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Theorem sadcadd 14315
Description: Non-recursive definition of the carry sequence. (Contributed by Mario Carneiro, 8-Sep-2016.)
Hypotheses
Ref Expression
sadval.a  |-  ( ph  ->  A  C_  NN0 )
sadval.b  |-  ( ph  ->  B  C_  NN0 )
sadval.c  |-  C  =  seq 0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  - 
1 ) ) ) )
sadcp1.n  |-  ( ph  ->  N  e.  NN0 )
sadcadd.k  |-  K  =  `' (bits  |`  NN0 )
Assertion
Ref Expression
sadcadd  |-  ( ph  ->  ( (/)  e.  ( C `  N )  <->  ( 2 ^ N )  <_  ( ( K `
 ( A  i^i  ( 0..^ N ) ) )  +  ( K `
 ( B  i^i  ( 0..^ N ) ) ) ) ) )
Distinct variable groups:    m, c, n    A, c, m    B, c, m    n, N
Allowed substitution hints:    ph( m, n, c)    A( n)    B( n)    C( m, n, c)    K( m, n, c)    N( m, c)

Proof of Theorem sadcadd
Dummy variables  k  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sadcp1.n . 2  |-  ( ph  ->  N  e.  NN0 )
2 fveq2 5848 . . . . . 6  |-  ( x  =  0  ->  ( C `  x )  =  ( C ` 
0 ) )
32eleq2d 2472 . . . . 5  |-  ( x  =  0  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  0 ) ) )
4 oveq2 6285 . . . . . . 7  |-  ( x  =  0  ->  (
2 ^ x )  =  ( 2 ^ 0 ) )
5 2cn 10646 . . . . . . . 8  |-  2  e.  CC
6 exp0 12212 . . . . . . . 8  |-  ( 2  e.  CC  ->  (
2 ^ 0 )  =  1 )
75, 6ax-mp 5 . . . . . . 7  |-  ( 2 ^ 0 )  =  1
84, 7syl6eq 2459 . . . . . 6  |-  ( x  =  0  ->  (
2 ^ x )  =  1 )
9 oveq2 6285 . . . . . . . . . . . . 13  |-  ( x  =  0  ->  (
0..^ x )  =  ( 0..^ 0 ) )
10 fzo0 11879 . . . . . . . . . . . . 13  |-  ( 0..^ 0 )  =  (/)
119, 10syl6eq 2459 . . . . . . . . . . . 12  |-  ( x  =  0  ->  (
0..^ x )  =  (/) )
1211ineq2d 3640 . . . . . . . . . . 11  |-  ( x  =  0  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  (/) ) )
13 in0 3764 . . . . . . . . . . 11  |-  ( A  i^i  (/) )  =  (/)
1412, 13syl6eq 2459 . . . . . . . . . 10  |-  ( x  =  0  ->  ( A  i^i  ( 0..^ x ) )  =  (/) )
1514fveq2d 5852 . . . . . . . . 9  |-  ( x  =  0  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  (/) ) )
16 sadcadd.k . . . . . . . . . . 11  |-  K  =  `' (bits  |`  NN0 )
17 0nn0 10850 . . . . . . . . . . . . 13  |-  0  e.  NN0
18 fvres 5862 . . . . . . . . . . . . 13  |-  ( 0  e.  NN0  ->  ( (bits  |`  NN0 ) `  0
)  =  (bits ` 
0 ) )
1917, 18ax-mp 5 . . . . . . . . . . . 12  |-  ( (bits  |`  NN0 ) `  0
)  =  (bits ` 
0 )
20 0bits 14296 . . . . . . . . . . . 12  |-  (bits ` 
0 )  =  (/)
2119, 20eqtr2i 2432 . . . . . . . . . . 11  |-  (/)  =  ( (bits  |`  NN0 ) ` 
0 )
2216, 21fveq12i 5853 . . . . . . . . . 10  |-  ( K `
 (/) )  =  ( `' (bits  |`  NN0 ) `  ( (bits  |`  NN0 ) `  0 ) )
23 bitsf1o 14302 . . . . . . . . . . 11  |-  (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )
24 f1ocnvfv1 6162 . . . . . . . . . . 11  |-  ( ( (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )  /\  0  e.  NN0 )  ->  ( `' (bits  |`  NN0 ) `  (
(bits  |`  NN0 ) ` 
0 ) )  =  0 )
2523, 17, 24mp2an 670 . . . . . . . . . 10  |-  ( `' (bits  |`  NN0 ) `  ( (bits  |`  NN0 ) `  0 ) )  =  0
2622, 25eqtri 2431 . . . . . . . . 9  |-  ( K `
 (/) )  =  0
2715, 26syl6eq 2459 . . . . . . . 8  |-  ( x  =  0  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  0 )
2811ineq2d 3640 . . . . . . . . . . 11  |-  ( x  =  0  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  (/) ) )
29 in0 3764 . . . . . . . . . . 11  |-  ( B  i^i  (/) )  =  (/)
3028, 29syl6eq 2459 . . . . . . . . . 10  |-  ( x  =  0  ->  ( B  i^i  ( 0..^ x ) )  =  (/) )
3130fveq2d 5852 . . . . . . . . 9  |-  ( x  =  0  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  (/) ) )
3231, 26syl6eq 2459 . . . . . . . 8  |-  ( x  =  0  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  0 )
3327, 32oveq12d 6295 . . . . . . 7  |-  ( x  =  0  ->  (
( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  =  ( 0  +  0 ) )
34 00id 9788 . . . . . . 7  |-  ( 0  +  0 )  =  0
3533, 34syl6eq 2459 . . . . . 6  |-  ( x  =  0  ->  (
( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  =  0 )
368, 35breq12d 4407 . . . . 5  |-  ( x  =  0  ->  (
( 2 ^ x
)  <_  ( ( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  <->  1  <_  0 ) )
373, 36bibi12d 319 . . . 4  |-  ( x  =  0  ->  (
( (/)  e.  ( C `
 x )  <->  ( 2 ^ x )  <_ 
( ( K `  ( A  i^i  (
0..^ x ) ) )  +  ( K `
 ( B  i^i  ( 0..^ x ) ) ) ) )  <->  ( (/)  e.  ( C `  0 )  <->  1  <_  0 ) ) )
3837imbi2d 314 . . 3  |-  ( x  =  0  ->  (
( ph  ->  ( (/)  e.  ( C `  x
)  <->  ( 2 ^ x )  <_  (
( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) ) ) )  <->  ( ph  ->  ( (/)  e.  ( C `  0 )  <->  1  <_  0 ) ) ) )
39 fveq2 5848 . . . . . 6  |-  ( x  =  k  ->  ( C `  x )  =  ( C `  k ) )
4039eleq2d 2472 . . . . 5  |-  ( x  =  k  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  k ) ) )
41 oveq2 6285 . . . . . 6  |-  ( x  =  k  ->  (
2 ^ x )  =  ( 2 ^ k ) )
42 oveq2 6285 . . . . . . . . 9  |-  ( x  =  k  ->  (
0..^ x )  =  ( 0..^ k ) )
4342ineq2d 3640 . . . . . . . 8  |-  ( x  =  k  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ k ) ) )
4443fveq2d 5852 . . . . . . 7  |-  ( x  =  k  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ k ) ) ) )
4542ineq2d 3640 . . . . . . . 8  |-  ( x  =  k  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ k ) ) )
4645fveq2d 5852 . . . . . . 7  |-  ( x  =  k  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ k ) ) ) )
4744, 46oveq12d 6295 . . . . . 6  |-  ( x  =  k  ->  (
( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  =  ( ( K `
 ( A  i^i  ( 0..^ k ) ) )  +  ( K `
 ( B  i^i  ( 0..^ k ) ) ) ) )
4841, 47breq12d 4407 . . . . 5  |-  ( x  =  k  ->  (
( 2 ^ x
)  <_  ( ( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  <-> 
( 2 ^ k
)  <_  ( ( K `  ( A  i^i  ( 0..^ k ) ) )  +  ( K `  ( B  i^i  ( 0..^ k ) ) ) ) ) )
4940, 48bibi12d 319 . . . 4  |-  ( x  =  k  ->  (
( (/)  e.  ( C `
 x )  <->  ( 2 ^ x )  <_ 
( ( K `  ( A  i^i  (
0..^ x ) ) )  +  ( K `
 ( B  i^i  ( 0..^ x ) ) ) ) )  <->  ( (/)  e.  ( C `  k )  <-> 
( 2 ^ k
)  <_  ( ( K `  ( A  i^i  ( 0..^ k ) ) )  +  ( K `  ( B  i^i  ( 0..^ k ) ) ) ) ) ) )
5049imbi2d 314 . . 3  |-  ( x  =  k  ->  (
( ph  ->  ( (/)  e.  ( C `  x
)  <->  ( 2 ^ x )  <_  (
( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) ) ) )  <->  ( ph  ->  ( (/)  e.  ( C `  k )  <->  ( 2 ^ k )  <_  ( ( K `
 ( A  i^i  ( 0..^ k ) ) )  +  ( K `
 ( B  i^i  ( 0..^ k ) ) ) ) ) ) ) )
51 fveq2 5848 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( C `  x )  =  ( C `  ( k  +  1 ) ) )
5251eleq2d 2472 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  ( k  +  1 ) ) ) )
53 oveq2 6285 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  (
2 ^ x )  =  ( 2 ^ ( k  +  1 ) ) )
54 oveq2 6285 . . . . . . . . 9  |-  ( x  =  ( k  +  1 )  ->  (
0..^ x )  =  ( 0..^ ( k  +  1 ) ) )
5554ineq2d 3640 . . . . . . . 8  |-  ( x  =  ( k  +  1 )  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) )
5655fveq2d 5852 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) ) )
5754ineq2d 3640 . . . . . . . 8  |-  ( x  =  ( k  +  1 )  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) )
5857fveq2d 5852 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) ) )
5956, 58oveq12d 6295 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  (
( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  =  ( ( K `
 ( A  i^i  ( 0..^ ( k  +  1 ) ) ) )  +  ( K `
 ( B  i^i  ( 0..^ ( k  +  1 ) ) ) ) ) )
6053, 59breq12d 4407 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  (
( 2 ^ x
)  <_  ( ( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  <-> 
( 2 ^ (
k  +  1 ) )  <_  ( ( K `  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) )  +  ( K `  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) ) ) ) )
6152, 60bibi12d 319 . . . 4  |-  ( x  =  ( k  +  1 )  ->  (
( (/)  e.  ( C `
 x )  <->  ( 2 ^ x )  <_ 
( ( K `  ( A  i^i  (
0..^ x ) ) )  +  ( K `
 ( B  i^i  ( 0..^ x ) ) ) ) )  <->  ( (/)  e.  ( C `  ( k  +  1 ) )  <-> 
( 2 ^ (
k  +  1 ) )  <_  ( ( K `  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) )  +  ( K `  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) ) ) ) ) )
6261imbi2d 314 . . 3  |-  ( x  =  ( k  +  1 )  ->  (
( ph  ->  ( (/)  e.  ( C `  x
)  <->  ( 2 ^ x )  <_  (
( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) ) ) )  <->  ( ph  ->  ( (/)  e.  ( C `  ( k  +  1 ) )  <-> 
( 2 ^ (
k  +  1 ) )  <_  ( ( K `  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) )  +  ( K `  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) ) ) ) ) ) )
63 fveq2 5848 . . . . . 6  |-  ( x  =  N  ->  ( C `  x )  =  ( C `  N ) )
6463eleq2d 2472 . . . . 5  |-  ( x  =  N  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  N ) ) )
65 oveq2 6285 . . . . . 6  |-  ( x  =  N  ->  (
2 ^ x )  =  ( 2 ^ N ) )
66 oveq2 6285 . . . . . . . . 9  |-  ( x  =  N  ->  (
0..^ x )  =  ( 0..^ N ) )
6766ineq2d 3640 . . . . . . . 8  |-  ( x  =  N  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ N ) ) )
6867fveq2d 5852 . . . . . . 7  |-  ( x  =  N  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ N ) ) ) )
6966ineq2d 3640 . . . . . . . 8  |-  ( x  =  N  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ N ) ) )
7069fveq2d 5852 . . . . . . 7  |-  ( x  =  N  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ N ) ) ) )
7168, 70oveq12d 6295 . . . . . 6  |-  ( x  =  N  ->  (
( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  =  ( ( K `
 ( A  i^i  ( 0..^ N ) ) )  +  ( K `
 ( B  i^i  ( 0..^ N ) ) ) ) )
7265, 71breq12d 4407 . . . . 5  |-  ( x  =  N  ->  (
( 2 ^ x
)  <_  ( ( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  <-> 
( 2 ^ N
)  <_  ( ( K `  ( A  i^i  ( 0..^ N ) ) )  +  ( K `  ( B  i^i  ( 0..^ N ) ) ) ) ) )
7364, 72bibi12d 319 . . . 4  |-  ( x  =  N  ->  (
( (/)  e.  ( C `
 x )  <->  ( 2 ^ x )  <_ 
( ( K `  ( A  i^i  (
0..^ x ) ) )  +  ( K `
 ( B  i^i  ( 0..^ x ) ) ) ) )  <->  ( (/)  e.  ( C `  N )  <-> 
( 2 ^ N
)  <_  ( ( K `  ( A  i^i  ( 0..^ N ) ) )  +  ( K `  ( B  i^i  ( 0..^ N ) ) ) ) ) ) )
7473imbi2d 314 . . 3  |-  ( x  =  N  ->  (
( ph  ->  ( (/)  e.  ( C `  x
)  <->  ( 2 ^ x )  <_  (
( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) ) ) )  <->  ( ph  ->  ( (/)  e.  ( C `  N )  <->  ( 2 ^ N )  <_  ( ( K `
 ( A  i^i  ( 0..^ N ) ) )  +  ( K `
 ( B  i^i  ( 0..^ N ) ) ) ) ) ) ) )
75 sadval.a . . . . 5  |-  ( ph  ->  A  C_  NN0 )
76 sadval.b . . . . 5  |-  ( ph  ->  B  C_  NN0 )
77 sadval.c . . . . 5  |-  C  =  seq 0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  - 
1 ) ) ) )
7875, 76, 77sadc0 14311 . . . 4  |-  ( ph  ->  -.  (/)  e.  ( C `
 0 ) )
79 0lt1 10114 . . . . . 6  |-  0  <  1
80 0re 9625 . . . . . . 7  |-  0  e.  RR
81 1re 9624 . . . . . . 7  |-  1  e.  RR
8280, 81ltnlei 9736 . . . . . 6  |-  ( 0  <  1  <->  -.  1  <_  0 )
8379, 82mpbi 208 . . . . 5  |-  -.  1  <_  0
8483a1i 11 . . . 4  |-  ( ph  ->  -.  1  <_  0
)
8578, 842falsed 349 . . 3  |-  ( ph  ->  ( (/)  e.  ( C `  0 )  <->  1  <_  0 ) )
8675ad2antrr 724 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  ( (/) 
e.  ( C `  k )  <->  ( 2 ^ k )  <_ 
( ( K `  ( A  i^i  (
0..^ k ) ) )  +  ( K `
 ( B  i^i  ( 0..^ k ) ) ) ) ) )  ->  A  C_  NN0 )
8776ad2antrr 724 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  ( (/) 
e.  ( C `  k )  <->  ( 2 ^ k )  <_ 
( ( K `  ( A  i^i  (
0..^ k ) ) )  +  ( K `
 ( B  i^i  ( 0..^ k ) ) ) ) ) )  ->  B  C_  NN0 )
88 simplr 754 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  ( (/) 
e.  ( C `  k )  <->  ( 2 ^ k )  <_ 
( ( K `  ( A  i^i  (
0..^ k ) ) )  +  ( K `
 ( B  i^i  ( 0..^ k ) ) ) ) ) )  ->  k  e.  NN0 )
89 simpr 459 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  ( (/) 
e.  ( C `  k )  <->  ( 2 ^ k )  <_ 
( ( K `  ( A  i^i  (
0..^ k ) ) )  +  ( K `
 ( B  i^i  ( 0..^ k ) ) ) ) ) )  ->  ( (/)  e.  ( C `  k )  <-> 
( 2 ^ k
)  <_  ( ( K `  ( A  i^i  ( 0..^ k ) ) )  +  ( K `  ( B  i^i  ( 0..^ k ) ) ) ) ) )
9086, 87, 77, 88, 16, 89sadcaddlem 14314 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  ( (/) 
e.  ( C `  k )  <->  ( 2 ^ k )  <_ 
( ( K `  ( A  i^i  (
0..^ k ) ) )  +  ( K `
 ( B  i^i  ( 0..^ k ) ) ) ) ) )  ->  ( (/)  e.  ( C `  ( k  +  1 ) )  <-> 
( 2 ^ (
k  +  1 ) )  <_  ( ( K `  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) )  +  ( K `  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) ) ) ) )
9190ex 432 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( (/) 
e.  ( C `  k )  <->  ( 2 ^ k )  <_ 
( ( K `  ( A  i^i  (
0..^ k ) ) )  +  ( K `
 ( B  i^i  ( 0..^ k ) ) ) ) )  -> 
( (/)  e.  ( C `
 ( k  +  1 ) )  <->  ( 2 ^ ( k  +  1 ) )  <_ 
( ( K `  ( A  i^i  (
0..^ ( k  +  1 ) ) ) )  +  ( K `
 ( B  i^i  ( 0..^ ( k  +  1 ) ) ) ) ) ) ) )
9291expcom 433 . . . 4  |-  ( k  e.  NN0  ->  ( ph  ->  ( ( (/)  e.  ( C `  k )  <-> 
( 2 ^ k
)  <_  ( ( K `  ( A  i^i  ( 0..^ k ) ) )  +  ( K `  ( B  i^i  ( 0..^ k ) ) ) ) )  ->  ( (/)  e.  ( C `  ( k  +  1 ) )  <-> 
( 2 ^ (
k  +  1 ) )  <_  ( ( K `  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) )  +  ( K `  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) ) ) ) ) ) )
9392a2d 26 . . 3  |-  ( k  e.  NN0  ->  ( (
ph  ->  ( (/)  e.  ( C `  k )  <-> 
( 2 ^ k
)  <_  ( ( K `  ( A  i^i  ( 0..^ k ) ) )  +  ( K `  ( B  i^i  ( 0..^ k ) ) ) ) ) )  ->  ( ph  ->  ( (/)  e.  ( C `  ( k  +  1 ) )  <-> 
( 2 ^ (
k  +  1 ) )  <_  ( ( K `  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) )  +  ( K `  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) ) ) ) ) ) )
9438, 50, 62, 74, 85, 93nn0ind 10997 . 2  |-  ( N  e.  NN0  ->  ( ph  ->  ( (/)  e.  ( C `  N )  <->  ( 2 ^ N )  <_  ( ( K `
 ( A  i^i  ( 0..^ N ) ) )  +  ( K `
 ( B  i^i  ( 0..^ N ) ) ) ) ) ) )
951, 94mpcom 34 1  |-  ( ph  ->  ( (/)  e.  ( C `  N )  <->  ( 2 ^ N )  <_  ( ( K `
 ( A  i^i  ( 0..^ N ) ) )  +  ( K `
 ( B  i^i  ( 0..^ N ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405  caddwcad 1460    e. wcel 1842    i^i cin 3412    C_ wss 3413   (/)c0 3737   ifcif 3884   ~Pcpw 3954   class class class wbr 4394    |-> cmpt 4452   `'ccnv 4821    |` cres 4824   -1-1-onto->wf1o 5567   ` cfv 5568  (class class class)co 6277    |-> cmpt2 6279   1oc1o 7159   2oc2o 7160   Fincfn 7553   CCcc 9519   0cc0 9521   1c1 9522    + caddc 9524    < clt 9657    <_ cle 9658    - cmin 9840   2c2 10625   NN0cn0 10835  ..^cfzo 11852    seqcseq 12149   ^cexp 12208  bitscbits 14276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-xor 1367  df-tru 1408  df-fal 1411  df-cad 1462  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-disj 4366  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-2o 7167  df-oadd 7170  df-er 7347  df-map 7458  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-sup 7934  df-oi 7968  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-n0 10836  df-z 10905  df-uz 11127  df-rp 11265  df-fz 11725  df-fzo 11853  df-fl 11964  df-mod 12033  df-seq 12150  df-exp 12209  df-hash 12451  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-clim 13458  df-sum 13656  df-dvds 14194  df-bits 14279
This theorem is referenced by: (None)
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