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Theorem sadadd2 12962
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 6081 . . . . . . . . . . 11  |-  ( x  =  0  ->  (
0..^ x )  =  ( 0..^ 0 ) )
3 fzo0 11149 . . . . . . . . . . 11  |-  ( 0..^ 0 )  =  (/)
42, 3syl6eq 2483 . . . . . . . . . 10  |-  ( x  =  0  ->  (
0..^ x )  =  (/) )
54ineq2d 3534 . . . . . . . . 9  |-  ( x  =  0  ->  (
( A sadd  B )  i^i  ( 0..^ x ) )  =  ( ( A sadd  B )  i^i  (/) ) )
6 in0 3645 . . . . . . . . 9  |-  ( ( A sadd  B )  i^i  (/) )  =  (/)
75, 6syl6eq 2483 . . . . . . . 8  |-  ( x  =  0  ->  (
( A sadd  B )  i^i  ( 0..^ x ) )  =  (/) )
87fveq2d 5724 . . . . . . 7  |-  ( x  =  0  ->  ( K `  ( ( A sadd  B )  i^i  (
0..^ x ) ) )  =  ( K `
 (/) ) )
9 sadcadd.k . . . . . . . . 9  |-  K  =  `' (bits  |`  NN0 )
10 0nn0 10226 . . . . . . . . . . 11  |-  0  e.  NN0
11 fvres 5737 . . . . . . . . . . 11  |-  ( 0  e.  NN0  ->  ( (bits  |`  NN0 ) `  0
)  =  (bits ` 
0 ) )
1210, 11ax-mp 8 . . . . . . . . . 10  |-  ( (bits  |`  NN0 ) `  0
)  =  (bits ` 
0 )
13 0bits 12941 . . . . . . . . . 10  |-  (bits ` 
0 )  =  (/)
1412, 13eqtr2i 2456 . . . . . . . . 9  |-  (/)  =  ( (bits  |`  NN0 ) ` 
0 )
159, 14fveq12i 5725 . . . . . . . 8  |-  ( K `
 (/) )  =  ( `' (bits  |`  NN0 ) `  ( (bits  |`  NN0 ) `  0 ) )
16 bitsf1o 12947 . . . . . . . . 9  |-  (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )
17 f1ocnvfv1 6006 . . . . . . . . 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 2455 . . . . . . 7  |-  ( K `
 (/) )  =  0
208, 19syl6eq 2483 . . . . . 6  |-  ( x  =  0  ->  ( K `  ( ( A sadd  B )  i^i  (
0..^ x ) ) )  =  0 )
21 fveq2 5720 . . . . . . . 8  |-  ( x  =  0  ->  ( C `  x )  =  ( C ` 
0 ) )
2221eleq2d 2502 . . . . . . 7  |-  ( x  =  0  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  0 ) ) )
23 oveq2 6081 . . . . . . 7  |-  ( x  =  0  ->  (
2 ^ x )  =  ( 2 ^ 0 ) )
24 eqidd 2436 . . . . . . 7  |-  ( x  =  0  ->  0  =  0 )
2522, 23, 24ifbieq12d 3753 . . . . . 6  |-  ( x  =  0  ->  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 )  =  if ( (/)  e.  ( C `  0
) ,  ( 2 ^ 0 ) ,  0 ) )
2620, 25oveq12d 6091 . . . . 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 3534 . . . . . . . . . 10  |-  ( x  =  0  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  (/) ) )
28 in0 3645 . . . . . . . . . 10  |-  ( A  i^i  (/) )  =  (/)
2927, 28syl6eq 2483 . . . . . . . . 9  |-  ( x  =  0  ->  ( A  i^i  ( 0..^ x ) )  =  (/) )
3029fveq2d 5724 . . . . . . . 8  |-  ( x  =  0  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  (/) ) )
3130, 19syl6eq 2483 . . . . . . 7  |-  ( x  =  0  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  0 )
324ineq2d 3534 . . . . . . . . . 10  |-  ( x  =  0  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  (/) ) )
33 in0 3645 . . . . . . . . . 10  |-  ( B  i^i  (/) )  =  (/)
3432, 33syl6eq 2483 . . . . . . . . 9  |-  ( x  =  0  ->  ( B  i^i  ( 0..^ x ) )  =  (/) )
3534fveq2d 5724 . . . . . . . 8  |-  ( x  =  0  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  (/) ) )
3635, 19syl6eq 2483 . . . . . . 7  |-  ( x  =  0  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  0 )
3731, 36oveq12d 6091 . . . . . 6  |-  ( x  =  0  ->  (
( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  =  ( 0  +  0 ) )
38 00id 9231 . . . . . 6  |-  ( 0  +  0 )  =  0
3937, 38syl6eq 2483 . . . . 5  |-  ( x  =  0  ->  (
( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  =  0 )
4026, 39eqeq12d 2449 . . . 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 6081 . . . . . . . 8  |-  ( x  =  k  ->  (
0..^ x )  =  ( 0..^ k ) )
4342ineq2d 3534 . . . . . . 7  |-  ( x  =  k  ->  (
( A sadd  B )  i^i  ( 0..^ x ) )  =  ( ( A sadd  B )  i^i  ( 0..^ k ) ) )
4443fveq2d 5724 . . . . . 6  |-  ( x  =  k  ->  ( K `  ( ( A sadd  B )  i^i  (
0..^ x ) ) )  =  ( K `
 ( ( A sadd 
B )  i^i  (
0..^ k ) ) ) )
45 fveq2 5720 . . . . . . . 8  |-  ( x  =  k  ->  ( C `  x )  =  ( C `  k ) )
4645eleq2d 2502 . . . . . . 7  |-  ( x  =  k  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  k ) ) )
47 oveq2 6081 . . . . . . 7  |-  ( x  =  k  ->  (
2 ^ x )  =  ( 2 ^ k ) )
48 eqidd 2436 . . . . . . 7  |-  ( x  =  k  ->  0  =  0 )
4946, 47, 48ifbieq12d 3753 . . . . . 6  |-  ( x  =  k  ->  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 )  =  if ( (/)  e.  ( C `  k
) ,  ( 2 ^ k ) ,  0 ) )
5044, 49oveq12d 6091 . . . . 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 3534 . . . . . . 7  |-  ( x  =  k  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ k ) ) )
5251fveq2d 5724 . . . . . 6  |-  ( x  =  k  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ k ) ) ) )
5342ineq2d 3534 . . . . . . 7  |-  ( x  =  k  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ k ) ) )
5453fveq2d 5724 . . . . . 6  |-  ( x  =  k  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ k ) ) ) )
5552, 54oveq12d 6091 . . . . 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 2449 . . . 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 6081 . . . . . . . 8  |-  ( x  =  ( k  +  1 )  ->  (
0..^ x )  =  ( 0..^ ( k  +  1 ) ) )
5958ineq2d 3534 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  (
( A sadd  B )  i^i  ( 0..^ x ) )  =  ( ( A sadd  B )  i^i  ( 0..^ ( k  +  1 ) ) ) )
6059fveq2d 5724 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( K `  ( ( A sadd  B )  i^i  (
0..^ x ) ) )  =  ( K `
 ( ( A sadd 
B )  i^i  (
0..^ ( k  +  1 ) ) ) ) )
61 fveq2 5720 . . . . . . . 8  |-  ( x  =  ( k  +  1 )  ->  ( C `  x )  =  ( C `  ( k  +  1 ) ) )
6261eleq2d 2502 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  ( k  +  1 ) ) ) )
63 oveq2 6081 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  (
2 ^ x )  =  ( 2 ^ ( k  +  1 ) ) )
64 eqidd 2436 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  0  =  0 )
6562, 63, 64ifbieq12d 3753 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 )  =  if ( (/)  e.  ( C `  (
k  +  1 ) ) ,  ( 2 ^ ( k  +  1 ) ) ,  0 ) )
6660, 65oveq12d 6091 . . . . 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 3534 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) )
6867fveq2d 5724 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) ) )
6958ineq2d 3534 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) )
7069fveq2d 5724 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) ) )
7168, 70oveq12d 6091 . . . . 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 2449 . . . 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 6081 . . . . . . . 8  |-  ( x  =  N  ->  (
0..^ x )  =  ( 0..^ N ) )
7574ineq2d 3534 . . . . . . 7  |-  ( x  =  N  ->  (
( A sadd  B )  i^i  ( 0..^ x ) )  =  ( ( A sadd  B )  i^i  ( 0..^ N ) ) )
7675fveq2d 5724 . . . . . 6  |-  ( x  =  N  ->  ( K `  ( ( A sadd  B )  i^i  (
0..^ x ) ) )  =  ( K `
 ( ( A sadd 
B )  i^i  (
0..^ N ) ) ) )
77 fveq2 5720 . . . . . . . 8  |-  ( x  =  N  ->  ( C `  x )  =  ( C `  N ) )
7877eleq2d 2502 . . . . . . 7  |-  ( x  =  N  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  N ) ) )
79 oveq2 6081 . . . . . . 7  |-  ( x  =  N  ->  (
2 ^ x )  =  ( 2 ^ N ) )
80 eqidd 2436 . . . . . . 7  |-  ( x  =  N  ->  0  =  0 )
8178, 79, 80ifbieq12d 3753 . . . . . 6  |-  ( x  =  N  ->  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 )  =  if ( (/)  e.  ( C `  N
) ,  ( 2 ^ N ) ,  0 ) )
8276, 81oveq12d 6091 . . . . 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 3534 . . . . . . 7  |-  ( x  =  N  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ N ) ) )
8483fveq2d 5724 . . . . . 6  |-  ( x  =  N  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ N ) ) ) )
8574ineq2d 3534 . . . . . . 7  |-  ( x  =  N  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ N ) ) )
8685fveq2d 5724 . . . . . 6  |-  ( x  =  N  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ N ) ) ) )
8784, 86oveq12d 6091 . . . . 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 2449 . . . 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 12956 . . . . . 6  |-  ( ph  ->  -.  (/)  e.  ( C `
 0 ) )
94 iffalse 3738 . . . . . 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 6089 . . . 4  |-  ( ph  ->  ( 0  +  if ( (/)  e.  ( C `
 0 ) ,  ( 2 ^ 0 ) ,  0 ) )  =  ( 0  +  0 ) )
9796, 38syl6eq 2483 . . 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 12961 . . . . . 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 10356 . 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 1388    = wceq 1652    e. wcel 1725    i^i cin 3311    C_ wss 3312   (/)c0 3620   ifcif 3731   ~Pcpw 3791    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   0cc0 8980   1c1 8981    + caddc 8983    - cmin 9281   2c2 10039   NN0cn0 10211  ..^cfzo 11125    seq cseq 11313   ^cexp 11372  bitscbits 12921   sadd csad 12922
This theorem is referenced by:  sadadd3  12963
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 7586  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058
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-had 1389  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 7469  df-card 7816  df-cda 8038  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-n0 10212  df-z 10273  df-uz 10479  df-rp 10603  df-fz 11034  df-fzo 11126  df-fl 11192  df-mod 11241  df-seq 11314  df-exp 11373  df-hash 11609  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-clim 12272  df-sum 12470  df-dvds 12843  df-bits 12924  df-sad 12953
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