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Theorem sadcadd 13767
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 5794 . . . . . 6  |-  ( x  =  0  ->  ( C `  x )  =  ( C ` 
0 ) )
32eleq2d 2522 . . . . 5  |-  ( x  =  0  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  0 ) ) )
4 oveq2 6203 . . . . . . 7  |-  ( x  =  0  ->  (
2 ^ x )  =  ( 2 ^ 0 ) )
5 2cn 10498 . . . . . . . 8  |-  2  e.  CC
6 exp0 11981 . . . . . . . 8  |-  ( 2  e.  CC  ->  (
2 ^ 0 )  =  1 )
75, 6ax-mp 5 . . . . . . 7  |-  ( 2 ^ 0 )  =  1
84, 7syl6eq 2509 . . . . . 6  |-  ( x  =  0  ->  (
2 ^ x )  =  1 )
9 oveq2 6203 . . . . . . . . . . . . 13  |-  ( x  =  0  ->  (
0..^ x )  =  ( 0..^ 0 ) )
10 fzo0 11685 . . . . . . . . . . . . 13  |-  ( 0..^ 0 )  =  (/)
119, 10syl6eq 2509 . . . . . . . . . . . 12  |-  ( x  =  0  ->  (
0..^ x )  =  (/) )
1211ineq2d 3655 . . . . . . . . . . 11  |-  ( x  =  0  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  (/) ) )
13 in0 3766 . . . . . . . . . . 11  |-  ( A  i^i  (/) )  =  (/)
1412, 13syl6eq 2509 . . . . . . . . . 10  |-  ( x  =  0  ->  ( A  i^i  ( 0..^ x ) )  =  (/) )
1514fveq2d 5798 . . . . . . . . 9  |-  ( x  =  0  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  (/) ) )
16 sadcadd.k . . . . . . . . . . 11  |-  K  =  `' (bits  |`  NN0 )
17 0nn0 10700 . . . . . . . . . . . . 13  |-  0  e.  NN0
18 fvres 5808 . . . . . . . . . . . . 13  |-  ( 0  e.  NN0  ->  ( (bits  |`  NN0 ) `  0
)  =  (bits ` 
0 ) )
1917, 18ax-mp 5 . . . . . . . . . . . 12  |-  ( (bits  |`  NN0 ) `  0
)  =  (bits ` 
0 )
20 0bits 13748 . . . . . . . . . . . 12  |-  (bits ` 
0 )  =  (/)
2119, 20eqtr2i 2482 . . . . . . . . . . 11  |-  (/)  =  ( (bits  |`  NN0 ) ` 
0 )
2216, 21fveq12i 5799 . . . . . . . . . 10  |-  ( K `
 (/) )  =  ( `' (bits  |`  NN0 ) `  ( (bits  |`  NN0 ) `  0 ) )
23 bitsf1o 13754 . . . . . . . . . . 11  |-  (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )
24 f1ocnvfv1 6087 . . . . . . . . . . 11  |-  ( ( (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )  /\  0  e.  NN0 )  ->  ( `' (bits  |`  NN0 ) `  (
(bits  |`  NN0 ) ` 
0 ) )  =  0 )
2523, 17, 24mp2an 672 . . . . . . . . . 10  |-  ( `' (bits  |`  NN0 ) `  ( (bits  |`  NN0 ) `  0 ) )  =  0
2622, 25eqtri 2481 . . . . . . . . 9  |-  ( K `
 (/) )  =  0
2715, 26syl6eq 2509 . . . . . . . 8  |-  ( x  =  0  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  0 )
2811ineq2d 3655 . . . . . . . . . . 11  |-  ( x  =  0  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  (/) ) )
29 in0 3766 . . . . . . . . . . 11  |-  ( B  i^i  (/) )  =  (/)
3028, 29syl6eq 2509 . . . . . . . . . 10  |-  ( x  =  0  ->  ( B  i^i  ( 0..^ x ) )  =  (/) )
3130fveq2d 5798 . . . . . . . . 9  |-  ( x  =  0  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  (/) ) )
3231, 26syl6eq 2509 . . . . . . . 8  |-  ( x  =  0  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  0 )
3327, 32oveq12d 6213 . . . . . . 7  |-  ( x  =  0  ->  (
( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  =  ( 0  +  0 ) )
34 00id 9650 . . . . . . 7  |-  ( 0  +  0 )  =  0
3533, 34syl6eq 2509 . . . . . 6  |-  ( x  =  0  ->  (
( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  =  0 )
368, 35breq12d 4408 . . . . 5  |-  ( x  =  0  ->  (
( 2 ^ x
)  <_  ( ( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  <->  1  <_  0 ) )
373, 36bibi12d 321 . . . 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 316 . . 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 5794 . . . . . 6  |-  ( x  =  k  ->  ( C `  x )  =  ( C `  k ) )
4039eleq2d 2522 . . . . 5  |-  ( x  =  k  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  k ) ) )
41 oveq2 6203 . . . . . 6  |-  ( x  =  k  ->  (
2 ^ x )  =  ( 2 ^ k ) )
42 oveq2 6203 . . . . . . . . 9  |-  ( x  =  k  ->  (
0..^ x )  =  ( 0..^ k ) )
4342ineq2d 3655 . . . . . . . 8  |-  ( x  =  k  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ k ) ) )
4443fveq2d 5798 . . . . . . 7  |-  ( x  =  k  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ k ) ) ) )
4542ineq2d 3655 . . . . . . . 8  |-  ( x  =  k  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ k ) ) )
4645fveq2d 5798 . . . . . . 7  |-  ( x  =  k  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ k ) ) ) )
4744, 46oveq12d 6213 . . . . . 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 4408 . . . . 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 321 . . . 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 316 . . 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 5794 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( C `  x )  =  ( C `  ( k  +  1 ) ) )
5251eleq2d 2522 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  ( k  +  1 ) ) ) )
53 oveq2 6203 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  (
2 ^ x )  =  ( 2 ^ ( k  +  1 ) ) )
54 oveq2 6203 . . . . . . . . 9  |-  ( x  =  ( k  +  1 )  ->  (
0..^ x )  =  ( 0..^ ( k  +  1 ) ) )
5554ineq2d 3655 . . . . . . . 8  |-  ( x  =  ( k  +  1 )  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) )
5655fveq2d 5798 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) ) )
5754ineq2d 3655 . . . . . . . 8  |-  ( x  =  ( k  +  1 )  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) )
5857fveq2d 5798 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) ) )
5956, 58oveq12d 6213 . . . . . 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 4408 . . . . 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 321 . . . 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 316 . . 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 5794 . . . . . 6  |-  ( x  =  N  ->  ( C `  x )  =  ( C `  N ) )
6463eleq2d 2522 . . . . 5  |-  ( x  =  N  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  N ) ) )
65 oveq2 6203 . . . . . 6  |-  ( x  =  N  ->  (
2 ^ x )  =  ( 2 ^ N ) )
66 oveq2 6203 . . . . . . . . 9  |-  ( x  =  N  ->  (
0..^ x )  =  ( 0..^ N ) )
6766ineq2d 3655 . . . . . . . 8  |-  ( x  =  N  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ N ) ) )
6867fveq2d 5798 . . . . . . 7  |-  ( x  =  N  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ N ) ) ) )
6966ineq2d 3655 . . . . . . . 8  |-  ( x  =  N  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ N ) ) )
7069fveq2d 5798 . . . . . . 7  |-  ( x  =  N  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ N ) ) ) )
7168, 70oveq12d 6213 . . . . . 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 4408 . . . . 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 321 . . . 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 316 . . 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 13763 . . . 4  |-  ( ph  ->  -.  (/)  e.  ( C `
 0 ) )
79 0lt1 9968 . . . . . 6  |-  0  <  1
80 0re 9492 . . . . . . 7  |-  0  e.  RR
81 1re 9491 . . . . . . 7  |-  1  e.  RR
8280, 81ltnlei 9601 . . . . . 6  |-  ( 0  <  1  <->  -.  1  <_  0 )
8379, 82mpbi 208 . . . . 5  |-  -.  1  <_  0
8483a1i 11 . . . 4  |-  ( ph  ->  -.  1  <_  0
)
8578, 842falsed 351 . . 3  |-  ( ph  ->  ( (/)  e.  ( C `  0 )  <->  1  <_  0 ) )
8675ad2antrr 725 . . . . . . 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 725 . . . . . . 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 461 . . . . . . 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 13766 . . . . . 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 434 . . . . 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 435 . . . 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 10844 . 2  |-  ( N  e.  NN0  ->  ( ph  ->  ( (/)  e.  ( C `  N )  <->  ( 2 ^ N )  <_  ( ( K `
 ( A  i^i  ( 0..^ N ) ) )  +  ( K `
 ( B  i^i  ( 0..^ N ) ) ) ) ) ) )
951, 94mpcom 36 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 369    = wceq 1370  caddwcad 1421    e. wcel 1758    i^i cin 3430    C_ wss 3431   (/)c0 3740   ifcif 3894   ~Pcpw 3963   class class class wbr 4395    |-> cmpt 4453   `'ccnv 4942    |` cres 4945   -1-1-onto->wf1o 5520   ` cfv 5521  (class class class)co 6195    |-> cmpt2 6197   1oc1o 7018   2oc2o 7019   Fincfn 7415   CCcc 9386   0cc0 9388   1c1 9389    + caddc 9391    < clt 9524    <_ cle 9525    - cmin 9701   2c2 10477   NN0cn0 10685  ..^cfzo 11660    seqcseq 11918   ^cexp 11977  bitscbits 13728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-xor 1352  df-tru 1373  df-fal 1376  df-cad 1423  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-disj 4366  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-2o 7026  df-oadd 7029  df-er 7206  df-map 7321  df-pm 7322  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-sup 7797  df-oi 7830  df-card 8215  df-cda 8443  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-n0 10686  df-z 10753  df-uz 10968  df-rp 11098  df-fz 11550  df-fzo 11661  df-fl 11754  df-mod 11821  df-seq 11919  df-exp 11978  df-hash 12216  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-clim 13079  df-sum 13277  df-dvds 13649  df-bits 13731
This theorem is referenced by: (None)
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