x  e.  CC  ->  (
2  x.  x )  e.  CC )
73 addass 8840 . . . . . . . . . . . 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 1269 . . . . . . . . . . 11  |-  ( ( 2  x.  x )  e.  CC  ->  (
( ( 2  x.  x )  +  1 )  +  1 )  =  ( ( 2  x.  x )  +  ( 1  +  1 ) ) )
7572, 74syl 15 . . . . . . . . . 10  |-  ( x  e.  CC  ->  (
( ( 2  x.  x )  +  1 )  +  1 )  =  ( ( 2  x.  x )  +  ( 1  +  1 ) ) )
7667, 70, 753eqtr4d 2338 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
( x  +  1 )  x.  2 )  =  ( ( ( 2  x.  x )  +  1 )  +  1 ) )
7759, 76syl 15 . . . . . . . 8  |-  ( x  e.  ZZ  ->  (
( x  +  1 )  x.  2 )  =  ( ( ( 2  x.  x )  +  1 )  +  1 ) )
7877adantl 452 . . . . . . 7  |-  ( ( m  e.  NN0  /\  x  e.  ZZ )  ->  ( ( x  + 
1 )  x.  2 )  =  ( ( ( 2  x.  x
)  +  1 )  +  1 ) )
79 oveq1 5881 . . . . . . . . 9  |-  ( k  =  ( x  + 
1 )  ->  (
k  x.  2 )  =  ( ( x  +  1 )  x.  2 ) )
8079eqeq1d 2304 . . . . . . . 8  |-  ( k  =  ( x  + 
1 )  ->  (
( k  x.  2 )  =  ( ( ( 2  x.  x
)  +  1 )  +  1 )  <->  ( (
x  +  1 )  x.  2 )  =  ( ( ( 2  x.  x )  +  1 )  +  1 ) ) )
8180rspcev 2897 . . . . . . 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 642 . . . . . 6  |-  ( ( m  e.  NN0  /\  x  e.  ZZ )  ->  E. k  e.  ZZ  ( k  x.  2 )  =  ( ( ( 2  x.  x
)  +  1 )  +  1 ) )
83 oveq1 5881 . . . . . . . 8  |-  ( ( ( 2  x.  x
)  +  1 )  =  m  ->  (
( ( 2  x.  x )  +  1 )  +  1 )  =  ( m  + 
1 ) )
8483eqeq2d 2307 . . . . . . 7  |-  ( ( ( 2  x.  x
)  +  1 )  =  m  ->  (
( k  x.  2 )  =  ( ( ( 2  x.  x
)  +  1 )  +  1 )  <->  ( k  x.  2 )  =  ( m  +  1 ) ) )
8584rexbidv 2577 . . . . . 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 211 . . . . 5  |-  ( ( m  e.  NN0  /\  x  e.  ZZ )  ->  ( ( ( 2  x.  x )  +  1 )  =  m  ->  E. k  e.  ZZ  ( k  x.  2 )  =  ( m  +  1 ) ) )
8786rexlimdva 2680 . . . 4  |-  ( m  e.  NN0  ->  ( E. x  e.  ZZ  (
( 2  x.  x
)  +  1 )  =  m  ->  E. k  e.  ZZ  ( k  x.  2 )  =  ( m  +  1 ) ) )
8856, 87orim12d 811 . . 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 208 . 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 10124 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 set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758   2c2 9811   NN0cn0 9981   ZZcz 10040
This theorem is referenced by:  odd2np1  12603
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041
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