ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pw2dvdslemn Unicode version

Theorem pw2dvdslemn 11482
Description: Lemma for pw2dvds 11483. If a natural number has some power of two which does not divide it, there is a highest power of two which does divide it. (Contributed by Jim Kingdon, 14-Nov-2021.)
Assertion
Ref Expression
pw2dvdslemn  |-  ( ( N  e.  NN  /\  A  e.  NN  /\  -.  ( 2 ^ A
)  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) )
Distinct variable group:    m, N
Allowed substitution hint:    A( m)

Proof of Theorem pw2dvdslemn
Dummy variables  w  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3simpb 942 . 2  |-  ( ( N  e.  NN  /\  A  e.  NN  /\  -.  ( 2 ^ A
)  ||  N )  ->  ( N  e.  NN  /\ 
-.  ( 2 ^ A )  ||  N
) )
2 oveq2 5674 . . . . . . . 8  |-  ( w  =  1  ->  (
2 ^ w )  =  ( 2 ^ 1 ) )
32breq1d 3861 . . . . . . 7  |-  ( w  =  1  ->  (
( 2 ^ w
)  ||  N  <->  ( 2 ^ 1 )  ||  N ) )
43notbid 628 . . . . . 6  |-  ( w  =  1  ->  ( -.  ( 2 ^ w
)  ||  N  <->  -.  (
2 ^ 1 ) 
||  N ) )
54anbi2d 453 . . . . 5  |-  ( w  =  1  ->  (
( N  e.  NN  /\ 
-.  ( 2 ^ w )  ||  N
)  <->  ( N  e.  NN  /\  -.  (
2 ^ 1 ) 
||  N ) ) )
65imbi1d 230 . . . 4  |-  ( w  =  1  ->  (
( ( N  e.  NN  /\  -.  (
2 ^ w ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) )  <->  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) ) )
7 oveq2 5674 . . . . . . . 8  |-  ( w  =  k  ->  (
2 ^ w )  =  ( 2 ^ k ) )
87breq1d 3861 . . . . . . 7  |-  ( w  =  k  ->  (
( 2 ^ w
)  ||  N  <->  ( 2 ^ k )  ||  N ) )
98notbid 628 . . . . . 6  |-  ( w  =  k  ->  ( -.  ( 2 ^ w
)  ||  N  <->  -.  (
2 ^ k ) 
||  N ) )
109anbi2d 453 . . . . 5  |-  ( w  =  k  ->  (
( N  e.  NN  /\ 
-.  ( 2 ^ w )  ||  N
)  <->  ( N  e.  NN  /\  -.  (
2 ^ k ) 
||  N ) ) )
1110imbi1d 230 . . . 4  |-  ( w  =  k  ->  (
( ( N  e.  NN  /\  -.  (
2 ^ w ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) )  <->  ( ( N  e.  NN  /\  -.  ( 2 ^ k
)  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) ) )
12 oveq2 5674 . . . . . . . 8  |-  ( w  =  ( k  +  1 )  ->  (
2 ^ w )  =  ( 2 ^ ( k  +  1 ) ) )
1312breq1d 3861 . . . . . . 7  |-  ( w  =  ( k  +  1 )  ->  (
( 2 ^ w
)  ||  N  <->  ( 2 ^ ( k  +  1 ) )  ||  N ) )
1413notbid 628 . . . . . 6  |-  ( w  =  ( k  +  1 )  ->  ( -.  ( 2 ^ w
)  ||  N  <->  -.  (
2 ^ ( k  +  1 ) ) 
||  N ) )
1514anbi2d 453 . . . . 5  |-  ( w  =  ( k  +  1 )  ->  (
( N  e.  NN  /\ 
-.  ( 2 ^ w )  ||  N
)  <->  ( N  e.  NN  /\  -.  (
2 ^ ( k  +  1 ) ) 
||  N ) ) )
1615imbi1d 230 . . . 4  |-  ( w  =  ( k  +  1 )  ->  (
( ( N  e.  NN  /\  -.  (
2 ^ w ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) )  <->  ( ( N  e.  NN  /\  -.  ( 2 ^ (
k  +  1 ) )  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) ) )
17 oveq2 5674 . . . . . . . 8  |-  ( w  =  A  ->  (
2 ^ w )  =  ( 2 ^ A ) )
1817breq1d 3861 . . . . . . 7  |-  ( w  =  A  ->  (
( 2 ^ w
)  ||  N  <->  ( 2 ^ A )  ||  N ) )
1918notbid 628 . . . . . 6  |-  ( w  =  A  ->  ( -.  ( 2 ^ w
)  ||  N  <->  -.  (
2 ^ A ) 
||  N ) )
2019anbi2d 453 . . . . 5  |-  ( w  =  A  ->  (
( N  e.  NN  /\ 
-.  ( 2 ^ w )  ||  N
)  <->  ( N  e.  NN  /\  -.  (
2 ^ A ) 
||  N ) ) )
2120imbi1d 230 . . . 4  |-  ( w  =  A  ->  (
( ( N  e.  NN  /\  -.  (
2 ^ w ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) )  <->  ( ( N  e.  NN  /\  -.  ( 2 ^ A
)  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) ) )
22 0nn0 8749 . . . . . 6  |-  0  e.  NN0
2322a1i 9 . . . . 5  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  0  e.  NN0 )
24 oveq2 5674 . . . . . . . 8  |-  ( m  =  0  ->  (
2 ^ m )  =  ( 2 ^ 0 ) )
2524breq1d 3861 . . . . . . 7  |-  ( m  =  0  ->  (
( 2 ^ m
)  ||  N  <->  ( 2 ^ 0 )  ||  N ) )
26 oveq1 5673 . . . . . . . . . 10  |-  ( m  =  0  ->  (
m  +  1 )  =  ( 0  +  1 ) )
2726oveq2d 5682 . . . . . . . . 9  |-  ( m  =  0  ->  (
2 ^ ( m  +  1 ) )  =  ( 2 ^ ( 0  +  1 ) ) )
2827breq1d 3861 . . . . . . . 8  |-  ( m  =  0  ->  (
( 2 ^ (
m  +  1 ) )  ||  N  <->  ( 2 ^ ( 0  +  1 ) )  ||  N ) )
2928notbid 628 . . . . . . 7  |-  ( m  =  0  ->  ( -.  ( 2 ^ (
m  +  1 ) )  ||  N  <->  -.  (
2 ^ ( 0  +  1 ) ) 
||  N ) )
3025, 29anbi12d 458 . . . . . 6  |-  ( m  =  0  ->  (
( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
)  <->  ( ( 2 ^ 0 )  ||  N  /\  -.  ( 2 ^ ( 0  +  1 ) )  ||  N ) ) )
3130adantl 272 . . . . 5  |-  ( ( ( N  e.  NN  /\ 
-.  ( 2 ^ 1 )  ||  N
)  /\  m  = 
0 )  ->  (
( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
)  <->  ( ( 2 ^ 0 )  ||  N  /\  -.  ( 2 ^ ( 0  +  1 ) )  ||  N ) ) )
32 2cnd 8556 . . . . . . . 8  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  2  e.  CC )
3332exp0d 10141 . . . . . . 7  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  ( 2 ^ 0 )  =  1 )
34 simpl 108 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  N  e.  NN )
3534nnzd 8928 . . . . . . . 8  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  N  e.  ZZ )
36 1dvds 11149 . . . . . . . 8  |-  ( N  e.  ZZ  ->  1  ||  N )
3735, 36syl 14 . . . . . . 7  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  1  ||  N
)
3833, 37eqbrtrd 3871 . . . . . 6  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  ( 2 ^ 0 )  ||  N
)
39 simpr 109 . . . . . . 7  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  -.  ( 2 ^ 1 )  ||  N )
40 0p1e1 8597 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
4140oveq2i 5677 . . . . . . . 8  |-  ( 2 ^ ( 0  +  1 ) )  =  ( 2 ^ 1 )
4241breq1i 3858 . . . . . . 7  |-  ( ( 2 ^ ( 0  +  1 ) ) 
||  N  <->  ( 2 ^ 1 )  ||  N )
4339, 42sylnibr 638 . . . . . 6  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  -.  ( 2 ^ ( 0  +  1 ) )  ||  N )
4438, 43jca 301 . . . . 5  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  ( ( 2 ^ 0 )  ||  N  /\  -.  ( 2 ^ ( 0  +  1 ) )  ||  N ) )
4523, 31, 44rspcedvd 2729 . . . 4  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) )
46 simpll 497 . . . . . . . . 9  |-  ( ( ( k  e.  NN  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  -> 
k  e.  NN )
4746nnnn0d 8787 . . . . . . . 8  |-  ( ( ( k  e.  NN  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  -> 
k  e.  NN0 )
48 oveq2 5674 . . . . . . . . . . 11  |-  ( m  =  k  ->  (
2 ^ m )  =  ( 2 ^ k ) )
4948breq1d 3861 . . . . . . . . . 10  |-  ( m  =  k  ->  (
( 2 ^ m
)  ||  N  <->  ( 2 ^ k )  ||  N ) )
50 oveq1 5673 . . . . . . . . . . . . 13  |-  ( m  =  k  ->  (
m  +  1 )  =  ( k  +  1 ) )
5150oveq2d 5682 . . . . . . . . . . . 12  |-  ( m  =  k  ->  (
2 ^ ( m  +  1 ) )  =  ( 2 ^ ( k  +  1 ) ) )
5251breq1d 3861 . . . . . . . . . . 11  |-  ( m  =  k  ->  (
( 2 ^ (
m  +  1 ) )  ||  N  <->  ( 2 ^ ( k  +  1 ) )  ||  N ) )
5352notbid 628 . . . . . . . . . 10  |-  ( m  =  k  ->  ( -.  ( 2 ^ (
m  +  1 ) )  ||  N  <->  -.  (
2 ^ ( k  +  1 ) ) 
||  N ) )
5449, 53anbi12d 458 . . . . . . . . 9  |-  ( m  =  k  ->  (
( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
)  <->  ( ( 2 ^ k )  ||  N  /\  -.  ( 2 ^ ( k  +  1 ) )  ||  N ) ) )
5554adantl 272 . . . . . . . 8  |-  ( ( ( ( k  e.  NN  /\  ( N  e.  NN  /\  -.  ( 2 ^ (
k  +  1 ) )  ||  N ) )  /\  ( 2 ^ k )  ||  N )  /\  m  =  k )  -> 
( ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  + 
1 ) )  ||  N )  <->  ( (
2 ^ k ) 
||  N  /\  -.  ( 2 ^ (
k  +  1 ) )  ||  N ) ) )
56 simpr 109 . . . . . . . . 9  |-  ( ( ( k  e.  NN  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  -> 
( 2 ^ k
)  ||  N )
57 simplrr 504 . . . . . . . . 9  |-  ( ( ( k  e.  NN  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  ->  -.  ( 2 ^ (
k  +  1 ) )  ||  N )
5856, 57jca 301 . . . . . . . 8  |-  ( ( ( k  e.  NN  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  -> 
( ( 2 ^ k )  ||  N  /\  -.  ( 2 ^ ( k  +  1 ) )  ||  N
) )
5947, 55, 58rspcedvd 2729 . . . . . . 7  |-  ( ( ( k  e.  NN  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) )
6059adantllr 466 . . . . . 6  |-  ( ( ( ( k  e.  NN  /\  ( ( N  e.  NN  /\  -.  ( 2 ^ k
)  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) )
61 simprl 499 . . . . . . . 8  |-  ( ( ( k  e.  NN  /\  ( ( N  e.  NN  /\  -.  (
2 ^ k ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  ->  N  e.  NN )
6261anim1i 334 . . . . . . 7  |-  ( ( ( ( k  e.  NN  /\  ( ( N  e.  NN  /\  -.  ( 2 ^ k
)  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  -.  ( 2 ^ k
)  ||  N )  ->  ( N  e.  NN  /\ 
-.  ( 2 ^ k )  ||  N
) )
63 simpllr 502 . . . . . . 7  |-  ( ( ( ( k  e.  NN  /\  ( ( N  e.  NN  /\  -.  ( 2 ^ k
)  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  -.  ( 2 ^ k
)  ||  N )  ->  ( ( N  e.  NN  /\  -.  (
2 ^ k ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )
6462, 63mpd 13 . . . . . 6  |-  ( ( ( ( k  e.  NN  /\  ( ( N  e.  NN  /\  -.  ( 2 ^ k
)  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  -.  ( 2 ^ k
)  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) )
65 2nn 8638 . . . . . . . . 9  |-  2  e.  NN
66 simpll 497 . . . . . . . . . 10  |-  ( ( ( k  e.  NN  /\  ( ( N  e.  NN  /\  -.  (
2 ^ k ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  ->  k  e.  NN )
6766nnnn0d 8787 . . . . . . . . 9  |-  ( ( ( k  e.  NN  /\  ( ( N  e.  NN  /\  -.  (
2 ^ k ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  ->  k  e.  NN0 )
68 nnexpcl 10029 . . . . . . . . 9  |-  ( ( 2  e.  NN  /\  k  e.  NN0 )  -> 
( 2 ^ k
)  e.  NN )
6965, 67, 68sylancr 406 . . . . . . . 8  |-  ( ( ( k  e.  NN  /\  ( ( N  e.  NN  /\  -.  (
2 ^ k ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  ->  (
2 ^ k )  e.  NN )
7061nnzd 8928 . . . . . . . 8  |-  ( ( ( k  e.  NN  /\  ( ( N  e.  NN  /\  -.  (
2 ^ k ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  ->  N  e.  ZZ )
71 dvdsdc 11143 . . . . . . . 8  |-  ( ( ( 2 ^ k
)  e.  NN  /\  N  e.  ZZ )  -> DECID  ( 2 ^ k ) 
||  N )
7269, 70, 71syl2anc 404 . . . . . . 7  |-  ( ( ( k  e.  NN  /\  ( ( N  e.  NN  /\  -.  (
2 ^ k ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  -> DECID  ( 2 ^ k
)  ||  N )
73 exmiddc 783 . . . . . . 7  |-  (DECID  ( 2 ^ k )  ||  N  ->  ( ( 2 ^ k )  ||  N  \/  -.  (
2 ^ k ) 
||  N ) )
7472, 73syl 14 . . . . . 6  |-  ( ( ( k  e.  NN  /\  ( ( N  e.  NN  /\  -.  (
2 ^ k ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  ->  (
( 2 ^ k
)  ||  N  \/  -.  ( 2 ^ k
)  ||  N )
)
7560, 64, 74mpjaodan 748 . . . . 5  |-  ( ( ( k  e.  NN  /\  ( ( N  e.  NN  /\  -.  (
2 ^ k ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  + 
1 ) )  ||  N ) )
7675exp31 357 . . . 4  |-  ( k  e.  NN  ->  (
( ( N  e.  NN  /\  -.  (
2 ^ k ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) )  ->  (
( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
)  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  + 
1 ) )  ||  N ) ) ) )
776, 11, 16, 21, 45, 76nnind 8499 . . 3  |-  ( A  e.  NN  ->  (
( N  e.  NN  /\ 
-.  ( 2 ^ A )  ||  N
)  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  + 
1 ) )  ||  N ) ) )
78773ad2ant2 966 . 2  |-  ( ( N  e.  NN  /\  A  e.  NN  /\  -.  ( 2 ^ A
)  ||  N )  ->  ( ( N  e.  NN  /\  -.  (
2 ^ A ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )
791, 78mpd 13 1  |-  ( ( N  e.  NN  /\  A  e.  NN  /\  -.  ( 2 ^ A
)  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 665  DECID wdc 781    /\ w3a 925    = wceq 1290    e. wcel 1439   E.wrex 2361   class class class wbr 3851  (class class class)co 5666   0cc0 7411   1c1 7412    + caddc 7414   NNcn 8483   2c2 8534   NN0cn0 8734   ZZcz 8811   ^cexp 10015    || cdvds 11135
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-mulrcl 7505  ax-addcom 7506  ax-mulcom 7507  ax-addass 7508  ax-mulass 7509  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-1rid 7513  ax-0id 7514  ax-rnegex 7515  ax-precex 7516  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-apti 7521  ax-pre-ltadd 7522  ax-pre-mulgt0 7523  ax-pre-mulext 7524  ax-arch 7525
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-frec 6170  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-reap 8113  df-ap 8120  df-div 8201  df-inn 8484  df-2 8542  df-n0 8735  df-z 8812  df-uz 9081  df-q 9166  df-rp 9196  df-fl 9738  df-mod 9791  df-iseq 9914  df-seq3 9915  df-exp 10016  df-dvds 11136
This theorem is referenced by:  pw2dvds  11483
  Copyright terms: Public domain W3C validator