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Theorem sadadd2 12899
Description: Sum of initial segments of the sadd 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
sadadd2  |-  ( ph  ->  ( ( K `  ( ( A sadd  B
)  i^i  ( 0..^ N ) ) )  +  if ( (/)  e.  ( C `  N
) ,  ( 2 ^ N ) ,  0 ) )  =  ( ( 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 sadadd2
Dummy variables  k  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sadcp1.n . 2  |-  ( ph  ->  N  e.  NN0 )
2 oveq2 6028 . . . . . . . . . . 11  |-  ( x  =  0  ->  (
0..^ x )  =  ( 0..^ 0 ) )
3 fzo0 11089 . . . . . . . . . . 11  |-  ( 0..^ 0 )  =  (/)
42, 3syl6eq 2435 . . . . . . . . . 10  |-  ( x  =  0  ->  (
0..^ x )  =  (/) )
54ineq2d 3485 . . . . . . . . 9  |-  ( x  =  0  ->  (
( A sadd  B )  i^i  ( 0..^ x ) )  =  ( ( A sadd  B )  i^i  (/) ) )
6 in0 3596 . . . . . . . . 9  |-  ( ( A sadd  B )  i^i  (/) )  =  (/)
75, 6syl6eq 2435 . . . . . . . 8  |-  ( x  =  0  ->  (
( A sadd  B )  i^i  ( 0..^ x ) )  =  (/) )
87fveq2d 5672 . . . . . . 7  |-  ( x  =  0  ->  ( K `  ( ( A sadd  B )  i^i  (
0..^ x ) ) )  =  ( K `
 (/) ) )
9 sadcadd.k . . . . . . . . 9  |-  K  =  `' (bits  |`  NN0 )
10 0nn0 10168 . . . . . . . . . . 11  |-  0  e.  NN0
11 fvres 5685 . . . . . . . . . . 11  |-  ( 0  e.  NN0  ->  ( (bits  |`  NN0 ) `  0
)  =  (bits ` 
0 ) )
1210, 11ax-mp 8 . . . . . . . . . 10  |-  ( (bits  |`  NN0 ) `  0
)  =  (bits ` 
0 )
13 0bits 12878 . . . . . . . . . 10  |-  (bits ` 
0 )  =  (/)
1412, 13eqtr2i 2408 . . . . . . . . 9  |-  (/)  =  ( (bits  |`  NN0 ) ` 
0 )
159, 14fveq12i 5673 . . . . . . . 8  |-  ( K `
 (/) )  =  ( `' (bits  |`  NN0 ) `  ( (bits  |`  NN0 ) `  0 ) )
16 bitsf1o 12884 . . . . . . . . 9  |-  (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )
17 f1ocnvfv1 5953 . . . . . . . . 9  |-  ( ( (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )  /\  0  e.  NN0 )  ->  ( `' (bits  |`  NN0 ) `  (
(bits  |`  NN0 ) ` 
0 ) )  =  0 )
1816, 10, 17mp2an 654 . . . . . . . 8  |-  ( `' (bits  |`  NN0 ) `  ( (bits  |`  NN0 ) `  0 ) )  =  0
1915, 18eqtri 2407 . . . . . . 7  |-  ( K `
 (/) )  =  0
208, 19syl6eq 2435 . . . . . 6  |-  ( x  =  0  ->  ( K `  ( ( A sadd  B )  i^i  (
0..^ x ) ) )  =  0 )
21 fveq2 5668 . . . . . . . 8  |-  ( x  =  0  ->  ( C `  x )  =  ( C ` 
0 ) )
2221eleq2d 2454 . . . . . . 7  |-  ( x  =  0  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  0 ) ) )
23 oveq2 6028 . . . . . . 7  |-  ( x  =  0  ->  (
2 ^ x )  =  ( 2 ^ 0 ) )
24 eqidd 2388 . . . . . . 7  |-  ( x  =  0  ->  0  =  0 )
2522, 23, 24ifbieq12d 3704 . . . . . 6  |-  ( x  =  0  ->  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 )  =  if ( (/)  e.  ( C `  0
) ,  ( 2 ^ 0 ) ,  0 ) )
2620, 25oveq12d 6038 . . . . 5  |-  ( x  =  0  ->  (
( K `  (
( A sadd  B )  i^i  ( 0..^ x ) ) )  +  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 ) )  =  ( 0  +  if ( (/)  e.  ( C `  0
) ,  ( 2 ^ 0 ) ,  0 ) ) )
274ineq2d 3485 . . . . . . . . . 10  |-  ( x  =  0  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  (/) ) )
28 in0 3596 . . . . . . . . . 10  |-  ( A  i^i  (/) )  =  (/)
2927, 28syl6eq 2435 . . . . . . . . 9  |-  ( x  =  0  ->  ( A  i^i  ( 0..^ x ) )  =  (/) )
3029fveq2d 5672 . . . . . . . 8  |-  ( x  =  0  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  (/) ) )
3130, 19syl6eq 2435 . . . . . . 7  |-  ( x  =  0  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  0 )
324ineq2d 3485 . . . . . . . . . 10  |-  ( x  =  0  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  (/) ) )
33 in0 3596 . . . . . . . . . 10  |-  ( B  i^i  (/) )  =  (/)
3432, 33syl6eq 2435 . . . . . . . . 9  |-  ( x  =  0  ->  ( B  i^i  ( 0..^ x ) )  =  (/) )
3534fveq2d 5672 . . . . . . . 8  |-  ( x  =  0  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  (/) ) )
3635, 19syl6eq 2435 . . . . . . 7  |-  ( x  =  0  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  0 )
3731, 36oveq12d 6038 . . . . . 6  |-  ( x  =  0  ->  (
( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  =  ( 0  +  0 ) )
38 00id 9173 . . . . . 6  |-  ( 0  +  0 )  =  0
3937, 38syl6eq 2435 . . . . 5  |-  ( x  =  0  ->  (
( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  =  0 )
4026, 39eqeq12d 2401 . . . 4  |-  ( x  =  0  ->  (
( ( K `  ( ( A sadd  B
)  i^i  ( 0..^ x ) ) )  +  if ( (/)  e.  ( C `  x
) ,  ( 2 ^ x ) ,  0 ) )  =  ( ( K `  ( A  i^i  (
0..^ x ) ) )  +  ( K `
 ( B  i^i  ( 0..^ x ) ) ) )  <->  ( 0  +  if ( (/)  e.  ( C `  0
) ,  ( 2 ^ 0 ) ,  0 ) )  =  0 ) )
4140imbi2d 308 . . 3  |-  ( x  =  0  ->  (
( ph  ->  ( ( K `  ( ( A sadd  B )  i^i  ( 0..^ x ) ) )  +  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 ) )  =  ( ( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) ) )  <->  ( ph  ->  ( 0  +  if (
(/)  e.  ( C `  0 ) ,  ( 2 ^ 0 ) ,  0 ) )  =  0 ) ) )
42 oveq2 6028 . . . . . . . 8  |-  ( x  =  k  ->  (
0..^ x )  =  ( 0..^ k ) )
4342ineq2d 3485 . . . . . . 7  |-  ( x  =  k  ->  (
( A sadd  B )  i^i  ( 0..^ x ) )  =  ( ( A sadd  B )  i^i  ( 0..^ k ) ) )
4443fveq2d 5672 . . . . . 6  |-  ( x  =  k  ->  ( K `  ( ( A sadd  B )  i^i  (
0..^ x ) ) )  =  ( K `
 ( ( A sadd 
B )  i^i  (
0..^ k ) ) ) )
45 fveq2 5668 . . . . . . . 8  |-  ( x  =  k  ->  ( C `  x )  =  ( C `  k ) )
4645eleq2d 2454 . . . . . . 7  |-  ( x  =  k  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  k ) ) )
47 oveq2 6028 . . . . . . 7  |-  ( x  =  k  ->  (
2 ^ x )  =  ( 2 ^ k ) )
48 eqidd 2388 . . . . . . 7  |-  ( x  =  k  ->  0  =  0 )
4946, 47, 48ifbieq12d 3704 . . . . . 6  |-  ( x  =  k  ->  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 )  =  if ( (/)  e.  ( C `  k
) ,  ( 2 ^ k ) ,  0 ) )
5044, 49oveq12d 6038 . . . . 5  |-  ( x  =  k  ->  (
( K `  (
( A sadd  B )  i^i  ( 0..^ x ) ) )  +  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 ) )  =  ( ( K `  ( ( A sadd  B )  i^i  ( 0..^ k ) ) )  +  if ( (/)  e.  ( C `
 k ) ,  ( 2 ^ k
) ,  0 ) ) )
5142ineq2d 3485 . . . . . . 7  |-  ( x  =  k  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ k ) ) )
5251fveq2d 5672 . . . . . 6  |-  ( x  =  k  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ k ) ) ) )
5342ineq2d 3485 . . . . . . 7  |-  ( x  =  k  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ k ) ) )
5453fveq2d 5672 . . . . . 6  |-  ( x  =  k  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ k ) ) ) )
5552, 54oveq12d 6038 . . . . 5  |-  ( 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 ) ) ) ) )
5650, 55eqeq12d 2401 . . . 4  |-  ( x  =  k  ->  (
( ( K `  ( ( A sadd  B
)  i^i  ( 0..^ x ) ) )  +  if ( (/)  e.  ( C `  x
) ,  ( 2 ^ x ) ,  0 ) )  =  ( ( K `  ( A  i^i  (
0..^ x ) ) )  +  ( K `
 ( B  i^i  ( 0..^ x ) ) ) )  <->  ( ( K `  ( ( A sadd  B )  i^i  (
0..^ k ) ) )  +  if (
(/)  e.  ( C `  k ) ,  ( 2 ^ k ) ,  0 ) )  =  ( ( K `
 ( A  i^i  ( 0..^ k ) ) )  +  ( K `
 ( B  i^i  ( 0..^ k ) ) ) ) ) )
5756imbi2d 308 . . 3  |-  ( x  =  k  ->  (
( ph  ->  ( ( K `  ( ( A sadd  B )  i^i  ( 0..^ x ) ) )  +  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 ) )  =  ( ( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) ) )  <->  ( ph  ->  ( ( K `  (
( A sadd  B )  i^i  ( 0..^ k ) ) )  +  if ( (/)  e.  ( C `
 k ) ,  ( 2 ^ k
) ,  0 ) )  =  ( ( K `  ( A  i^i  ( 0..^ k ) ) )  +  ( K `  ( B  i^i  ( 0..^ k ) ) ) ) ) ) )
58 oveq2 6028 . . . . . . . 8  |-  ( x  =  ( k  +  1 )  ->  (
0..^ x )  =  ( 0..^ ( k  +  1 ) ) )
5958ineq2d 3485 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  (
( A sadd  B )  i^i  ( 0..^ x ) )  =  ( ( A sadd  B )  i^i  ( 0..^ ( k  +  1 ) ) ) )
6059fveq2d 5672 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( K `  ( ( A sadd  B )  i^i  (
0..^ x ) ) )  =  ( K `
 ( ( A sadd 
B )  i^i  (
0..^ ( k  +  1 ) ) ) ) )
61 fveq2 5668 . . . . . . . 8  |-  ( x  =  ( k  +  1 )  ->  ( C `  x )  =  ( C `  ( k  +  1 ) ) )
6261eleq2d 2454 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  ( k  +  1 ) ) ) )
63 oveq2 6028 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  (
2 ^ x )  =  ( 2 ^ ( k  +  1 ) ) )
64 eqidd 2388 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  0  =  0 )
6562, 63, 64ifbieq12d 3704 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 )  =  if ( (/)  e.  ( C `  (
k  +  1 ) ) ,  ( 2 ^ ( k  +  1 ) ) ,  0 ) )
6660, 65oveq12d 6038 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  (
( K `  (
( A sadd  B )  i^i  ( 0..^ x ) ) )  +  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 ) )  =  ( ( K `  ( ( A sadd  B )  i^i  ( 0..^ ( k  +  1 ) ) ) )  +  if ( (/)  e.  ( C `
 ( k  +  1 ) ) ,  ( 2 ^ (
k  +  1 ) ) ,  0 ) ) )
6758ineq2d 3485 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) )
6867fveq2d 5672 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) ) )
6958ineq2d 3485 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) )
7069fveq2d 5672 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) ) )
7168, 70oveq12d 6038 . . . . 5  |-  ( 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 ) ) ) ) ) )
7266, 71eqeq12d 2401 . . . 4  |-  ( x  =  ( k  +  1 )  ->  (
( ( K `  ( ( A sadd  B
)  i^i  ( 0..^ x ) ) )  +  if ( (/)  e.  ( C `  x
) ,  ( 2 ^ x ) ,  0 ) )  =  ( ( K `  ( A  i^i  (
0..^ x ) ) )  +  ( K `
 ( B  i^i  ( 0..^ x ) ) ) )  <->  ( ( K `  ( ( A sadd  B )  i^i  (
0..^ ( k  +  1 ) ) ) )  +  if (
(/)  e.  ( C `  ( k  +  1 ) ) ,  ( 2 ^ ( k  +  1 ) ) ,  0 ) )  =  ( ( K `
 ( A  i^i  ( 0..^ ( k  +  1 ) ) ) )  +  ( K `
 ( B  i^i  ( 0..^ ( k  +  1 ) ) ) ) ) ) )
7372imbi2d 308 . . 3  |-  ( x  =  ( k  +  1 )  ->  (
( ph  ->  ( ( K `  ( ( A sadd  B )  i^i  ( 0..^ x ) ) )  +  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 ) )  =  ( ( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) ) )  <->  ( ph  ->  ( ( K `  (
( A sadd  B )  i^i  ( 0..^ ( k  +  1 ) ) ) )  +  if ( (/)  e.  ( C `
 ( k  +  1 ) ) ,  ( 2 ^ (
k  +  1 ) ) ,  0 ) )  =  ( ( K `  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) )  +  ( K `  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) ) ) ) ) )
74 oveq2 6028 . . . . . . . 8  |-  ( x  =  N  ->  (
0..^ x )  =  ( 0..^ N ) )
7574ineq2d 3485 . . . . . . 7  |-  ( x  =  N  ->  (
( A sadd  B )  i^i  ( 0..^ x ) )  =  ( ( A sadd  B )  i^i  ( 0..^ N ) ) )
7675fveq2d 5672 . . . . . 6  |-  ( x  =  N  ->  ( K `  ( ( A sadd  B )  i^i  (
0..^ x ) ) )  =  ( K `
 ( ( A sadd 
B )  i^i  (
0..^ N ) ) ) )
77 fveq2 5668 . . . . . . . 8  |-  ( x  =  N  ->  ( C `  x )  =  ( C `  N ) )
7877eleq2d 2454 . . . . . . 7  |-  ( x  =  N  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  N ) ) )
79 oveq2 6028 . . . . . . 7  |-  ( x  =  N  ->  (
2 ^ x )  =  ( 2 ^ N ) )
80 eqidd 2388 . . . . . . 7  |-  ( x  =  N  ->  0  =  0 )
8178, 79, 80ifbieq12d 3704 . . . . . 6  |-  ( x  =  N  ->  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 )  =  if ( (/)  e.  ( C `  N
) ,  ( 2 ^ N ) ,  0 ) )
8276, 81oveq12d 6038 . . . . 5  |-  ( x  =  N  ->  (
( K `  (
( A sadd  B )  i^i  ( 0..^ x ) ) )  +  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 ) )  =  ( ( K `  ( ( A sadd  B )  i^i  ( 0..^ N ) ) )  +  if ( (/)  e.  ( C `
 N ) ,  ( 2 ^ N
) ,  0 ) ) )
8374ineq2d 3485 . . . . . . 7  |-  ( x  =  N  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ N ) ) )
8483fveq2d 5672 . . . . . 6  |-  ( x  =  N  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ N ) ) ) )
8574ineq2d 3485 . . . . . . 7  |-  ( x  =  N  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ N ) ) )
8685fveq2d 5672 . . . . . 6  |-  ( x  =  N  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ N ) ) ) )
8784, 86oveq12d 6038 . . . . 5  |-  ( 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 ) ) ) ) )
8882, 87eqeq12d 2401 . . . 4  |-  ( x  =  N  ->  (
( ( K `  ( ( A sadd  B
)  i^i  ( 0..^ x ) ) )  +  if ( (/)  e.  ( C `  x
) ,  ( 2 ^ x ) ,  0 ) )  =  ( ( K `  ( A  i^i  (
0..^ x ) ) )  +  ( K `
 ( B  i^i  ( 0..^ x ) ) ) )  <->  ( ( K `  ( ( A sadd  B )  i^i  (
0..^ N ) ) )  +  if (
(/)  e.  ( C `  N ) ,  ( 2 ^ N ) ,  0 ) )  =  ( ( K `
 ( A  i^i  ( 0..^ N ) ) )  +  ( K `
 ( B  i^i  ( 0..^ N ) ) ) ) ) )
8988imbi2d 308 . . 3  |-  ( x  =  N  ->  (
( ph  ->  ( ( K `  ( ( A sadd  B )  i^i  ( 0..^ x ) ) )  +  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 ) )  =  ( ( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) ) )  <->  ( ph  ->  ( ( K `  (
( A sadd  B )  i^i  ( 0..^ N ) ) )  +  if ( (/)  e.  ( C `
 N ) ,  ( 2 ^ N
) ,  0 ) )  =  ( ( K `  ( A  i^i  ( 0..^ N ) ) )  +  ( K `  ( B  i^i  ( 0..^ N ) ) ) ) ) ) )
90 sadval.a . . . . . . 7  |-  ( ph  ->  A  C_  NN0 )
91 sadval.b . . . . . . 7  |-  ( ph  ->  B  C_  NN0 )
92 sadval.c . . . . . . 7  |-  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 ) ) ) )
9390, 91, 92sadc0 12893 . . . . . 6  |-  ( ph  ->  -.  (/)  e.  ( C `
 0 ) )
94 iffalse 3689 . . . . . 6  |-  ( -.  (/)  e.  ( C ` 
0 )  ->  if ( (/)  e.  ( C `
 0 ) ,  ( 2 ^ 0 ) ,  0 )  =  0 )
9593, 94syl 16 . . . . 5  |-  ( ph  ->  if ( (/)  e.  ( C `  0 ) ,  ( 2 ^ 0 ) ,  0 )  =  0 )
9695oveq2d 6036 . . . 4  |-  ( ph  ->  ( 0  +  if ( (/)  e.  ( C `
 0 ) ,  ( 2 ^ 0 ) ,  0 ) )  =  ( 0  +  0 ) )
9796, 38syl6eq 2435 . . 3  |-  ( ph  ->  ( 0  +  if ( (/)  e.  ( C `
 0 ) ,  ( 2 ^ 0 ) ,  0 ) )  =  0 )
9890ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  (
( K `  (
( A sadd  B )  i^i  ( 0..^ k ) ) )  +  if ( (/)  e.  ( C `
 k ) ,  ( 2 ^ k
) ,  0 ) )  =  ( ( K `  ( A  i^i  ( 0..^ k ) ) )  +  ( K `  ( B  i^i  ( 0..^ k ) ) ) ) )  ->  A  C_  NN0 )
9991ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  (
( K `  (
( A sadd  B )  i^i  ( 0..^ k ) ) )  +  if ( (/)  e.  ( C `
 k ) ,  ( 2 ^ k
) ,  0 ) )  =  ( ( K `  ( A  i^i  ( 0..^ k ) ) )  +  ( K `  ( B  i^i  ( 0..^ k ) ) ) ) )  ->  B  C_  NN0 )
100 simplr 732 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  (
( K `  (
( A sadd  B )  i^i  ( 0..^ k ) ) )  +  if ( (/)  e.  ( C `
 k ) ,  ( 2 ^ k
) ,  0 ) )  =  ( ( K `  ( A  i^i  ( 0..^ k ) ) )  +  ( K `  ( B  i^i  ( 0..^ k ) ) ) ) )  ->  k  e.  NN0 )
101 simpr 448 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  (
( K `  (
( A sadd  B )  i^i  ( 0..^ k ) ) )  +  if ( (/)  e.  ( C `
 k ) ,  ( 2 ^ k
) ,  0 ) )  =  ( ( K `  ( A  i^i  ( 0..^ k ) ) )  +  ( K `  ( B  i^i  ( 0..^ k ) ) ) ) )  ->  ( ( K `  ( ( A sadd  B )  i^i  (
0..^ k ) ) )  +  if (
(/)  e.  ( C `  k ) ,  ( 2 ^ k ) ,  0 ) )  =  ( ( K `
 ( A  i^i  ( 0..^ k ) ) )  +  ( K `
 ( B  i^i  ( 0..^ k ) ) ) ) )
10298, 99, 92, 100, 9, 101sadadd2lem 12898 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  (
( K `  (
( A sadd  B )  i^i  ( 0..^ k ) ) )  +  if ( (/)  e.  ( C `
 k ) ,  ( 2 ^ k
) ,  0 ) )  =  ( ( K `  ( A  i^i  ( 0..^ k ) ) )  +  ( K `  ( B  i^i  ( 0..^ k ) ) ) ) )  ->  ( ( K `  ( ( A sadd  B )  i^i  (
0..^ ( k  +  1 ) ) ) )  +  if (
(/)  e.  ( C `  ( k  +  1 ) ) ,  ( 2 ^ ( k  +  1 ) ) ,  0 ) )  =  ( ( K `
 ( A  i^i  ( 0..^ ( k  +  1 ) ) ) )  +  ( K `
 ( B  i^i  ( 0..^ ( k  +  1 ) ) ) ) ) )
103102ex 424 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( (
( K `  (
( A sadd  B )  i^i  ( 0..^ k ) ) )  +  if ( (/)  e.  ( C `
 k ) ,  ( 2 ^ k
) ,  0 ) )  =  ( ( K `  ( A  i^i  ( 0..^ k ) ) )  +  ( K `  ( B  i^i  ( 0..^ k ) ) ) )  ->  ( ( K `
 ( ( A sadd 
B )  i^i  (
0..^ ( k  +  1 ) ) ) )  +  if (
(/)  e.  ( C `  ( k  +  1 ) ) ,  ( 2 ^ ( k  +  1 ) ) ,  0 ) )  =  ( ( K `
 ( A  i^i  ( 0..^ ( k  +  1 ) ) ) )  +  ( K `
 ( B  i^i  ( 0..^ ( k  +  1 ) ) ) ) ) ) )
104103expcom 425 . . . 4  |-  ( k  e.  NN0  ->  ( ph  ->  ( ( ( K `
 ( ( A sadd 
B )  i^i  (
0..^ k ) ) )  +  if (
(/)  e.  ( C `  k ) ,  ( 2 ^ k ) ,  0 ) )  =  ( ( K `
 ( A  i^i  ( 0..^ k ) ) )  +  ( K `
 ( B  i^i  ( 0..^ k ) ) ) )  ->  (
( K `  (
( A sadd  B )  i^i  ( 0..^ ( k  +  1 ) ) ) )  +  if ( (/)  e.  ( C `
 ( k  +  1 ) ) ,  ( 2 ^ (
k  +  1 ) ) ,  0 ) )  =  ( ( K `  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) )  +  ( K `  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) ) ) ) ) )
105104a2d 24 . . 3  |-  ( k  e.  NN0  ->  ( (
ph  ->  ( ( K `
 ( ( A sadd 
B )  i^i  (
0..^ k ) ) )  +  if (
(/)  e.  ( C `  k ) ,  ( 2 ^ k ) ,  0 ) )  =  ( ( K `
 ( A  i^i  ( 0..^ k ) ) )  +  ( K `
 ( B  i^i  ( 0..^ k ) ) ) ) )  -> 
( ph  ->  ( ( K `  ( ( A sadd  B )  i^i  ( 0..^ ( k  +  1 ) ) ) )  +  if ( (/)  e.  ( C `
 ( k  +  1 ) ) ,  ( 2 ^ (
k  +  1 ) ) ,  0 ) )  =  ( ( K `  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) )  +  ( K `  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) ) ) ) ) )
10641, 57, 73, 89, 97, 105nn0ind 10298 . 2  |-  ( N  e.  NN0  ->  ( ph  ->  ( ( K `  ( ( A sadd  B
)  i^i  ( 0..^ N ) ) )  +  if ( (/)  e.  ( C `  N
) ,  ( 2 ^ N ) ,  0 ) )  =  ( ( K `  ( A  i^i  (
0..^ N ) ) )  +  ( K `
 ( B  i^i  ( 0..^ N ) ) ) ) ) )
1071, 106mpcom 34 1  |-  ( ph  ->  ( ( K `  ( ( A sadd  B
)  i^i  ( 0..^ N ) ) )  +  if ( (/)  e.  ( C `  N
) ,  ( 2 ^ N ) ,  0 ) )  =  ( ( K `  ( A  i^i  (
0..^ N ) ) )  +  ( K `
 ( B  i^i  ( 0..^ N ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359  caddwcad 1385    = wceq 1649    e. wcel 1717    i^i cin 3262    C_ wss 3263   (/)c0 3571   ifcif 3682   ~Pcpw 3742    e. cmpt 4207   `'ccnv 4817    |` cres 4820   -1-1-onto->wf1o 5393   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022   1oc1o 6653   2oc2o 6654   Fincfn 7045   0cc0 8923   1c1 8924    + caddc 8926    - cmin 9223   2c2 9981   NN0cn0 10153  ..^cfzo 11065    seq cseq 11250   ^cexp 11309  bitscbits 12858   sadd csad 12859
This theorem is referenced by:  sadadd3  12900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-xor 1311  df-tru 1325  df-had 1386  df-cad 1387  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-disj 4124  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-map 6956  df-pm 6957  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-cda 7981  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-fz 10976  df-fzo 11066  df-fl 11129  df-mod 11178  df-seq 11251  df-exp 11310  df-hash 11546  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-clim 12209  df-sum 12407  df-dvds 12780  df-bits 12861  df-sad 12890
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