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Theorem pw2dvdslemn 11417
Description: Lemma for pw2dvds 11418. 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 941 . 2  |-  ( ( N  e.  NN  /\  A  e.  NN  /\  -.  ( 2 ^ A
)  ||  N )  ->  ( N  e.  NN  /\ 
-.  ( 2 ^ A )  ||  N
) )
2 oveq2 5660 . . . . . . . 8  |-  ( w  =  1  ->  (
2 ^ w )  =  ( 2 ^ 1 ) )
32breq1d 3855 . . . . . . 7  |-  ( w  =  1  ->  (
( 2 ^ w
)  ||  N  <->  ( 2 ^ 1 )  ||  N ) )
43notbid 627 . . . . . 6  |-  ( w  =  1  ->  ( -.  ( 2 ^ w
)  ||  N  <->  -.  (
2 ^ 1 ) 
||  N ) )
54anbi2d 452 . . . . 5  |-  ( w  =  1  ->  (
( N  e.  NN  /\ 
-.  ( 2 ^ w )  ||  N
)  <->  ( N  e.  NN  /\  -.  (
2 ^ 1 ) 
||  N ) ) )
65imbi1d 229 . . . 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 5660 . . . . . . . 8  |-  ( w  =  k  ->  (
2 ^ w )  =  ( 2 ^ k ) )
87breq1d 3855 . . . . . . 7  |-  ( w  =  k  ->  (
( 2 ^ w
)  ||  N  <->  ( 2 ^ k )  ||  N ) )
98notbid 627 . . . . . 6  |-  ( w  =  k  ->  ( -.  ( 2 ^ w
)  ||  N  <->  -.  (
2 ^ k ) 
||  N ) )
109anbi2d 452 . . . . 5  |-  ( w  =  k  ->  (
( N  e.  NN  /\ 
-.  ( 2 ^ w )  ||  N
)  <->  ( N  e.  NN  /\  -.  (
2 ^ k ) 
||  N ) ) )
1110imbi1d 229 . . . 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 5660 . . . . . . . 8  |-  ( w  =  ( k  +  1 )  ->  (
2 ^ w )  =  ( 2 ^ ( k  +  1 ) ) )
1312breq1d 3855 . . . . . . 7  |-  ( w  =  ( k  +  1 )  ->  (
( 2 ^ w
)  ||  N  <->  ( 2 ^ ( k  +  1 ) )  ||  N ) )
1413notbid 627 . . . . . 6  |-  ( w  =  ( k  +  1 )  ->  ( -.  ( 2 ^ w
)  ||  N  <->  -.  (
2 ^ ( k  +  1 ) ) 
||  N ) )
1514anbi2d 452 . . . . 5  |-  ( w  =  ( k  +  1 )  ->  (
( N  e.  NN  /\ 
-.  ( 2 ^ w )  ||  N
)  <->  ( N  e.  NN  /\  -.  (
2 ^ ( k  +  1 ) ) 
||  N ) ) )
1615imbi1d 229 . . . 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 5660 . . . . . . . 8  |-  ( w  =  A  ->  (
2 ^ w )  =  ( 2 ^ A ) )
1817breq1d 3855 . . . . . . 7  |-  ( w  =  A  ->  (
( 2 ^ w
)  ||  N  <->  ( 2 ^ A )  ||  N ) )
1918notbid 627 . . . . . 6  |-  ( w  =  A  ->  ( -.  ( 2 ^ w
)  ||  N  <->  -.  (
2 ^ A ) 
||  N ) )
2019anbi2d 452 . . . . 5  |-  ( w  =  A  ->  (
( N  e.  NN  /\ 
-.  ( 2 ^ w )  ||  N
)  <->  ( N  e.  NN  /\  -.  (
2 ^ A ) 
||  N ) ) )
2120imbi1d 229 . . . 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 8686 . . . . . 6  |-  0  e.  NN0
2322a1i 9 . . . . 5  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  0  e.  NN0 )
24 oveq2 5660 . . . . . . . 8  |-  ( m  =  0  ->  (
2 ^ m )  =  ( 2 ^ 0 ) )
2524breq1d 3855 . . . . . . 7  |-  ( m  =  0  ->  (
( 2 ^ m
)  ||  N  <->  ( 2 ^ 0 )  ||  N ) )
26 oveq1 5659 . . . . . . . . . 10  |-  ( m  =  0  ->  (
m  +  1 )  =  ( 0  +  1 ) )
2726oveq2d 5668 . . . . . . . . 9  |-  ( m  =  0  ->  (
2 ^ ( m  +  1 ) )  =  ( 2 ^ ( 0  +  1 ) ) )
2827breq1d 3855 . . . . . . . 8  |-  ( m  =  0  ->  (
( 2 ^ (
m  +  1 ) )  ||  N  <->  ( 2 ^ ( 0  +  1 ) )  ||  N ) )
2928notbid 627 . . . . . . 7  |-  ( m  =  0  ->  ( -.  ( 2 ^ (
m  +  1 ) )  ||  N  <->  -.  (
2 ^ ( 0  +  1 ) ) 
||  N ) )
3025, 29anbi12d 457 . . . . . 6  |-  ( m  =  0  ->  (
( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
)  <->  ( ( 2 ^ 0 )  ||  N  /\  -.  ( 2 ^ ( 0  +  1 ) )  ||  N ) ) )
3130adantl 271 . . . . 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 8493 . . . . . . . 8  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  2  e.  CC )
3332exp0d 10076 . . . . . . 7  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  ( 2 ^ 0 )  =  1 )
34 simpl 107 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  N  e.  NN )
3534nnzd 8865 . . . . . . . 8  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  N  e.  ZZ )
36 1dvds 11084 . . . . . . . 8  |-  ( N  e.  ZZ  ->  1  ||  N )
3735, 36syl 14 . . . . . . 7  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  1  ||  N
)
3833, 37eqbrtrd 3865 . . . . . 6  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  ( 2 ^ 0 )  ||  N
)
39 simpr 108 . . . . . . 7  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  -.  ( 2 ^ 1 )  ||  N )
40 0p1e1 8534 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
4140oveq2i 5663 . . . . . . . 8  |-  ( 2 ^ ( 0  +  1 ) )  =  ( 2 ^ 1 )
4241breq1i 3852 . . . . . . 7  |-  ( ( 2 ^ ( 0  +  1 ) ) 
||  N  <->  ( 2 ^ 1 )  ||  N )
4339, 42sylnibr 637 . . . . . 6  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  -.  ( 2 ^ ( 0  +  1 ) )  ||  N )
4438, 43jca 300 . . . . 5  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  ( ( 2 ^ 0 )  ||  N  /\  -.  ( 2 ^ ( 0  +  1 ) )  ||  N ) )
4523, 31, 44rspcedvd 2728 . . . 4  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) )
46 simpll 496 . . . . . . . . 9  |-  ( ( ( k  e.  NN  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  -> 
k  e.  NN )
4746nnnn0d 8724 . . . . . . . 8  |-  ( ( ( k  e.  NN  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  -> 
k  e.  NN0 )
48 oveq2 5660 . . . . . . . . . . 11  |-  ( m  =  k  ->  (
2 ^ m )  =  ( 2 ^ k ) )
4948breq1d 3855 . . . . . . . . . 10  |-  ( m  =  k  ->  (
( 2 ^ m
)  ||  N  <->  ( 2 ^ k )  ||  N ) )
50 oveq1 5659 . . . . . . . . . . . . 13  |-  ( m  =  k  ->  (
m  +  1 )  =  ( k  +  1 ) )
5150oveq2d 5668 . . . . . . . . . . . 12  |-  ( m  =  k  ->  (
2 ^ ( m  +  1 ) )  =  ( 2 ^ ( k  +  1 ) ) )
5251breq1d 3855 . . . . . . . . . . 11  |-  ( m  =  k  ->  (
( 2 ^ (
m  +  1 ) )  ||  N  <->  ( 2 ^ ( k  +  1 ) )  ||  N ) )
5352notbid 627 . . . . . . . . . 10  |-  ( m  =  k  ->  ( -.  ( 2 ^ (
m  +  1 ) )  ||  N  <->  -.  (
2 ^ ( k  +  1 ) ) 
||  N ) )
5449, 53anbi12d 457 . . . . . . . . 9  |-  ( m  =  k  ->  (
( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
)  <->  ( ( 2 ^ k )  ||  N  /\  -.  ( 2 ^ ( k  +  1 ) )  ||  N ) ) )
5554adantl 271 . . . . . . . 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 108 . . . . . . . . 9  |-  ( ( ( k  e.  NN  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  -> 
( 2 ^ k
)  ||  N )
57 simplrr 503 . . . . . . . . 9  |-  ( ( ( k  e.  NN  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  ->  -.  ( 2 ^ (
k  +  1 ) )  ||  N )
5856, 57jca 300 . . . . . . . 8  |-  ( ( ( k  e.  NN  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  -> 
( ( 2 ^ k )  ||  N  /\  -.  ( 2 ^ ( k  +  1 ) )  ||  N
) )
5947, 55, 58rspcedvd 2728 . . . . . . 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 465 . . . . . 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 498 . . . . . . . 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 333 . . . . . . 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 501 . . . . . . 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 8575 . . . . . . . . 9  |-  2  e.  NN
66 simpll 496 . . . . . . . . . 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 8724 . . . . . . . . 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 9964 . . . . . . . . 9  |-  ( ( 2  e.  NN  /\  k  e.  NN0 )  -> 
( 2 ^ k
)  e.  NN )
6965, 67, 68sylancr 405 . . . . . . . 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 8865 . . . . . . . 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 11078 . . . . . . . 8  |-  ( ( ( 2 ^ k
)  e.  NN  /\  N  e.  ZZ )  -> DECID  ( 2 ^ k ) 
||  N )
7269, 70, 71syl2anc 403 . . . . . . 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 782 . . . . . . 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 747 . . . . 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 356 . . . 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 8436 . . 3  |-  ( A  e.  NN  ->  (
( N  e.  NN  /\ 
-.  ( 2 ^ A )  ||  N
)  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  + 
1 ) )  ||  N ) ) )
78773ad2ant2 965 . 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 102    <-> wb 103    \/ wo 664  DECID wdc 780    /\ w3a 924    = wceq 1289    e. wcel 1438   E.wrex 2360   class class class wbr 3845  (class class class)co 5652   0cc0 7348   1c1 7349    + caddc 7351   NNcn 8420   2c2 8471   NN0cn0 8671   ZZcz 8748   ^cexp 9950    || cdvds 11070
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-mulrcl 7442  ax-addcom 7443  ax-mulcom 7444  ax-addass 7445  ax-mulass 7446  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-1rid 7450  ax-0id 7451  ax-rnegex 7452  ax-precex 7453  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-apti 7458  ax-pre-ltadd 7459  ax-pre-mulgt0 7460  ax-pre-mulext 7461  ax-arch 7462
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-reap 8050  df-ap 8057  df-div 8138  df-inn 8421  df-2 8479  df-n0 8672  df-z 8749  df-uz 9018  df-q 9103  df-rp 9133  df-fl 9673  df-mod 9726  df-iseq 9849  df-seq3 9850  df-exp 9951  df-dvds 11071
This theorem is referenced by:  pw2dvds  11418
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