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Theorem sadadd2 12651
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 5866 . . . . . . . . . . 11  |-  ( x  =  0  ->  (
0..^ x )  =  ( 0..^ 0 ) )
3 fzo0 10893 . . . . . . . . . . 11  |-  ( 0..^ 0 )  =  (/)
42, 3syl6eq 2331 . . . . . . . . . 10  |-  ( x  =  0  ->  (
0..^ x )  =  (/) )
54ineq2d 3370 . . . . . . . . 9  |-  ( x  =  0  ->  (
( A sadd  B )  i^i  ( 0..^ x ) )  =  ( ( A sadd  B )  i^i  (/) ) )
6 in0 3480 . . . . . . . . 9  |-  ( ( A sadd  B )  i^i  (/) )  =  (/)
75, 6syl6eq 2331 . . . . . . . 8  |-  ( x  =  0  ->  (
( A sadd  B )  i^i  ( 0..^ x ) )  =  (/) )
87fveq2d 5529 . . . . . . 7  |-  ( x  =  0  ->  ( K `  ( ( A sadd  B )  i^i  (
0..^ x ) ) )  =  ( K `
 (/) ) )
9 sadcadd.k . . . . . . . . 9  |-  K  =  `' (bits  |`  NN0 )
10 0nn0 9980 . . . . . . . . . . 11  |-  0  e.  NN0
11 fvres 5542 . . . . . . . . . . 11  |-  ( 0  e.  NN0  ->  ( (bits  |`  NN0 ) `  0
)  =  (bits ` 
0 ) )
1210, 11ax-mp 8 . . . . . . . . . 10  |-  ( (bits  |`  NN0 ) `  0
)  =  (bits ` 
0 )
13 0bits 12630 . . . . . . . . . 10  |-  (bits ` 
0 )  =  (/)
1412, 13eqtr2i 2304 . . . . . . . . 9  |-  (/)  =  ( (bits  |`  NN0 ) ` 
0 )
159, 14fveq12i 5530 . . . . . . . 8  |-  ( K `
 (/) )  =  ( `' (bits  |`  NN0 ) `  ( (bits  |`  NN0 ) `  0 ) )
16 bitsf1o 12636 . . . . . . . . 9  |-  (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )
17 f1ocnvfv1 5792 . . . . . . . . 9  |-  ( ( (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )  /\  0  e.  NN0 )  ->  ( `' (bits  |`  NN0 ) `  (
(bits  |`  NN0 ) ` 
0 ) )  =  0 )
1816, 10, 17mp2an 653 . . . . . . . 8  |-  ( `' (bits  |`  NN0 ) `  ( (bits  |`  NN0 ) `  0 ) )  =  0
1915, 18eqtri 2303 . . . . . . 7  |-  ( K `
 (/) )  =  0
208, 19syl6eq 2331 . . . . . 6  |-  ( x  =  0  ->  ( K `  ( ( A sadd  B )  i^i  (
0..^ x ) ) )  =  0 )
21 fveq2 5525 . . . . . . . 8  |-  ( x  =  0  ->  ( C `  x )  =  ( C ` 
0 ) )
2221eleq2d 2350 . . . . . . 7  |-  ( x  =  0  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  0 ) ) )
23 oveq2 5866 . . . . . . 7  |-  ( x  =  0  ->  (
2 ^ x )  =  ( 2 ^ 0 ) )
24 eqidd 2284 . . . . . . 7  |-  ( x  =  0  ->  0  =  0 )
2522, 23, 24ifbieq12d 3587 . . . . . 6  |-  ( x  =  0  ->  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 )  =  if ( (/)  e.  ( C `  0
) ,  ( 2 ^ 0 ) ,  0 ) )
2620, 25oveq12d 5876 . . . . 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 3370 . . . . . . . . . 10  |-  ( x  =  0  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  (/) ) )
28 in0 3480 . . . . . . . . . 10  |-  ( A  i^i  (/) )  =  (/)
2927, 28syl6eq 2331 . . . . . . . . 9  |-  ( x  =  0  ->  ( A  i^i  ( 0..^ x ) )  =  (/) )
3029fveq2d 5529 . . . . . . . 8  |-  ( x  =  0  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  (/) ) )
3130, 19syl6eq 2331 . . . . . . 7  |-  ( x  =  0  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  0 )
324ineq2d 3370 . . . . . . . . . 10  |-  ( x  =  0  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  (/) ) )
33 in0 3480 . . . . . . . . . 10  |-  ( B  i^i  (/) )  =  (/)
3432, 33syl6eq 2331 . . . . . . . . 9  |-  ( x  =  0  ->  ( B  i^i  ( 0..^ x ) )  =  (/) )
3534fveq2d 5529 . . . . . . . 8  |-  ( x  =  0  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  (/) ) )
3635, 19syl6eq 2331 . . . . . . 7  |-  ( x  =  0  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  0 )
3731, 36oveq12d 5876 . . . . . 6  |-  ( x  =  0  ->  (
( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  =  ( 0  +  0 ) )
38 00id 8987 . . . . . 6  |-  ( 0  +  0 )  =  0
3937, 38syl6eq 2331 . . . . 5  |-  ( x  =  0  ->  (
( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  =  0 )
4026, 39eqeq12d 2297 . . . 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 307 . . 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 5866 . . . . . . . 8  |-  ( x  =  k  ->  (
0..^ x )  =  ( 0..^ k ) )
4342ineq2d 3370 . . . . . . 7  |-  ( x  =  k  ->  (
( A sadd  B )  i^i  ( 0..^ x ) )  =  ( ( A sadd  B )  i^i  ( 0..^ k ) ) )
4443fveq2d 5529 . . . . . 6  |-  ( x  =  k  ->  ( K `  ( ( A sadd  B )  i^i  (
0..^ x ) ) )  =  ( K `
 ( ( A sadd 
B )  i^i  (
0..^ k ) ) ) )
45 fveq2 5525 . . . . . . . 8  |-  ( x  =  k  ->  ( C `  x )  =  ( C `  k ) )
4645eleq2d 2350 . . . . . . 7  |-  ( x  =  k  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  k ) ) )
47 oveq2 5866 . . . . . . 7  |-  ( x  =  k  ->  (
2 ^ x )  =  ( 2 ^ k ) )
48 eqidd 2284 . . . . . . 7  |-  ( x  =  k  ->  0  =  0 )
4946, 47, 48ifbieq12d 3587 . . . . . 6  |-  ( x  =  k  ->  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 )  =  if ( (/)  e.  ( C `  k
) ,  ( 2 ^ k ) ,  0 ) )
5044, 49oveq12d 5876 . . . . 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 3370 . . . . . . 7  |-  ( x  =  k  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ k ) ) )
5251fveq2d 5529 . . . . . 6  |-  ( x  =  k  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ k ) ) ) )
5342ineq2d 3370 . . . . . . 7  |-  ( x  =  k  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ k ) ) )
5453fveq2d 5529 . . . . . 6  |-  ( x  =  k  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ k ) ) ) )
5552, 54oveq12d 5876 . . . . 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 2297 . . . 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 307 . . 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 5866 . . . . . . . 8  |-  ( x  =  ( k  +  1 )  ->  (
0..^ x )  =  ( 0..^ ( k  +  1 ) ) )
5958ineq2d 3370 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  (
( A sadd  B )  i^i  ( 0..^ x ) )  =  ( ( A sadd  B )  i^i  ( 0..^ ( k  +  1 ) ) ) )
6059fveq2d 5529 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( K `  ( ( A sadd  B )  i^i  (
0..^ x ) ) )  =  ( K `
 ( ( A sadd 
B )  i^i  (
0..^ ( k  +  1 ) ) ) ) )
61 fveq2 5525 . . . . . . . 8  |-  ( x  =  ( k  +  1 )  ->  ( C `  x )  =  ( C `  ( k  +  1 ) ) )
6261eleq2d 2350 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  ( k  +  1 ) ) ) )
63 oveq2 5866 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  (
2 ^ x )  =  ( 2 ^ ( k  +  1 ) ) )
64 eqidd 2284 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  0  =  0 )
6562, 63, 64ifbieq12d 3587 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 )  =  if ( (/)  e.  ( C `  (
k  +  1 ) ) ,  ( 2 ^ ( k  +  1 ) ) ,  0 ) )
6660, 65oveq12d 5876 . . . . 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 3370 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) )
6867fveq2d 5529 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) ) )
6958ineq2d 3370 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) )
7069fveq2d 5529 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) ) )
7168, 70oveq12d 5876 . . . . 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 2297 . . . 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 307 . . 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 5866 . . . . . . . 8  |-  ( x  =  N  ->  (
0..^ x )  =  ( 0..^ N ) )
7574ineq2d 3370 . . . . . . 7  |-  ( x  =  N  ->  (
( A sadd  B )  i^i  ( 0..^ x ) )  =  ( ( A sadd  B )  i^i  ( 0..^ N ) ) )
7675fveq2d 5529 . . . . . 6  |-  ( x  =  N  ->  ( K `  ( ( A sadd  B )  i^i  (
0..^ x ) ) )  =  ( K `
 ( ( A sadd 
B )  i^i  (
0..^ N ) ) ) )
77 fveq2 5525 . . . . . . . 8  |-  ( x  =  N  ->  ( C `  x )  =  ( C `  N ) )
7877eleq2d 2350 . . . . . . 7  |-  ( x  =  N  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  N ) ) )
79 oveq2 5866 . . . . . . 7  |-  ( x  =  N  ->  (
2 ^ x )  =  ( 2 ^ N ) )
80 eqidd 2284 . . . . . . 7  |-  ( x  =  N  ->  0  =  0 )
8178, 79, 80ifbieq12d 3587 . . . . . 6  |-  ( x  =  N  ->  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 )  =  if ( (/)  e.  ( C `  N
) ,  ( 2 ^ N ) ,  0 ) )
8276, 81oveq12d 5876 . . . . 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 3370 . . . . . . 7  |-  ( x  =  N  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ N ) ) )
8483fveq2d 5529 . . . . . 6  |-  ( x  =  N  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ N ) ) ) )
8574ineq2d 3370 . . . . . . 7  |-  ( x  =  N  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ N ) ) )
8685fveq2d 5529 . . . . . 6  |-  ( x  =  N  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ N ) ) ) )
8784, 86oveq12d 5876 . . . . 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 2297 . . . 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 307 . . 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 12645 . . . . . 6  |-  ( ph  ->  -.  (/)  e.  ( C `
 0 ) )
94 iffalse 3572 . . . . . 6  |-  ( -.  (/)  e.  ( C ` 
0 )  ->  if ( (/)  e.  ( C `
 0 ) ,  ( 2 ^ 0 ) ,  0 )  =  0 )
9593, 94syl 15 . . . . 5  |-  ( ph  ->  if ( (/)  e.  ( C `  0 ) ,  ( 2 ^ 0 ) ,  0 )  =  0 )
9695oveq2d 5874 . . . 4  |-  ( ph  ->  ( 0  +  if ( (/)  e.  ( C `
 0 ) ,  ( 2 ^ 0 ) ,  0 ) )  =  ( 0  +  0 ) )
9796, 38syl6eq 2331 . . 3  |-  ( ph  ->  ( 0  +  if ( (/)  e.  ( C `
 0 ) ,  ( 2 ^ 0 ) ,  0 ) )  =  0 )
9890ad2antrr 706 . . . . . . 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 706 . . . . . . 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 731 . . . . . . 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 447 . . . . . . 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 12650 . . . . . 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 423 . . . . 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 424 . . . 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 23 . . 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 10108 . 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 32 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 358  caddwcad 1369    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   (/)c0 3455   ifcif 3565   ~Pcpw 3625    e. cmpt 4077   `'ccnv 4688    |` cres 4691   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1oc1o 6472   2oc2o 6473   Fincfn 6863   0cc0 8737   1c1 8738    + caddc 8740    - cmin 9037   2c2 9795   NN0cn0 9965  ..^cfzo 10870    seq cseq 11046   ^cexp 11104  bitscbits 12610   sadd csad 12611
This theorem is referenced by:  sadadd3  12652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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-had 1370  df-cad 1371  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-disj 3994  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-dvds 12532  df-bits 12613  df-sad 12642
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