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Theorem sadcadd 12962
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 5720 . . . . . 6  |-  ( x  =  0  ->  ( C `  x )  =  ( C ` 
0 ) )
32eleq2d 2502 . . . . 5  |-  ( x  =  0  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  0 ) ) )
4 oveq2 6081 . . . . . . 7  |-  ( x  =  0  ->  (
2 ^ x )  =  ( 2 ^ 0 ) )
5 2cn 10062 . . . . . . . 8  |-  2  e.  CC
6 exp0 11378 . . . . . . . 8  |-  ( 2  e.  CC  ->  (
2 ^ 0 )  =  1 )
75, 6ax-mp 8 . . . . . . 7  |-  ( 2 ^ 0 )  =  1
84, 7syl6eq 2483 . . . . . 6  |-  ( x  =  0  ->  (
2 ^ x )  =  1 )
9 oveq2 6081 . . . . . . . . . . . . 13  |-  ( x  =  0  ->  (
0..^ x )  =  ( 0..^ 0 ) )
10 fzo0 11151 . . . . . . . . . . . . 13  |-  ( 0..^ 0 )  =  (/)
119, 10syl6eq 2483 . . . . . . . . . . . 12  |-  ( x  =  0  ->  (
0..^ x )  =  (/) )
1211ineq2d 3534 . . . . . . . . . . 11  |-  ( x  =  0  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  (/) ) )
13 in0 3645 . . . . . . . . . . 11  |-  ( A  i^i  (/) )  =  (/)
1412, 13syl6eq 2483 . . . . . . . . . 10  |-  ( x  =  0  ->  ( A  i^i  ( 0..^ x ) )  =  (/) )
1514fveq2d 5724 . . . . . . . . 9  |-  ( x  =  0  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  (/) ) )
16 sadcadd.k . . . . . . . . . . 11  |-  K  =  `' (bits  |`  NN0 )
17 0nn0 10228 . . . . . . . . . . . . 13  |-  0  e.  NN0
18 fvres 5737 . . . . . . . . . . . . 13  |-  ( 0  e.  NN0  ->  ( (bits  |`  NN0 ) `  0
)  =  (bits ` 
0 ) )
1917, 18ax-mp 8 . . . . . . . . . . . 12  |-  ( (bits  |`  NN0 ) `  0
)  =  (bits ` 
0 )
20 0bits 12943 . . . . . . . . . . . 12  |-  (bits ` 
0 )  =  (/)
2119, 20eqtr2i 2456 . . . . . . . . . . 11  |-  (/)  =  ( (bits  |`  NN0 ) ` 
0 )
2216, 21fveq12i 5725 . . . . . . . . . 10  |-  ( K `
 (/) )  =  ( `' (bits  |`  NN0 ) `  ( (bits  |`  NN0 ) `  0 ) )
23 bitsf1o 12949 . . . . . . . . . . 11  |-  (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )
24 f1ocnvfv1 6006 . . . . . . . . . . 11  |-  ( ( (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )  /\  0  e.  NN0 )  ->  ( `' (bits  |`  NN0 ) `  (
(bits  |`  NN0 ) ` 
0 ) )  =  0 )
2523, 17, 24mp2an 654 . . . . . . . . . 10  |-  ( `' (bits  |`  NN0 ) `  ( (bits  |`  NN0 ) `  0 ) )  =  0
2622, 25eqtri 2455 . . . . . . . . 9  |-  ( K `
 (/) )  =  0
2715, 26syl6eq 2483 . . . . . . . 8  |-  ( x  =  0  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  0 )
2811ineq2d 3534 . . . . . . . . . . 11  |-  ( x  =  0  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  (/) ) )
29 in0 3645 . . . . . . . . . . 11  |-  ( B  i^i  (/) )  =  (/)
3028, 29syl6eq 2483 . . . . . . . . . 10  |-  ( x  =  0  ->  ( B  i^i  ( 0..^ x ) )  =  (/) )
3130fveq2d 5724 . . . . . . . . 9  |-  ( x  =  0  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  (/) ) )
3231, 26syl6eq 2483 . . . . . . . 8  |-  ( x  =  0  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  0 )
3327, 32oveq12d 6091 . . . . . . 7  |-  ( x  =  0  ->  (
( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  =  ( 0  +  0 ) )
34 00id 9233 . . . . . . 7  |-  ( 0  +  0 )  =  0
3533, 34syl6eq 2483 . . . . . 6  |-  ( x  =  0  ->  (
( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  =  0 )
368, 35breq12d 4217 . . . . 5  |-  ( x  =  0  ->  (
( 2 ^ x
)  <_  ( ( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  <->  1  <_  0 ) )
373, 36bibi12d 313 . . . 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 308 . . 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 5720 . . . . . 6  |-  ( x  =  k  ->  ( C `  x )  =  ( C `  k ) )
4039eleq2d 2502 . . . . 5  |-  ( x  =  k  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  k ) ) )
41 oveq2 6081 . . . . . 6  |-  ( x  =  k  ->  (
2 ^ x )  =  ( 2 ^ k ) )
42 oveq2 6081 . . . . . . . . 9  |-  ( x  =  k  ->  (
0..^ x )  =  ( 0..^ k ) )
4342ineq2d 3534 . . . . . . . 8  |-  ( x  =  k  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ k ) ) )
4443fveq2d 5724 . . . . . . 7  |-  ( x  =  k  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ k ) ) ) )
4542ineq2d 3534 . . . . . . . 8  |-  ( x  =  k  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ k ) ) )
4645fveq2d 5724 . . . . . . 7  |-  ( x  =  k  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ k ) ) ) )
4744, 46oveq12d 6091 . . . . . 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 4217 . . . . 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 313 . . . 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 308 . . 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 5720 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( C `  x )  =  ( C `  ( k  +  1 ) ) )
5251eleq2d 2502 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  ( k  +  1 ) ) ) )
53 oveq2 6081 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  (
2 ^ x )  =  ( 2 ^ ( k  +  1 ) ) )
54 oveq2 6081 . . . . . . . . 9  |-  ( x  =  ( k  +  1 )  ->  (
0..^ x )  =  ( 0..^ ( k  +  1 ) ) )
5554ineq2d 3534 . . . . . . . 8  |-  ( x  =  ( k  +  1 )  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) )
5655fveq2d 5724 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) ) )
5754ineq2d 3534 . . . . . . . 8  |-  ( x  =  ( k  +  1 )  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) )
5857fveq2d 5724 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) ) )
5956, 58oveq12d 6091 . . . . . 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 4217 . . . . 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 313 . . . 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 308 . . 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 5720 . . . . . 6  |-  ( x  =  N  ->  ( C `  x )  =  ( C `  N ) )
6463eleq2d 2502 . . . . 5  |-  ( x  =  N  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  N ) ) )
65 oveq2 6081 . . . . . 6  |-  ( x  =  N  ->  (
2 ^ x )  =  ( 2 ^ N ) )
66 oveq2 6081 . . . . . . . . 9  |-  ( x  =  N  ->  (
0..^ x )  =  ( 0..^ N ) )
6766ineq2d 3534 . . . . . . . 8  |-  ( x  =  N  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ N ) ) )
6867fveq2d 5724 . . . . . . 7  |-  ( x  =  N  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ N ) ) ) )
6966ineq2d 3534 . . . . . . . 8  |-  ( x  =  N  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ N ) ) )
7069fveq2d 5724 . . . . . . 7  |-  ( x  =  N  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ N ) ) ) )
7168, 70oveq12d 6091 . . . . . 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 4217 . . . . 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 313 . . . 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 308 . . 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 12958 . . . 4  |-  ( ph  ->  -.  (/)  e.  ( C `
 0 ) )
79 0lt1 9542 . . . . . 6  |-  0  <  1
80 0re 9083 . . . . . . 7  |-  0  e.  RR
81 1re 9082 . . . . . . 7  |-  1  e.  RR
8280, 81ltnlei 9186 . . . . . 6  |-  ( 0  <  1  <->  -.  1  <_  0 )
8379, 82mpbi 200 . . . . 5  |-  -.  1  <_  0
8483a1i 11 . . . 4  |-  ( ph  ->  -.  1  <_  0
)
8578, 842falsed 341 . . 3  |-  ( ph  ->  ( (/)  e.  ( C `  0 )  <->  1  <_  0 ) )
8675ad2antrr 707 . . . . . . 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 707 . . . . . . 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 732 . . . . . . 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 448 . . . . . . 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 12961 . . . . . 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 424 . . . . 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 425 . . . 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 24 . . 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 10358 . 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359  caddwcad 1388    = wceq 1652    e. wcel 1725    i^i cin 3311    C_ wss 3312   (/)c0 3620   ifcif 3731   ~Pcpw 3791   class class class wbr 4204    e. cmpt 4258   `'ccnv 4869    |` cres 4872   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   1oc1o 6709   2oc2o 6710   Fincfn 7101   CCcc 8980   0cc0 8982   1c1 8983    + caddc 8985    < clt 9112    <_ cle 9113    - cmin 9283   2c2 10041   NN0cn0 10213  ..^cfzo 11127    seq cseq 11315   ^cexp 11374  bitscbits 12923
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-xor 1314  df-tru 1328  df-cad 1390  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-disj 4175  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-fzo 11128  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472  df-dvds 12845  df-bits 12926
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