Theorem sadadd2 15182

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.

Step	Hyp	Ref	Expression
1		sadcp1.n	. 2 |- ( ph -> N e. NN0 )
2		oveq2 6658	. . . . . . . . . . 11 |- ( x = 0 -> ( 0..^ x ) = ( 0..^ 0 ) )
3		fzo0 12492	. . . . . . . . . . 11 |- ( 0..^ 0 ) = (/)
4	2, 3	syl6eq 2672	. . . . . . . . . 10 |- ( x = 0 -> ( 0..^ x ) = (/) )
5	4	ineq2d 3814	. . . . . . . . 9 |- ( x = 0 -> ( ( A sadd B ) i^i ( 0..^ x ) ) = ( ( A sadd B ) i^i (/) ) )
6		in0 3968	. . . . . . . . 9 |- ( ( A sadd B ) i^i (/) ) = (/)
7	5, 6	syl6eq 2672	. . . . . . . 8 |- ( x = 0 -> ( ( A sadd B ) i^i ( 0..^ x ) ) = (/) )
8	7	fveq2d 6195	. . . . . . 7 |- ( x = 0 -> ( K ` ( ( A sadd B ) i^i ( 0..^ x ) ) ) = ( K ` (/) ) )
9		sadcadd.k	. . . . . . . . 9 |- K = `' (bits |` NN0 )
10		0nn0 11307	. . . . . . . . . . 11 |- 0 e. NN0
11		fvres 6207	. . . . . . . . . . 11 |- ( 0 e. NN0 -> ( (bits |` NN0 ) ` 0 ) = (bits ` 0 ) )
12	10, 11	ax-mp 5	. . . . . . . . . 10 |- ( (bits |` NN0 ) ` 0 ) = (bits ` 0 )
13		0bits 15161	. . . . . . . . . 10 |- (bits ` 0 ) = (/)
14	12, 13	eqtr2i 2645	. . . . . . . . 9 |- (/) = ( (bits |` NN0 ) ` 0 )
15	9, 14	fveq12i 6196	. . . . . . . 8 |- ( K ` (/) ) = ( `' (bits |` NN0 ) ` ( (bits |` NN0 ) ` 0 ) )
16		bitsf1o 15167	. . . . . . . . 9 |- (bits |` NN0 ) : NN0 -1-1-onto-> ( ~P NN0 i^i Fin )
17		f1ocnvfv1 6532	. . . . . . . . 9 |- ( ( (bits |` NN0 ) : NN0 -1-1-onto-> ( ~P NN0 i^i Fin ) /\ 0 e. NN0 ) -> ( `' (bits |` NN0 ) ` ( (bits |` NN0 ) ` 0 ) ) = 0 )
18	16, 10, 17	mp2an 708	. . . . . . . 8 |- ( `' (bits |` NN0 ) ` ( (bits |` NN0 ) ` 0 ) ) = 0
19	15, 18	eqtri 2644	. . . . . . 7 |- ( K ` (/) ) = 0
20	8, 19	syl6eq 2672	. . . . . 6 |- ( x = 0 -> ( K ` ( ( A sadd B ) i^i ( 0..^ x ) ) ) = 0 )
21		fveq2 6191	. . . . . . . 8 |- ( x = 0 -> ( C ` x ) = ( C ` 0 ) )
22	21	eleq2d 2687	. . . . . . 7 |- ( x = 0 -> ( (/) e. ( C ` x ) <-> (/) e. ( C ` 0 ) ) )
23		oveq2 6658	. . . . . . 7 |- ( x = 0 -> ( 2 ^ x ) = ( 2 ^ 0 ) )
24	22, 23	ifbieq1d 4109	. . . . . 6 |- ( x = 0 -> if ( (/) e. ( C ` x ) , ( 2 ^ x ) , 0 ) = if ( (/) e. ( C ` 0 ) , ( 2 ^ 0 ) , 0 ) )
25	20, 24	oveq12d 6668	. . . . 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 ) ) )
26	4	ineq2d 3814	. . . . . . . . . 10 |- ( x = 0 -> ( A i^i ( 0..^ x ) ) = ( A i^i (/) ) )
27		in0 3968	. . . . . . . . . 10 |- ( A i^i (/) ) = (/)
28	26, 27	syl6eq 2672	. . . . . . . . 9 |- ( x = 0 -> ( A i^i ( 0..^ x ) ) = (/) )
29	28	fveq2d 6195	. . . . . . . 8 |- ( x = 0 -> ( K ` ( A i^i ( 0..^ x ) ) ) = ( K ` (/) ) )
30	29, 19	syl6eq 2672	. . . . . . 7 |- ( x = 0 -> ( K ` ( A i^i ( 0..^ x ) ) ) = 0 )
31	4	ineq2d 3814	. . . . . . . . . 10 |- ( x = 0 -> ( B i^i ( 0..^ x ) ) = ( B i^i (/) ) )
32		in0 3968	. . . . . . . . . 10 |- ( B i^i (/) ) = (/)
33	31, 32	syl6eq 2672	. . . . . . . . 9 |- ( x = 0 -> ( B i^i ( 0..^ x ) ) = (/) )
34	33	fveq2d 6195	. . . . . . . 8 |- ( x = 0 -> ( K ` ( B i^i ( 0..^ x ) ) ) = ( K ` (/) ) )
35	34, 19	syl6eq 2672	. . . . . . 7 |- ( x = 0 -> ( K ` ( B i^i ( 0..^ x ) ) ) = 0 )
36	30, 35	oveq12d 6668	. . . . . 6 |- ( x = 0 -> ( ( K ` ( A i^i ( 0..^ x ) ) ) + ( K ` ( B i^i ( 0..^ x ) ) ) ) = ( 0 + 0 ) )
37		00id 10211	. . . . . 6 |- ( 0 + 0 ) = 0
38	36, 37	syl6eq 2672	. . . . 5 |- ( x = 0 -> ( ( K ` ( A i^i ( 0..^ x ) ) ) + ( K ` ( B i^i ( 0..^ x ) ) ) ) = 0 )
39	25, 38	eqeq12d 2637	. . . 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 ) )
40	39	imbi2d 330	. . 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 6658	. . . . . . . 8 |- ( x = k -> ( 0..^ x ) = ( 0..^ k ) )
42	41	ineq2d 3814	. . . . . . 7 |- ( x = k -> ( ( A sadd B ) i^i ( 0..^ x ) ) = ( ( A sadd B ) i^i ( 0..^ k ) ) )
43	42	fveq2d 6195	. . . . . 6 |- ( x = k -> ( K ` ( ( A sadd B ) i^i ( 0..^ x ) ) ) = ( K ` ( ( A sadd B ) i^i ( 0..^ k ) ) ) )
44		fveq2 6191	. . . . . . . 8 |- ( x = k -> ( C ` x ) = ( C ` k ) )
45	44	eleq2d 2687	. . . . . . 7 |- ( x = k -> ( (/) e. ( C ` x ) <-> (/) e. ( C ` k ) ) )
46		oveq2 6658	. . . . . . 7 |- ( x = k -> ( 2 ^ x ) = ( 2 ^ k ) )
47	45, 46	ifbieq1d 4109	. . . . . 6 |- ( x = k -> if ( (/) e. ( C ` x ) , ( 2 ^ x ) , 0 ) = if ( (/) e. ( C ` k ) , ( 2 ^ k ) , 0 ) )
48	43, 47	oveq12d 6668	. . . . 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 ) ) )
49	41	ineq2d 3814	. . . . . . 7 |- ( x = k -> ( A i^i ( 0..^ x ) ) = ( A i^i ( 0..^ k ) ) )
50	49	fveq2d 6195	. . . . . 6 |- ( x = k -> ( K ` ( A i^i ( 0..^ x ) ) ) = ( K ` ( A i^i ( 0..^ k ) ) ) )
51	41	ineq2d 3814	. . . . . . 7 |- ( x = k -> ( B i^i ( 0..^ x ) ) = ( B i^i ( 0..^ k ) ) )
52	51	fveq2d 6195	. . . . . 6 |- ( x = k -> ( K ` ( B i^i ( 0..^ x ) ) ) = ( K ` ( B i^i ( 0..^ k ) ) ) )
53	50, 52	oveq12d 6668	. . . . 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 ) ) ) ) )
54	48, 53	eqeq12d 2637	. . . 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 ) ) ) ) ) )
55	54	imbi2d 330	. . 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 6658	. . . . . . . 8 |- ( x = ( k + 1 ) -> ( 0..^ x ) = ( 0..^ ( k + 1 ) ) )
57	56	ineq2d 3814	. . . . . . 7 |- ( x = ( k + 1 ) -> ( ( A sadd B ) i^i ( 0..^ x ) ) = ( ( A sadd B ) i^i ( 0..^ ( k + 1 ) ) ) )
58	57	fveq2d 6195	. . . . . 6 |- ( x = ( k + 1 ) -> ( K ` ( ( A sadd B ) i^i ( 0..^ x ) ) ) = ( K ` ( ( A sadd B ) i^i ( 0..^ ( k + 1 ) ) ) ) )
59		fveq2 6191	. . . . . . . 8 |- ( x = ( k + 1 ) -> ( C ` x ) = ( C ` ( k + 1 ) ) )
60	59	eleq2d 2687	. . . . . . 7 |- ( x = ( k + 1 ) -> ( (/) e. ( C ` x ) <-> (/) e. ( C ` ( k + 1 ) ) ) )
61		oveq2 6658	. . . . . . 7 |- ( x = ( k + 1 ) -> ( 2 ^ x ) = ( 2 ^ ( k + 1 ) ) )
62	60, 61	ifbieq1d 4109	. . . . . 6 |- ( x = ( k + 1 ) -> if ( (/) e. ( C ` x ) , ( 2 ^ x ) , 0 ) = if ( (/) e. ( C ` ( k + 1 ) ) , ( 2 ^ ( k + 1 ) ) , 0 ) )
63	58, 62	oveq12d 6668	. . . . 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 ) ) )
64	56	ineq2d 3814	. . . . . . 7 |- ( x = ( k + 1 ) -> ( A i^i ( 0..^ x ) ) = ( A i^i ( 0..^ ( k + 1 ) ) ) )
65	64	fveq2d 6195	. . . . . 6 |- ( x = ( k + 1 ) -> ( K ` ( A i^i ( 0..^ x ) ) ) = ( K ` ( A i^i ( 0..^ ( k + 1 ) ) ) ) )
66	56	ineq2d 3814	. . . . . . 7 |- ( x = ( k + 1 ) -> ( B i^i ( 0..^ x ) ) = ( B i^i ( 0..^ ( k + 1 ) ) ) )
67	66	fveq2d 6195	. . . . . 6 |- ( x = ( k + 1 ) -> ( K ` ( B i^i ( 0..^ x ) ) ) = ( K ` ( B i^i ( 0..^ ( k + 1 ) ) ) ) )
68	65, 67	oveq12d 6668	. . . . 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 ) ) ) ) ) )
69	63, 68	eqeq12d 2637	. . . 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 ) ) ) ) ) ) )
70	69	imbi2d 330	. . 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 6658	. . . . . . . 8 |- ( x = N -> ( 0..^ x ) = ( 0..^ N ) )
72	71	ineq2d 3814	. . . . . . 7 |- ( x = N -> ( ( A sadd B ) i^i ( 0..^ x ) ) = ( ( A sadd B ) i^i ( 0..^ N ) ) )
73	72	fveq2d 6195	. . . . . 6 |- ( x = N -> ( K ` ( ( A sadd B ) i^i ( 0..^ x ) ) ) = ( K ` ( ( A sadd B ) i^i ( 0..^ N ) ) ) )
74		fveq2 6191	. . . . . . . 8 |- ( x = N -> ( C ` x ) = ( C ` N ) )
75	74	eleq2d 2687	. . . . . . 7 |- ( x = N -> ( (/) e. ( C ` x ) <-> (/) e. ( C ` N ) ) )
76		oveq2 6658	. . . . . . 7 |- ( x = N -> ( 2 ^ x ) = ( 2 ^ N ) )
77	75, 76	ifbieq1d 4109	. . . . . 6 |- ( x = N -> if ( (/) e. ( C ` x ) , ( 2 ^ x ) , 0 ) = if ( (/) e. ( C ` N ) , ( 2 ^ N ) , 0 ) )
78	73, 77	oveq12d 6668	. . . . 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 ) ) )
79	71	ineq2d 3814	. . . . . . 7 |- ( x = N -> ( A i^i ( 0..^ x ) ) = ( A i^i ( 0..^ N ) ) )
80	79	fveq2d 6195	. . . . . 6 |- ( x = N -> ( K ` ( A i^i ( 0..^ x ) ) ) = ( K ` ( A i^i ( 0..^ N ) ) ) )
81	71	ineq2d 3814	. . . . . . 7 |- ( x = N -> ( B i^i ( 0..^ x ) ) = ( B i^i ( 0..^ N ) ) )
82	81	fveq2d 6195	. . . . . 6 |- ( x = N -> ( K ` ( B i^i ( 0..^ x ) ) ) = ( K ` ( B i^i ( 0..^ N ) ) ) )
83	80, 82	oveq12d 6668	. . . . 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 ) ) ) ) )
84	78, 83	eqeq12d 2637	. . . 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 ) ) ) ) ) )
85	84	imbi2d 330	. . 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 ) ) ) )
89	86, 87, 88	sadc0 15176	. . . . . 6 |- ( ph -> -. (/) e. ( C ` 0 ) )
90	89	iffalsed 4097	. . . . 5 |- ( ph -> if ( (/) e. ( C ` 0 ) , ( 2 ^ 0 ) , 0 ) = 0 )
91	90	oveq2d 6666	. . . 4 |- ( ph -> ( 0 + if ( (/) e. ( C ` 0 ) , ( 2 ^ 0 ) , 0 ) ) = ( 0 + 0 ) )
92	91, 37	syl6eq 2672	. . 3 |- ( ph -> ( 0 + if ( (/) e. ( C ` 0 ) , ( 2 ^ 0 ) , 0 ) ) = 0 )
93	86	ad2antrr 762	. . . . . . 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 )
94	87	ad2antrr 762	. . . . . . 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 )
95		simplr 792	. . . . . . 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 )
96		simpr 477	. . . . . . 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 ) ) ) ) )
97	93, 94, 88, 95, 9, 96	sadadd2lem 15181	. . . . . 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 ) ) ) ) ) )
98	97	ex 450	. . . . 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 ) ) ) ) ) ) )
99	98	expcom 451	. . . 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 ) ) ) ) ) ) ) )
100	99	a2d 29	. . 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 ) ) ) ) ) ) ) )
101	40, 55, 70, 85, 92, 100	nn0ind 11472	. 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 ) ) ) ) ) )
102	1, 101	mpcom 38	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 ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483  caddwcad 1545   e. wcel 1990   i^i cin 3573   C_ wss 3574   (/)c0 3915   ifcif 4086   ~Pcpw 4158   |-> cmpt 4729   `'ccnv 5113   |` cres 5116   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1oc1o 7553   2oc2o 7554   Fincfn 7955   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   2c2 11070   NN0cn0 11292  ..^cfzo 12465   seqcseq 12801   ^cexp 12860  bitscbits 15141   sadd csad 15142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-xor 1465  df-tru 1486  df-fal 1489  df-had 1533  df-cad 1546  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984  df-bits 15144  df-sad 15173

This theorem is referenced by:  sadadd3  15183