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Theorem sadcadd 12665
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 5541 . . . . . 6  |-  ( x  =  0  ->  ( C `  x )  =  ( C ` 
0 ) )
32eleq2d 2363 . . . . 5  |-  ( x  =  0  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  0 ) ) )
4 oveq2 5882 . . . . . . 7  |-  ( x  =  0  ->  (
2 ^ x )  =  ( 2 ^ 0 ) )
5 2cn 9832 . . . . . . . 8  |-  2  e.  CC
6 exp0 11124 . . . . . . . 8  |-  ( 2  e.  CC  ->  (
2 ^ 0 )  =  1 )
75, 6ax-mp 8 . . . . . . 7  |-  ( 2 ^ 0 )  =  1
84, 7syl6eq 2344 . . . . . 6  |-  ( x  =  0  ->  (
2 ^ x )  =  1 )
9 oveq2 5882 . . . . . . . . . . . . 13  |-  ( x  =  0  ->  (
0..^ x )  =  ( 0..^ 0 ) )
10 fzo0 10909 . . . . . . . . . . . . 13  |-  ( 0..^ 0 )  =  (/)
119, 10syl6eq 2344 . . . . . . . . . . . 12  |-  ( x  =  0  ->  (
0..^ x )  =  (/) )
1211ineq2d 3383 . . . . . . . . . . 11  |-  ( x  =  0  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  (/) ) )
13 in0 3493 . . . . . . . . . . 11  |-  ( A  i^i  (/) )  =  (/)
1412, 13syl6eq 2344 . . . . . . . . . 10  |-  ( x  =  0  ->  ( A  i^i  ( 0..^ x ) )  =  (/) )
1514fveq2d 5545 . . . . . . . . 9  |-  ( x  =  0  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  (/) ) )
16 sadcadd.k . . . . . . . . . . 11  |-  K  =  `' (bits  |`  NN0 )
17 0nn0 9996 . . . . . . . . . . . . 13  |-  0  e.  NN0
18 fvres 5558 . . . . . . . . . . . . 13  |-  ( 0  e.  NN0  ->  ( (bits  |`  NN0 ) `  0
)  =  (bits ` 
0 ) )
1917, 18ax-mp 8 . . . . . . . . . . . 12  |-  ( (bits  |`  NN0 ) `  0
)  =  (bits ` 
0 )
20 0bits 12646 . . . . . . . . . . . 12  |-  (bits ` 
0 )  =  (/)
2119, 20eqtr2i 2317 . . . . . . . . . . 11  |-  (/)  =  ( (bits  |`  NN0 ) ` 
0 )
2216, 21fveq12i 5546 . . . . . . . . . 10  |-  ( K `
 (/) )  =  ( `' (bits  |`  NN0 ) `  ( (bits  |`  NN0 ) `  0 ) )
23 bitsf1o 12652 . . . . . . . . . . 11  |-  (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )
24 f1ocnvfv1 5808 . . . . . . . . . . 11  |-  ( ( (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )  /\  0  e.  NN0 )  ->  ( `' (bits  |`  NN0 ) `  (
(bits  |`  NN0 ) ` 
0 ) )  =  0 )
2523, 17, 24mp2an 653 . . . . . . . . . 10  |-  ( `' (bits  |`  NN0 ) `  ( (bits  |`  NN0 ) `  0 ) )  =  0
2622, 25eqtri 2316 . . . . . . . . 9  |-  ( K `
 (/) )  =  0
2715, 26syl6eq 2344 . . . . . . . 8  |-  ( x  =  0  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  0 )
2811ineq2d 3383 . . . . . . . . . . 11  |-  ( x  =  0  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  (/) ) )
29 in0 3493 . . . . . . . . . . 11  |-  ( B  i^i  (/) )  =  (/)
3028, 29syl6eq 2344 . . . . . . . . . 10  |-  ( x  =  0  ->  ( B  i^i  ( 0..^ x ) )  =  (/) )
3130fveq2d 5545 . . . . . . . . 9  |-  ( x  =  0  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  (/) ) )
3231, 26syl6eq 2344 . . . . . . . 8  |-  ( x  =  0  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  0 )
3327, 32oveq12d 5892 . . . . . . 7  |-  ( x  =  0  ->  (
( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  =  ( 0  +  0 ) )
34 00id 9003 . . . . . . 7  |-  ( 0  +  0 )  =  0
3533, 34syl6eq 2344 . . . . . 6  |-  ( x  =  0  ->  (
( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  =  0 )
368, 35breq12d 4052 . . . . 5  |-  ( x  =  0  ->  (
( 2 ^ x
)  <_  ( ( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  <->  1  <_  0 ) )
373, 36bibi12d 312 . . . 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 307 . . 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 5541 . . . . . 6  |-  ( x  =  k  ->  ( C `  x )  =  ( C `  k ) )
4039eleq2d 2363 . . . . 5  |-  ( x  =  k  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  k ) ) )
41 oveq2 5882 . . . . . 6  |-  ( x  =  k  ->  (
2 ^ x )  =  ( 2 ^ k ) )
42 oveq2 5882 . . . . . . . . 9  |-  ( x  =  k  ->  (
0..^ x )  =  ( 0..^ k ) )
4342ineq2d 3383 . . . . . . . 8  |-  ( x  =  k  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ k ) ) )
4443fveq2d 5545 . . . . . . 7  |-  ( x  =  k  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ k ) ) ) )
4542ineq2d 3383 . . . . . . . 8  |-  ( x  =  k  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ k ) ) )
4645fveq2d 5545 . . . . . . 7  |-  ( x  =  k  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ k ) ) ) )
4744, 46oveq12d 5892 . . . . . 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 4052 . . . . 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 312 . . . 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 307 . . 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 5541 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( C `  x )  =  ( C `  ( k  +  1 ) ) )
5251eleq2d 2363 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  ( k  +  1 ) ) ) )
53 oveq2 5882 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  (
2 ^ x )  =  ( 2 ^ ( k  +  1 ) ) )
54 oveq2 5882 . . . . . . . . 9  |-  ( x  =  ( k  +  1 )  ->  (
0..^ x )  =  ( 0..^ ( k  +  1 ) ) )
5554ineq2d 3383 . . . . . . . 8  |-  ( x  =  ( k  +  1 )  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) )
5655fveq2d 5545 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) ) )
5754ineq2d 3383 . . . . . . . 8  |-  ( x  =  ( k  +  1 )  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) )
5857fveq2d 5545 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) ) )
5956, 58oveq12d 5892 . . . . . 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 4052 . . . . 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 312 . . . 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 307 . . 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 5541 . . . . . 6  |-  ( x  =  N  ->  ( C `  x )  =  ( C `  N ) )
6463eleq2d 2363 . . . . 5  |-  ( x  =  N  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  N ) ) )
65 oveq2 5882 . . . . . 6  |-  ( x  =  N  ->  (
2 ^ x )  =  ( 2 ^ N ) )
66 oveq2 5882 . . . . . . . . 9  |-  ( x  =  N  ->  (
0..^ x )  =  ( 0..^ N ) )
6766ineq2d 3383 . . . . . . . 8  |-  ( x  =  N  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ N ) ) )
6867fveq2d 5545 . . . . . . 7  |-  ( x  =  N  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ N ) ) ) )
6966ineq2d 3383 . . . . . . . 8  |-  ( x  =  N  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ N ) ) )
7069fveq2d 5545 . . . . . . 7  |-  ( x  =  N  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ N ) ) ) )
7168, 70oveq12d 5892 . . . . . 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 4052 . . . . 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 312 . . . 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 307 . . 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 12661 . . . 4  |-  ( ph  ->  -.  (/)  e.  ( C `
 0 ) )
79 0lt1 9312 . . . . . 6  |-  0  <  1
80 0re 8854 . . . . . . 7  |-  0  e.  RR
81 1re 8853 . . . . . . 7  |-  1  e.  RR
8280, 81ltnlei 8955 . . . . . 6  |-  ( 0  <  1  <->  -.  1  <_  0 )
8379, 82mpbi 199 . . . . 5  |-  -.  1  <_  0
8483a1i 10 . . . 4  |-  ( ph  ->  -.  1  <_  0
)
8578, 842falsed 340 . . 3  |-  ( ph  ->  ( (/)  e.  ( C `  0 )  <->  1  <_  0 ) )
8675ad2antrr 706 . . . . . . 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 706 . . . . . . 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 731 . . . . . . 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 447 . . . . . . 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 12664 . . . . . 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 423 . . . . 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 424 . . . 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 23 . . 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 10124 . 2  |-  ( N  e.  NN0  ->  ( ph  ->  ( (/)  e.  ( C `  N )  <->  ( 2 ^ N )  <_  ( ( K `
 ( A  i^i  ( 0..^ N ) ) )  +  ( K `
 ( B  i^i  ( 0..^ N ) ) ) ) ) ) )
951, 94mpcom 32 1  |-  ( ph  ->  ( (/)  e.  ( C `  N )  <->  ( 2 ^ N )  <_  ( ( K `
 ( A  i^i  ( 0..^ N ) ) )  +  ( K `
 ( B  i^i  ( 0..^ N ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358  caddwcad 1369    = wceq 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   (/)c0 3468   ifcif 3578   ~Pcpw 3638   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704    |` cres 4707   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1oc1o 6488   2oc2o 6489   Fincfn 6879   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    < clt 8883    <_ cle 8884    - cmin 9053   2c2 9811   NN0cn0 9981  ..^cfzo 10886    seq cseq 11062   ^cexp 11120  bitscbits 12626
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-xor 1296  df-tru 1310  df-cad 1371  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-disj 4010  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-dvds 12548  df-bits 12629
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