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Theorem sadadd2 13775
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 6209 . . . . . . . . . . 11  |-  ( x  =  0  ->  (
0..^ x )  =  ( 0..^ 0 ) )
3 fzo0 11691 . . . . . . . . . . 11  |-  ( 0..^ 0 )  =  (/)
42, 3syl6eq 2511 . . . . . . . . . 10  |-  ( x  =  0  ->  (
0..^ x )  =  (/) )
54ineq2d 3661 . . . . . . . . 9  |-  ( x  =  0  ->  (
( A sadd  B )  i^i  ( 0..^ x ) )  =  ( ( A sadd  B )  i^i  (/) ) )
6 in0 3772 . . . . . . . . 9  |-  ( ( A sadd  B )  i^i  (/) )  =  (/)
75, 6syl6eq 2511 . . . . . . . 8  |-  ( x  =  0  ->  (
( A sadd  B )  i^i  ( 0..^ x ) )  =  (/) )
87fveq2d 5804 . . . . . . 7  |-  ( x  =  0  ->  ( K `  ( ( A sadd  B )  i^i  (
0..^ x ) ) )  =  ( K `
 (/) ) )
9 sadcadd.k . . . . . . . . 9  |-  K  =  `' (bits  |`  NN0 )
10 0nn0 10706 . . . . . . . . . . 11  |-  0  e.  NN0
11 fvres 5814 . . . . . . . . . . 11  |-  ( 0  e.  NN0  ->  ( (bits  |`  NN0 ) `  0
)  =  (bits ` 
0 ) )
1210, 11ax-mp 5 . . . . . . . . . 10  |-  ( (bits  |`  NN0 ) `  0
)  =  (bits ` 
0 )
13 0bits 13754 . . . . . . . . . 10  |-  (bits ` 
0 )  =  (/)
1412, 13eqtr2i 2484 . . . . . . . . 9  |-  (/)  =  ( (bits  |`  NN0 ) ` 
0 )
159, 14fveq12i 5805 . . . . . . . 8  |-  ( K `
 (/) )  =  ( `' (bits  |`  NN0 ) `  ( (bits  |`  NN0 ) `  0 ) )
16 bitsf1o 13760 . . . . . . . . 9  |-  (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )
17 f1ocnvfv1 6093 . . . . . . . . 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 2483 . . . . . . 7  |-  ( K `
 (/) )  =  0
208, 19syl6eq 2511 . . . . . 6  |-  ( x  =  0  ->  ( K `  ( ( A sadd  B )  i^i  (
0..^ x ) ) )  =  0 )
21 fveq2 5800 . . . . . . . 8  |-  ( x  =  0  ->  ( C `  x )  =  ( C ` 
0 ) )
2221eleq2d 2524 . . . . . . 7  |-  ( x  =  0  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  0 ) ) )
23 oveq2 6209 . . . . . . 7  |-  ( x  =  0  ->  (
2 ^ x )  =  ( 2 ^ 0 ) )
2422, 23ifbieq1d 3921 . . . . . 6  |-  ( x  =  0  ->  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 )  =  if ( (/)  e.  ( C `  0
) ,  ( 2 ^ 0 ) ,  0 ) )
2520, 24oveq12d 6219 . . . . 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 3661 . . . . . . . . . 10  |-  ( x  =  0  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  (/) ) )
27 in0 3772 . . . . . . . . . 10  |-  ( A  i^i  (/) )  =  (/)
2826, 27syl6eq 2511 . . . . . . . . 9  |-  ( x  =  0  ->  ( A  i^i  ( 0..^ x ) )  =  (/) )
2928fveq2d 5804 . . . . . . . 8  |-  ( x  =  0  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  (/) ) )
3029, 19syl6eq 2511 . . . . . . 7  |-  ( x  =  0  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  0 )
314ineq2d 3661 . . . . . . . . . 10  |-  ( x  =  0  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  (/) ) )
32 in0 3772 . . . . . . . . . 10  |-  ( B  i^i  (/) )  =  (/)
3331, 32syl6eq 2511 . . . . . . . . 9  |-  ( x  =  0  ->  ( B  i^i  ( 0..^ x ) )  =  (/) )
3433fveq2d 5804 . . . . . . . 8  |-  ( x  =  0  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  (/) ) )
3534, 19syl6eq 2511 . . . . . . 7  |-  ( x  =  0  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  0 )
3630, 35oveq12d 6219 . . . . . 6  |-  ( x  =  0  ->  (
( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  =  ( 0  +  0 ) )
37 00id 9656 . . . . . 6  |-  ( 0  +  0 )  =  0
3836, 37syl6eq 2511 . . . . 5  |-  ( x  =  0  ->  (
( K `  ( A  i^i  ( 0..^ x ) ) )  +  ( K `  ( B  i^i  ( 0..^ x ) ) ) )  =  0 )
3925, 38eqeq12d 2476 . . . 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 6209 . . . . . . . 8  |-  ( x  =  k  ->  (
0..^ x )  =  ( 0..^ k ) )
4241ineq2d 3661 . . . . . . 7  |-  ( x  =  k  ->  (
( A sadd  B )  i^i  ( 0..^ x ) )  =  ( ( A sadd  B )  i^i  ( 0..^ k ) ) )
4342fveq2d 5804 . . . . . 6  |-  ( x  =  k  ->  ( K `  ( ( A sadd  B )  i^i  (
0..^ x ) ) )  =  ( K `
 ( ( A sadd 
B )  i^i  (
0..^ k ) ) ) )
44 fveq2 5800 . . . . . . . 8  |-  ( x  =  k  ->  ( C `  x )  =  ( C `  k ) )
4544eleq2d 2524 . . . . . . 7  |-  ( x  =  k  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  k ) ) )
46 oveq2 6209 . . . . . . 7  |-  ( x  =  k  ->  (
2 ^ x )  =  ( 2 ^ k ) )
4745, 46ifbieq1d 3921 . . . . . 6  |-  ( x  =  k  ->  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 )  =  if ( (/)  e.  ( C `  k
) ,  ( 2 ^ k ) ,  0 ) )
4843, 47oveq12d 6219 . . . . 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 3661 . . . . . . 7  |-  ( x  =  k  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ k ) ) )
5049fveq2d 5804 . . . . . 6  |-  ( x  =  k  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ k ) ) ) )
5141ineq2d 3661 . . . . . . 7  |-  ( x  =  k  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ k ) ) )
5251fveq2d 5804 . . . . . 6  |-  ( x  =  k  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ k ) ) ) )
5350, 52oveq12d 6219 . . . . 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 2476 . . . 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 6209 . . . . . . . 8  |-  ( x  =  ( k  +  1 )  ->  (
0..^ x )  =  ( 0..^ ( k  +  1 ) ) )
5756ineq2d 3661 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  (
( A sadd  B )  i^i  ( 0..^ x ) )  =  ( ( A sadd  B )  i^i  ( 0..^ ( k  +  1 ) ) ) )
5857fveq2d 5804 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( K `  ( ( A sadd  B )  i^i  (
0..^ x ) ) )  =  ( K `
 ( ( A sadd 
B )  i^i  (
0..^ ( k  +  1 ) ) ) ) )
59 fveq2 5800 . . . . . . . 8  |-  ( x  =  ( k  +  1 )  ->  ( C `  x )  =  ( C `  ( k  +  1 ) ) )
6059eleq2d 2524 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  ( k  +  1 ) ) ) )
61 oveq2 6209 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  (
2 ^ x )  =  ( 2 ^ ( k  +  1 ) ) )
6260, 61ifbieq1d 3921 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 )  =  if ( (/)  e.  ( C `  (
k  +  1 ) ) ,  ( 2 ^ ( k  +  1 ) ) ,  0 ) )
6358, 62oveq12d 6219 . . . . 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 3661 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) )
6564fveq2d 5804 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ ( k  +  1 ) ) ) ) )
6656ineq2d 3661 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) )
6766fveq2d 5804 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ ( k  +  1 ) ) ) ) )
6865, 67oveq12d 6219 . . . . 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 2476 . . . 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 6209 . . . . . . . 8  |-  ( x  =  N  ->  (
0..^ x )  =  ( 0..^ N ) )
7271ineq2d 3661 . . . . . . 7  |-  ( x  =  N  ->  (
( A sadd  B )  i^i  ( 0..^ x ) )  =  ( ( A sadd  B )  i^i  ( 0..^ N ) ) )
7372fveq2d 5804 . . . . . 6  |-  ( x  =  N  ->  ( K `  ( ( A sadd  B )  i^i  (
0..^ x ) ) )  =  ( K `
 ( ( A sadd 
B )  i^i  (
0..^ N ) ) ) )
74 fveq2 5800 . . . . . . . 8  |-  ( x  =  N  ->  ( C `  x )  =  ( C `  N ) )
7574eleq2d 2524 . . . . . . 7  |-  ( x  =  N  ->  ( (/) 
e.  ( C `  x )  <->  (/)  e.  ( C `  N ) ) )
76 oveq2 6209 . . . . . . 7  |-  ( x  =  N  ->  (
2 ^ x )  =  ( 2 ^ N ) )
7775, 76ifbieq1d 3921 . . . . . 6  |-  ( x  =  N  ->  if ( (/)  e.  ( C `
 x ) ,  ( 2 ^ x
) ,  0 )  =  if ( (/)  e.  ( C `  N
) ,  ( 2 ^ N ) ,  0 ) )
7873, 77oveq12d 6219 . . . . 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 3661 . . . . . . 7  |-  ( x  =  N  ->  ( A  i^i  ( 0..^ x ) )  =  ( A  i^i  ( 0..^ N ) ) )
8079fveq2d 5804 . . . . . 6  |-  ( x  =  N  ->  ( K `  ( A  i^i  ( 0..^ x ) ) )  =  ( K `  ( A  i^i  ( 0..^ N ) ) ) )
8171ineq2d 3661 . . . . . . 7  |-  ( x  =  N  ->  ( B  i^i  ( 0..^ x ) )  =  ( B  i^i  ( 0..^ N ) ) )
8281fveq2d 5804 . . . . . 6  |-  ( x  =  N  ->  ( K `  ( B  i^i  ( 0..^ x ) ) )  =  ( K `  ( B  i^i  ( 0..^ N ) ) ) )
8380, 82oveq12d 6219 . . . . 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 2476 . . . 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 13769 . . . . . 6  |-  ( ph  ->  -.  (/)  e.  ( C `
 0 ) )
90 iffalse 3908 . . . . . 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 6217 . . . 4  |-  ( ph  ->  ( 0  +  if ( (/)  e.  ( C `
 0 ) ,  ( 2 ^ 0 ) ,  0 ) )  =  ( 0  +  0 ) )
9392, 37syl6eq 2511 . . 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 13774 . . . . . 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 10850 . 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 1370  caddwcad 1421    e. wcel 1758    i^i cin 3436    C_ wss 3437   (/)c0 3746   ifcif 3900   ~Pcpw 3969    |-> cmpt 4459   `'ccnv 4948    |` cres 4951   -1-1-onto->wf1o 5526   ` cfv 5527  (class class class)co 6201    |-> cmpt2 6203   1oc1o 7024   2oc2o 7025   Fincfn 7421   0cc0 9394   1c1 9395    + caddc 9397    - cmin 9707   2c2 10483   NN0cn0 10691  ..^cfzo 11666    seqcseq 11924   ^cexp 11983  bitscbits 13734   sadd csad 13735
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 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-pre-sup 9472
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-had 1422  df-cad 1423  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-disj 4372  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-2o 7032  df-oadd 7035  df-er 7212  df-map 7327  df-pm 7328  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-sup 7803  df-oi 7836  df-card 8221  df-cda 8449  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106  df-nn 10435  df-2 10492  df-3 10493  df-n0 10692  df-z 10759  df-uz 10974  df-rp 11104  df-fz 11556  df-fzo 11667  df-fl 11760  df-mod 11827  df-seq 11925  df-exp 11984  df-hash 12222  df-cj 12707  df-re 12708  df-im 12709  df-sqr 12843  df-abs 12844  df-clim 13085  df-sum 13283  df-dvds 13655  df-bits 13737  df-sad 13766
This theorem is referenced by:  sadadd3  13776
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