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Theorem sadadd2 13958
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 6283 . . . . . . . . . . 11  |-  ( x  =  0  ->  (
0..^ x )  =  ( 0..^ 0 ) )
3 fzo0 11806 . . . . . . . . . . 11  |-  ( 0..^ 0 )  =  (/)
42, 3syl6eq 2517 . . . . . . . . . 10  |-  ( x  =  0  ->  (
0..^ x )  =  (/) )
54ineq2d 3693 . . . . . . . . 9  |-  ( x  =  0  ->  (
( A sadd  B )  i^i  ( 0..^ x ) )  =  ( ( A sadd  B )  i^i  (/) ) )
6 in0 3804 . . . . . . . . 9  |-  ( ( A sadd  B )  i^i  (/) )  =  (/)
75, 6syl6eq 2517 . . . . . . . 8  |-  ( x  =  0  ->  (
( A sadd  B )  i^i  ( 0..^ x ) )  =  (/) )
87fveq2d 5861 . . . . . . 7  |-  ( x  =  0  ->  ( K `  ( ( A sadd  B )  i^i  (
0..^ x ) ) )  =  ( K `
 (/) ) )
9 sadcadd.k . . . . . . . . 9  |-  K  =  `' (bits  |`  NN0 )
10 0nn0 10799 . . . . . . . . . . 11  |-  0  e.  NN0
11 fvres 5871 . . . . . . . . . . 11  |-  ( 0  e.  NN0  ->  ( (bits  |`  NN0 ) `  0
)  =  (bits ` 
0 ) )
1210, 11ax-mp 5 . . . . . . . . . 10  |-  ( (bits  |`  NN0 ) `  0
)  =  (bits ` 
0 )
13 0bits 13937 . . . . . . . . . 10  |-  (bits ` 
0 )  =  (/)
1412, 13eqtr2i 2490 . . . . . . . . 9  |-  (/)  =  ( (bits  |`  NN0 ) ` 
0 )
159, 14fveq12i 5862 . . . . . . . 8  |-  ( K `
 (/) )  =  ( `' (bits  |`  NN0 ) `  ( (bits  |`  NN0 ) `  0 ) )
16 bitsf1o 13943 . . . . . . . . 9  |-  (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )
17 f1ocnvfv1 6161 . . . . . . . . 9  |-  ( ( (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )  /\  0  e.  NN0 )  ->  ( `' (bits  |`  NN0 ) `  (
(bits  |`  NN0 ) ` 
0 ) )  =  0 )
1816, 10, 17mp2an 672 . . . . . . . 8  |-  ( `' (bits  |`  NN0 ) `  ( (bits  |`  NN0 ) `  0 ) )  =  0
1915, 18eqtri 2489 . . . . . . 7  |-  ( K `
 (/) )  =  0
208, 19syl6eq 2517 . . . . . 6  |-  ( x  =  0  ->  ( K `  ( ( A sadd  B )  i^i  (
0..^ x ) ) )  =  0 )
21 fveq2 5857 . . . . . . . 8  |-  ( x  =  0  ->  ( C `  x )  =  ( C ` 
0 ) )
2221eleq2d 2530 . . . . . . 7  |-  ( x  =  0  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  0 ) ) )
23 oveq2 6283 . . . . . . 7  |-  ( x  =  0  ->  (
2 ^ x )  =  ( 2 ^ 0 ) )
2422, 23ifbieq1d 3955 . . . . . 6  |-  ( x  =  0  ->  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 )  =  if ( (/)  e.  ( C `  0
) ,  ( 2 ^ 0 ) ,  0 ) )
2520, 24oveq12d 6293 . . . . 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 ) ) )
264ineq2d 3693 . . . . . . . . . 10  |-  ( x  =  0  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  (/) ) )
27 in0 3804 . . . . . . . . . 10  |-  ( A  i^i  (/) )  =  (/)
2826, 27syl6eq 2517 . . . . . . . . 9  |-  ( x  =  0  ->  ( A  i^i  ( 0..^ x ) )  =  (/) )
2928fveq2d 5861 . . . . . . . 8  |-  ( x  =  0  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  (/) ) )
3029, 19syl6eq 2517 . . . . . . 7  |-  ( x  =  0  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  0 )
314ineq2d 3693 . . . . . . . . . 10  |-  ( x  =  0  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  (/) ) )
32 in0 3804 . . . . . . . . . 10  |-  ( B  i^i  (/) )  =  (/)
3331, 32syl6eq 2517 . . . . . . . . 9  |-  ( x  =  0  ->  ( B  i^i  ( 0..^ x ) )  =  (/) )
3433fveq2d 5861 . . . . . . . 8  |-  ( x  =  0  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  (/) ) )
3534, 19syl6eq 2517 . . . . . . 7  |-  ( x  =  0  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  0 )
3630, 35oveq12d 6293 . . . . . 6  |-  ( x  =  0  ->  (
( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  =  ( 0  +  0 ) )
37 00id 9743 . . . . . 6  |-  ( 0  +  0 )  =  0
3836, 37syl6eq 2517 . . . . 5  |-  ( x  =  0  ->  (
( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  =  0 )
3925, 38eqeq12d 2482 . . . 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 ) )
4039imbi2d 316 . . 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 ) ) )
41 oveq2 6283 . . . . . . . 8  |-  ( x  =  k  ->  (
0..^ x )  =  ( 0..^ k ) )
4241ineq2d 3693 . . . . . . 7  |-  ( x  =  k  ->  (
( A sadd  B )  i^i  ( 0..^ x ) )  =  ( ( A sadd  B )  i^i  ( 0..^ k ) ) )
4342fveq2d 5861 . . . . . 6  |-  ( x  =  k  ->  ( K `  ( ( A sadd  B )  i^i  (
0..^ x ) ) )  =  ( K `
 ( ( A sadd 
B )  i^i  (
0..^ k ) ) ) )
44 fveq2 5857 . . . . . . . 8  |-  ( x  =  k  ->  ( C `  x )  =  ( C `  k ) )
4544eleq2d 2530 . . . . . . 7  |-  ( x  =  k  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  k ) ) )
46 oveq2 6283 . . . . . . 7  |-  ( x  =  k  ->  (
2 ^ x )  =  ( 2 ^ k ) )
4745, 46ifbieq1d 3955 . . . . . 6  |-  ( x  =  k  ->  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 )  =  if ( (/)  e.  ( C `  k
) ,  ( 2 ^ k ) ,  0 ) )
4843, 47oveq12d 6293 . . . . 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 ) ) )
4941ineq2d 3693 . . . . . . 7  |-  ( x  =  k  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ k ) ) )
5049fveq2d 5861 . . . . . 6  |-  ( x  =  k  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ k ) ) ) )
5141ineq2d 3693 . . . . . . 7  |-  ( x  =  k  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ k ) ) )
5251fveq2d 5861 . . . . . 6  |-  ( x  =  k  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ k ) ) ) )
5350, 52oveq12d 6293 . . . . 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 ) ) ) ) )
5448, 53eqeq12d 2482 . . . 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 ) ) ) ) ) )
5554imbi2d 316 . . 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 ) ) ) ) ) ) )
56 oveq2 6283 . . . . . . . 8  |-  ( x  =  ( k  +  1 )  ->  (
0..^ x )  =  ( 0..^ ( k  +  1 ) ) )
5756ineq2d 3693 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  (
( A sadd  B )  i^i  ( 0..^ x ) )  =  ( ( A sadd  B )  i^i  ( 0..^ ( k  +  1 ) ) ) )
5857fveq2d 5861 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( K `  ( ( A sadd  B )  i^i  (
0..^ x ) ) )  =  ( K `
 ( ( A sadd 
B )  i^i  (
0..^ ( k  +  1 ) ) ) ) )
59 fveq2 5857 . . . . . . . 8  |-  ( x  =  ( k  +  1 )  ->  ( C `  x )  =  ( C `  ( k  +  1 ) ) )
6059eleq2d 2530 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  ( k  +  1 ) ) ) )
61 oveq2 6283 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  (
2 ^ x )  =  ( 2 ^ ( k  +  1 ) ) )
6260, 61ifbieq1d 3955 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 )  =  if ( (/)  e.  ( C `  (
k  +  1 ) ) ,  ( 2 ^ ( k  +  1 ) ) ,  0 ) )
6358, 62oveq12d 6293 . . . . 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 ) ) )
6456ineq2d 3693 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) )
6564fveq2d 5861 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) ) )
6656ineq2d 3693 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) )
6766fveq2d 5861 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) ) )
6865, 67oveq12d 6293 . . . . 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 ) ) ) ) ) )
6963, 68eqeq12d 2482 . . . 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 ) ) ) ) ) ) )
7069imbi2d 316 . . 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 ) ) ) ) ) ) ) )
71 oveq2 6283 . . . . . . . 8  |-  ( x  =  N  ->  (
0..^ x )  =  ( 0..^ N ) )
7271ineq2d 3693 . . . . . . 7  |-  ( x  =  N  ->  (
( A sadd  B )  i^i  ( 0..^ x ) )  =  ( ( A sadd  B )  i^i  ( 0..^ N ) ) )
7372fveq2d 5861 . . . . . 6  |-  ( x  =  N  ->  ( K `  ( ( A sadd  B )  i^i  (
0..^ x ) ) )  =  ( K `
 ( ( A sadd 
B )  i^i  (
0..^ N ) ) ) )
74 fveq2 5857 . . . . . . . 8  |-  ( x  =  N  ->  ( C `  x )  =  ( C `  N ) )
7574eleq2d 2530 . . . . . . 7  |-  ( x  =  N  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  N ) ) )
76 oveq2 6283 . . . . . . 7  |-  ( x  =  N  ->  (
2 ^ x )  =  ( 2 ^ N ) )
7775, 76ifbieq1d 3955 . . . . . 6  |-  ( x  =  N  ->  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 )  =  if ( (/)  e.  ( C `  N
) ,  ( 2 ^ N ) ,  0 ) )
7873, 77oveq12d 6293 . . . . 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 ) ) )
7971ineq2d 3693 . . . . . . 7  |-  ( x  =  N  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ N ) ) )
8079fveq2d 5861 . . . . . 6  |-  ( x  =  N  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ N ) ) ) )
8171ineq2d 3693 . . . . . . 7  |-  ( x  =  N  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ N ) ) )
8281fveq2d 5861 . . . . . 6  |-  ( x  =  N  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ N ) ) ) )
8380, 82oveq12d 6293 . . . . 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 ) ) ) ) )
8478, 83eqeq12d 2482 . . . 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 ) ) ) ) ) )
8584imbi2d 316 . . 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 ) ) ) ) ) ) )
86 sadval.a . . . . . . 7  |-  ( ph  ->  A  C_  NN0 )
87 sadval.b . . . . . . 7  |-  ( ph  ->  B  C_  NN0 )
88 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 ) ) ) )
8986, 87, 88sadc0 13952 . . . . . 6  |-  ( ph  ->  -.  (/)  e.  ( C `
 0 ) )
90 iffalse 3941 . . . . . 6  |-  ( -.  (/)  e.  ( C ` 
0 )  ->  if ( (/)  e.  ( C `
 0 ) ,  ( 2 ^ 0 ) ,  0 )  =  0 )
9189, 90syl 16 . . . . 5  |-  ( ph  ->  if ( (/)  e.  ( C `  0 ) ,  ( 2 ^ 0 ) ,  0 )  =  0 )
9291oveq2d 6291 . . . 4  |-  ( ph  ->  ( 0  +  if ( (/)  e.  ( C `
 0 ) ,  ( 2 ^ 0 ) ,  0 ) )  =  ( 0  +  0 ) )
9392, 37syl6eq 2517 . . 3  |-  ( ph  ->  ( 0  +  if ( (/)  e.  ( C `
 0 ) ,  ( 2 ^ 0 ) ,  0 ) )  =  0 )
9486ad2antrr 725 . . . . . . 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 )
9587ad2antrr 725 . . . . . . 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 )
96 simplr 754 . . . . . . 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 )
97 simpr 461 . . . . . . 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 ) ) ) ) )
9894, 95, 88, 96, 9, 97sadadd2lem 13957 . . . . . 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 ) ) ) ) ) )
9998ex 434 . . . . 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 ) ) ) ) ) ) )
10099expcom 435 . . . 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 ) ) ) ) ) ) ) )
101100a2d 26 . . 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 ) ) ) ) ) ) ) )
10240, 55, 70, 85, 93, 101nn0ind 10946 . 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 ) ) ) ) ) )
1031, 102mpcom 36 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1374  caddwcad 1425    e. wcel 1762    i^i cin 3468    C_ wss 3469   (/)c0 3778   ifcif 3932   ~Pcpw 4003    |-> cmpt 4498   `'ccnv 4991    |` cres 4994   -1-1-onto->wf1o 5578   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   1oc1o 7113   2oc2o 7114   Fincfn 7506   0cc0 9481   1c1 9482    + caddc 9484    - cmin 9794   2c2 10574   NN0cn0 10784  ..^cfzo 11781    seqcseq 12063   ^cexp 12122  bitscbits 13917   sadd csad 13918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-xor 1356  df-tru 1377  df-fal 1380  df-had 1426  df-cad 1427  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-disj 4411  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-fz 11662  df-fzo 11782  df-fl 11886  df-mod 11953  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-sum 13458  df-dvds 13837  df-bits 13920  df-sad 13949
This theorem is referenced by:  sadadd3  13959
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