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

Theorem pw2dvdslemn 11749
Description: Lemma for pw2dvds 11750. 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 962 . 2  |-  ( ( N  e.  NN  /\  A  e.  NN  /\  -.  ( 2 ^ A
)  ||  N )  ->  ( N  e.  NN  /\ 
-.  ( 2 ^ A )  ||  N
) )
2 oveq2 5748 . . . . . . . 8  |-  ( w  =  1  ->  (
2 ^ w )  =  ( 2 ^ 1 ) )
32breq1d 3907 . . . . . . 7  |-  ( w  =  1  ->  (
( 2 ^ w
)  ||  N  <->  ( 2 ^ 1 )  ||  N ) )
43notbid 639 . . . . . 6  |-  ( w  =  1  ->  ( -.  ( 2 ^ w
)  ||  N  <->  -.  (
2 ^ 1 ) 
||  N ) )
54anbi2d 457 . . . . 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 5748 . . . . . . . 8  |-  ( w  =  k  ->  (
2 ^ w )  =  ( 2 ^ k ) )
87breq1d 3907 . . . . . . 7  |-  ( w  =  k  ->  (
( 2 ^ w
)  ||  N  <->  ( 2 ^ k )  ||  N ) )
98notbid 639 . . . . . 6  |-  ( w  =  k  ->  ( -.  ( 2 ^ w
)  ||  N  <->  -.  (
2 ^ k ) 
||  N ) )
109anbi2d 457 . . . . 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 5748 . . . . . . . 8  |-  ( w  =  ( k  +  1 )  ->  (
2 ^ w )  =  ( 2 ^ ( k  +  1 ) ) )
1312breq1d 3907 . . . . . . 7  |-  ( w  =  ( k  +  1 )  ->  (
( 2 ^ w
)  ||  N  <->  ( 2 ^ ( k  +  1 ) )  ||  N ) )
1413notbid 639 . . . . . 6  |-  ( w  =  ( k  +  1 )  ->  ( -.  ( 2 ^ w
)  ||  N  <->  -.  (
2 ^ ( k  +  1 ) ) 
||  N ) )
1514anbi2d 457 . . . . 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 5748 . . . . . . . 8  |-  ( w  =  A  ->  (
2 ^ w )  =  ( 2 ^ A ) )
1817breq1d 3907 . . . . . . 7  |-  ( w  =  A  ->  (
( 2 ^ w
)  ||  N  <->  ( 2 ^ A )  ||  N ) )
1918notbid 639 . . . . . 6  |-  ( w  =  A  ->  ( -.  ( 2 ^ w
)  ||  N  <->  -.  (
2 ^ A ) 
||  N ) )
2019anbi2d 457 . . . . 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 8946 . . . . . 6  |-  0  e.  NN0
2322a1i 9 . . . . 5  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  0  e.  NN0 )
24 oveq2 5748 . . . . . . . 8  |-  ( m  =  0  ->  (
2 ^ m )  =  ( 2 ^ 0 ) )
2524breq1d 3907 . . . . . . 7  |-  ( m  =  0  ->  (
( 2 ^ m
)  ||  N  <->  ( 2 ^ 0 )  ||  N ) )
26 oveq1 5747 . . . . . . . . . 10  |-  ( m  =  0  ->  (
m  +  1 )  =  ( 0  +  1 ) )
2726oveq2d 5756 . . . . . . . . 9  |-  ( m  =  0  ->  (
2 ^ ( m  +  1 ) )  =  ( 2 ^ ( 0  +  1 ) ) )
2827breq1d 3907 . . . . . . . 8  |-  ( m  =  0  ->  (
( 2 ^ (
m  +  1 ) )  ||  N  <->  ( 2 ^ ( 0  +  1 ) )  ||  N ) )
2928notbid 639 . . . . . . 7  |-  ( m  =  0  ->  ( -.  ( 2 ^ (
m  +  1 ) )  ||  N  <->  -.  (
2 ^ ( 0  +  1 ) ) 
||  N ) )
3025, 29anbi12d 462 . . . . . 6  |-  ( m  =  0  ->  (
( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
)  <->  ( ( 2 ^ 0 )  ||  N  /\  -.  ( 2 ^ ( 0  +  1 ) )  ||  N ) ) )
3130adantl 273 . . . . 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 8753 . . . . . . . 8  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  2  e.  CC )
3332exp0d 10369 . . . . . . 7  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  ( 2 ^ 0 )  =  1 )
34 simpl 108 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  N  e.  NN )
3534nnzd 9126 . . . . . . . 8  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  N  e.  ZZ )
36 1dvds 11414 . . . . . . . 8  |-  ( N  e.  ZZ  ->  1  ||  N )
3735, 36syl 14 . . . . . . 7  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  1  ||  N
)
3833, 37eqbrtrd 3918 . . . . . 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 8794 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
4140oveq2i 5751 . . . . . . . 8  |-  ( 2 ^ ( 0  +  1 ) )  =  ( 2 ^ 1 )
4241breq1i 3904 . . . . . . 7  |-  ( ( 2 ^ ( 0  +  1 ) ) 
||  N  <->  ( 2 ^ 1 )  ||  N )
4339, 42sylnibr 649 . . . . . 6  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  -.  ( 2 ^ ( 0  +  1 ) )  ||  N )
4438, 43jca 302 . . . . 5  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  ( ( 2 ^ 0 )  ||  N  /\  -.  ( 2 ^ ( 0  +  1 ) )  ||  N ) )
4523, 31, 44rspcedvd 2767 . . . 4  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) )
46 simpll 501 . . . . . . . . 9  |-  ( ( ( k  e.  NN  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  -> 
k  e.  NN )
4746nnnn0d 8984 . . . . . . . 8  |-  ( ( ( k  e.  NN  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  -> 
k  e.  NN0 )
48 oveq2 5748 . . . . . . . . . . 11  |-  ( m  =  k  ->  (
2 ^ m )  =  ( 2 ^ k ) )
4948breq1d 3907 . . . . . . . . . 10  |-  ( m  =  k  ->  (
( 2 ^ m
)  ||  N  <->  ( 2 ^ k )  ||  N ) )
50 oveq1 5747 . . . . . . . . . . . . 13  |-  ( m  =  k  ->  (
m  +  1 )  =  ( k  +  1 ) )
5150oveq2d 5756 . . . . . . . . . . . 12  |-  ( m  =  k  ->  (
2 ^ ( m  +  1 ) )  =  ( 2 ^ ( k  +  1 ) ) )
5251breq1d 3907 . . . . . . . . . . 11  |-  ( m  =  k  ->  (
( 2 ^ (
m  +  1 ) )  ||  N  <->  ( 2 ^ ( k  +  1 ) )  ||  N ) )
5352notbid 639 . . . . . . . . . 10  |-  ( m  =  k  ->  ( -.  ( 2 ^ (
m  +  1 ) )  ||  N  <->  -.  (
2 ^ ( k  +  1 ) ) 
||  N ) )
5449, 53anbi12d 462 . . . . . . . . 9  |-  ( m  =  k  ->  (
( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
)  <->  ( ( 2 ^ k )  ||  N  /\  -.  ( 2 ^ ( k  +  1 ) )  ||  N ) ) )
5554adantl 273 . . . . . . . 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 508 . . . . . . . . 9  |-  ( ( ( k  e.  NN  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  ->  -.  ( 2 ^ (
k  +  1 ) )  ||  N )
5856, 57jca 302 . . . . . . . 8  |-  ( ( ( k  e.  NN  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  -> 
( ( 2 ^ k )  ||  N  /\  -.  ( 2 ^ ( k  +  1 ) )  ||  N
) )
5947, 55, 58rspcedvd 2767 . . . . . . 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 470 . . . . . 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 503 . . . . . . . 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 336 . . . . . . 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 506 . . . . . . 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 8835 . . . . . . . . 9  |-  2  e.  NN
66 simpll 501 . . . . . . . . . 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 8984 . . . . . . . . 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 10257 . . . . . . . . 9  |-  ( ( 2  e.  NN  /\  k  e.  NN0 )  -> 
( 2 ^ k
)  e.  NN )
6965, 67, 68sylancr 408 . . . . . . . 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 9126 . . . . . . . 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 11408 . . . . . . . 8  |-  ( ( ( 2 ^ k
)  e.  NN  /\  N  e.  ZZ )  -> DECID  ( 2 ^ k ) 
||  N )
7269, 70, 71syl2anc 406 . . . . . . 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 804 . . . . . . 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 770 . . . . 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 359 . . . 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 8696 . . 3  |-  ( A  e.  NN  ->  (
( N  e.  NN  /\ 
-.  ( 2 ^ A )  ||  N
)  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  + 
1 ) )  ||  N ) ) )
78773ad2ant2 986 . 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 680  DECID wdc 802    /\ w3a 945    = wceq 1314    e. wcel 1463   E.wrex 2392   class class class wbr 3897  (class class class)co 5740   0cc0 7584   1c1 7585    + caddc 7587   NNcn 8680   2c2 8731   NN0cn0 8931   ZZcz 9008   ^cexp 10243    || cdvds 11400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8396  df-inn 8681  df-2 8739  df-n0 8932  df-z 9009  df-uz 9279  df-q 9364  df-rp 9394  df-fl 9994  df-mod 10047  df-seqfrec 10170  df-exp 10244  df-dvds 11401
This theorem is referenced by:  pw2dvds  11750
  Copyright terms: Public domain W3C validator