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Theorem pw2dvdslemn 12148
Description: Lemma for pw2dvds 12149. 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 995 . 2  |-  ( ( N  e.  NN  /\  A  e.  NN  /\  -.  ( 2 ^ A
)  ||  N )  ->  ( N  e.  NN  /\ 
-.  ( 2 ^ A )  ||  N
) )
2 oveq2 5877 . . . . . . . 8  |-  ( w  =  1  ->  (
2 ^ w )  =  ( 2 ^ 1 ) )
32breq1d 4010 . . . . . . 7  |-  ( w  =  1  ->  (
( 2 ^ w
)  ||  N  <->  ( 2 ^ 1 )  ||  N ) )
43notbid 667 . . . . . 6  |-  ( w  =  1  ->  ( -.  ( 2 ^ w
)  ||  N  <->  -.  (
2 ^ 1 ) 
||  N ) )
54anbi2d 464 . . . . 5  |-  ( w  =  1  ->  (
( N  e.  NN  /\ 
-.  ( 2 ^ w )  ||  N
)  <->  ( N  e.  NN  /\  -.  (
2 ^ 1 ) 
||  N ) ) )
65imbi1d 231 . . . 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 5877 . . . . . . . 8  |-  ( w  =  k  ->  (
2 ^ w )  =  ( 2 ^ k ) )
87breq1d 4010 . . . . . . 7  |-  ( w  =  k  ->  (
( 2 ^ w
)  ||  N  <->  ( 2 ^ k )  ||  N ) )
98notbid 667 . . . . . 6  |-  ( w  =  k  ->  ( -.  ( 2 ^ w
)  ||  N  <->  -.  (
2 ^ k ) 
||  N ) )
109anbi2d 464 . . . . 5  |-  ( w  =  k  ->  (
( N  e.  NN  /\ 
-.  ( 2 ^ w )  ||  N
)  <->  ( N  e.  NN  /\  -.  (
2 ^ k ) 
||  N ) ) )
1110imbi1d 231 . . . 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 5877 . . . . . . . 8  |-  ( w  =  ( k  +  1 )  ->  (
2 ^ w )  =  ( 2 ^ ( k  +  1 ) ) )
1312breq1d 4010 . . . . . . 7  |-  ( w  =  ( k  +  1 )  ->  (
( 2 ^ w
)  ||  N  <->  ( 2 ^ ( k  +  1 ) )  ||  N ) )
1413notbid 667 . . . . . 6  |-  ( w  =  ( k  +  1 )  ->  ( -.  ( 2 ^ w
)  ||  N  <->  -.  (
2 ^ ( k  +  1 ) ) 
||  N ) )
1514anbi2d 464 . . . . 5  |-  ( w  =  ( k  +  1 )  ->  (
( N  e.  NN  /\ 
-.  ( 2 ^ w )  ||  N
)  <->  ( N  e.  NN  /\  -.  (
2 ^ ( k  +  1 ) ) 
||  N ) ) )
1615imbi1d 231 . . . 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 5877 . . . . . . . 8  |-  ( w  =  A  ->  (
2 ^ w )  =  ( 2 ^ A ) )
1817breq1d 4010 . . . . . . 7  |-  ( w  =  A  ->  (
( 2 ^ w
)  ||  N  <->  ( 2 ^ A )  ||  N ) )
1918notbid 667 . . . . . 6  |-  ( w  =  A  ->  ( -.  ( 2 ^ w
)  ||  N  <->  -.  (
2 ^ A ) 
||  N ) )
2019anbi2d 464 . . . . 5  |-  ( w  =  A  ->  (
( N  e.  NN  /\ 
-.  ( 2 ^ w )  ||  N
)  <->  ( N  e.  NN  /\  -.  (
2 ^ A ) 
||  N ) ) )
2120imbi1d 231 . . . 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 9180 . . . . . 6  |-  0  e.  NN0
2322a1i 9 . . . . 5  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  0  e.  NN0 )
24 oveq2 5877 . . . . . . . 8  |-  ( m  =  0  ->  (
2 ^ m )  =  ( 2 ^ 0 ) )
2524breq1d 4010 . . . . . . 7  |-  ( m  =  0  ->  (
( 2 ^ m
)  ||  N  <->  ( 2 ^ 0 )  ||  N ) )
26 oveq1 5876 . . . . . . . . . 10  |-  ( m  =  0  ->  (
m  +  1 )  =  ( 0  +  1 ) )
2726oveq2d 5885 . . . . . . . . 9  |-  ( m  =  0  ->  (
2 ^ ( m  +  1 ) )  =  ( 2 ^ ( 0  +  1 ) ) )
2827breq1d 4010 . . . . . . . 8  |-  ( m  =  0  ->  (
( 2 ^ (
m  +  1 ) )  ||  N  <->  ( 2 ^ ( 0  +  1 ) )  ||  N ) )
2928notbid 667 . . . . . . 7  |-  ( m  =  0  ->  ( -.  ( 2 ^ (
m  +  1 ) )  ||  N  <->  -.  (
2 ^ ( 0  +  1 ) ) 
||  N ) )
3025, 29anbi12d 473 . . . . . 6  |-  ( m  =  0  ->  (
( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
)  <->  ( ( 2 ^ 0 )  ||  N  /\  -.  ( 2 ^ ( 0  +  1 ) )  ||  N ) ) )
3130adantl 277 . . . . 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 8981 . . . . . . . 8  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  2  e.  CC )
3332exp0d 10633 . . . . . . 7  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  ( 2 ^ 0 )  =  1 )
34 simpl 109 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  N  e.  NN )
3534nnzd 9363 . . . . . . . 8  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  N  e.  ZZ )
36 1dvds 11796 . . . . . . . 8  |-  ( N  e.  ZZ  ->  1  ||  N )
3735, 36syl 14 . . . . . . 7  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  1  ||  N
)
3833, 37eqbrtrd 4022 . . . . . 6  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  ( 2 ^ 0 )  ||  N
)
39 simpr 110 . . . . . . 7  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  -.  ( 2 ^ 1 )  ||  N )
40 0p1e1 9022 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
4140oveq2i 5880 . . . . . . . 8  |-  ( 2 ^ ( 0  +  1 ) )  =  ( 2 ^ 1 )
4241breq1i 4007 . . . . . . 7  |-  ( ( 2 ^ ( 0  +  1 ) ) 
||  N  <->  ( 2 ^ 1 )  ||  N )
4339, 42sylnibr 677 . . . . . 6  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  -.  ( 2 ^ ( 0  +  1 ) )  ||  N )
4438, 43jca 306 . . . . 5  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  ( ( 2 ^ 0 )  ||  N  /\  -.  ( 2 ^ ( 0  +  1 ) )  ||  N ) )
4523, 31, 44rspcedvd 2847 . . . 4  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) )
46 simpll 527 . . . . . . . . 9  |-  ( ( ( k  e.  NN  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  -> 
k  e.  NN )
4746nnnn0d 9218 . . . . . . . 8  |-  ( ( ( k  e.  NN  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  -> 
k  e.  NN0 )
48 oveq2 5877 . . . . . . . . . . 11  |-  ( m  =  k  ->  (
2 ^ m )  =  ( 2 ^ k ) )
4948breq1d 4010 . . . . . . . . . 10  |-  ( m  =  k  ->  (
( 2 ^ m
)  ||  N  <->  ( 2 ^ k )  ||  N ) )
50 oveq1 5876 . . . . . . . . . . . . 13  |-  ( m  =  k  ->  (
m  +  1 )  =  ( k  +  1 ) )
5150oveq2d 5885 . . . . . . . . . . . 12  |-  ( m  =  k  ->  (
2 ^ ( m  +  1 ) )  =  ( 2 ^ ( k  +  1 ) ) )
5251breq1d 4010 . . . . . . . . . . 11  |-  ( m  =  k  ->  (
( 2 ^ (
m  +  1 ) )  ||  N  <->  ( 2 ^ ( k  +  1 ) )  ||  N ) )
5352notbid 667 . . . . . . . . . 10  |-  ( m  =  k  ->  ( -.  ( 2 ^ (
m  +  1 ) )  ||  N  <->  -.  (
2 ^ ( k  +  1 ) ) 
||  N ) )
5449, 53anbi12d 473 . . . . . . . . 9  |-  ( m  =  k  ->  (
( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
)  <->  ( ( 2 ^ k )  ||  N  /\  -.  ( 2 ^ ( k  +  1 ) )  ||  N ) ) )
5554adantl 277 . . . . . . . 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 110 . . . . . . . . 9  |-  ( ( ( k  e.  NN  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  -> 
( 2 ^ k
)  ||  N )
57 simplrr 536 . . . . . . . . 9  |-  ( ( ( k  e.  NN  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  ->  -.  ( 2 ^ (
k  +  1 ) )  ||  N )
5856, 57jca 306 . . . . . . . 8  |-  ( ( ( k  e.  NN  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  -> 
( ( 2 ^ k )  ||  N  /\  -.  ( 2 ^ ( k  +  1 ) )  ||  N
) )
5947, 55, 58rspcedvd 2847 . . . . . . 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 481 . . . . . 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 529 . . . . . . . 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 340 . . . . . . 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 534 . . . . . . 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 9069 . . . . . . . . 9  |-  2  e.  NN
66 simpll 527 . . . . . . . . . 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 9218 . . . . . . . . 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 10519 . . . . . . . . 9  |-  ( ( 2  e.  NN  /\  k  e.  NN0 )  -> 
( 2 ^ k
)  e.  NN )
6965, 67, 68sylancr 414 . . . . . . . 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 9363 . . . . . . . 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 11789 . . . . . . . 8  |-  ( ( ( 2 ^ k
)  e.  NN  /\  N  e.  ZZ )  -> DECID  ( 2 ^ k ) 
||  N )
7269, 70, 71syl2anc 411 . . . . . . 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 836 . . . . . . 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 798 . . . . 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 364 . . . 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 8924 . . 3  |-  ( A  e.  NN  ->  (
( N  e.  NN  /\ 
-.  ( 2 ^ A )  ||  N
)  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  + 
1 ) )  ||  N ) ) )
78773ad2ant2 1019 . 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 104    <-> wb 105    \/ wo 708  DECID wdc 834    /\ w3a 978    = wceq 1353    e. wcel 2148   E.wrex 2456   class class class wbr 4000  (class class class)co 5869   0cc0 7802   1c1 7803    + caddc 7805   NNcn 8908   2c2 8959   NN0cn0 9165   ZZcz 9242   ^cexp 10505    || cdvds 11778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-dvds 11779
This theorem is referenced by:  pw2dvds  12149
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