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.

Step	Hyp	Ref	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 ) = (/)
4	2, 3	syl6eq 2511	. . . . . . . . . 10 |- ( x = 0 -> ( 0..^ x ) = (/) )
5	4	ineq2d 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 (/) ) = (/)
7	5, 6	syl6eq 2511	. . . . . . . 8 |- ( x = 0 -> ( ( A sadd B ) i^i ( 0..^ x ) ) = (/) )
8	7	fveq2d 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 ) )
12	10, 11	ax-mp 5	. . . . . . . . . 10 |- ( (bits |` NN0 ) ` 0 ) = (bits ` 0 )
13		0bits 13754	. . . . . . . . . 10 |- (bits ` 0 ) = (/)
14	12, 13	eqtr2i 2484	. . . . . . . . 9 |- (/) = ( (bits |` NN0 ) ` 0 )
15	9, 14	fveq12i 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 )
18	16, 10, 17	mp2an 672	. . . . . . . 8 |- ( `' (bits |` NN0 ) ` ( (bits |` NN0 ) ` 0 ) ) = 0
19	15, 18	eqtri 2483	. . . . . . 7 |- ( K ` (/) ) = 0
20	8, 19	syl6eq 2511	. . . . . 6 |- ( x = 0 -> ( K ` ( ( A sadd B ) i^i ( 0..^ x ) ) ) = 0 )
21		fveq2 5800	. . . . . . . 8 |- ( x = 0 -> ( C ` x ) = ( C ` 0 ) )
22	21	eleq2d 2524	. . . . . . 7 |- ( x = 0 -> ( (/) e. ( C ` x ) <-> (/) e. ( C ` 0 ) ) )
23		oveq2 6209	. . . . . . 7 |- ( x = 0 -> ( 2 ^ x ) = ( 2 ^ 0 ) )
24	22, 23	ifbieq1d 3921	. . . . . 6 |- ( x = 0 -> if ( (/) e. ( C ` x ) , ( 2 ^ x ) , 0 ) = if ( (/) e. ( C ` 0 ) , ( 2 ^ 0 ) , 0 ) )
25	20, 24	oveq12d 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 ) ) )
26	4	ineq2d 3661	. . . . . . . . . 10 |- ( x = 0 -> ( A i^i ( 0..^ x ) ) = ( A i^i (/) ) )
27		in0 3772	. . . . . . . . . 10 |- ( A i^i (/) ) = (/)
28	26, 27	syl6eq 2511	. . . . . . . . 9 |- ( x = 0 -> ( A i^i ( 0..^ x ) ) = (/) )
29	28	fveq2d 5804	. . . . . . . 8 |- ( x = 0 -> ( K ` ( A i^i ( 0..^ x ) ) ) = ( K ` (/) ) )
30	29, 19	syl6eq 2511	. . . . . . 7 |- ( x = 0 -> ( K ` ( A i^i ( 0..^ x ) ) ) = 0 )
31	4	ineq2d 3661	. . . . . . . . . 10 |- ( x = 0 -> ( B i^i ( 0..^ x ) ) = ( B i^i (/) ) )
32		in0 3772	. . . . . . . . . 10 |- ( B i^i (/) ) = (/)
33	31, 32	syl6eq 2511	. . . . . . . . 9 |- ( x = 0 -> ( B i^i ( 0..^ x ) ) = (/) )
34	33	fveq2d 5804	. . . . . . . 8 |- ( x = 0 -> ( K ` ( B i^i ( 0..^ x ) ) ) = ( K ` (/) ) )
35	34, 19	syl6eq 2511	. . . . . . 7 |- ( x = 0 -> ( K ` ( B i^i ( 0..^ x ) ) ) = 0 )
36	30, 35	oveq12d 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
38	36, 37	syl6eq 2511	. . . . 5 |- ( x = 0 -> ( ( K ` ( A i^i ( 0..^ x ) ) ) + ( K ` ( B i^i ( 0..^ x ) ) ) ) = 0 )
39	25, 38	eqeq12d 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 ) )
40	39	imbi2d 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 ) )
42	41	ineq2d 3661	. . . . . . 7 |- ( x = k -> ( ( A sadd B ) i^i ( 0..^ x ) ) = ( ( A sadd B ) i^i ( 0..^ k ) ) )
43	42	fveq2d 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 ) )
45	44	eleq2d 2524	. . . . . . 7 |- ( x = k -> ( (/) e. ( C ` x ) <-> (/) e. ( C ` k ) ) )
46		oveq2 6209	. . . . . . 7 |- ( x = k -> ( 2 ^ x ) = ( 2 ^ k ) )
47	45, 46	ifbieq1d 3921	. . . . . 6 |- ( x = k -> if ( (/) e. ( C ` x ) , ( 2 ^ x ) , 0 ) = if ( (/) e. ( C ` k ) , ( 2 ^ k ) , 0 ) )
48	43, 47	oveq12d 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 ) ) )
49	41	ineq2d 3661	. . . . . . 7 |- ( x = k -> ( A i^i ( 0..^ x ) ) = ( A i^i ( 0..^ k ) ) )
50	49	fveq2d 5804	. . . . . 6 |- ( x = k -> ( K ` ( A i^i ( 0..^ x ) ) ) = ( K ` ( A i^i ( 0..^ k ) ) ) )
51	41	ineq2d 3661	. . . . . . 7 |- ( x = k -> ( B i^i ( 0..^ x ) ) = ( B i^i ( 0..^ k ) ) )
52	51	fveq2d 5804	. . . . . 6 |- ( x = k -> ( K ` ( B i^i ( 0..^ x ) ) ) = ( K ` ( B i^i ( 0..^ k ) ) ) )
53	50, 52	oveq12d 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 ) ) ) ) )
54	48, 53	eqeq12d 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 ) ) ) ) ) )
55	54	imbi2d 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 ) ) )
57	56	ineq2d 3661	. . . . . . 7 |- ( x = ( k + 1 ) -> ( ( A sadd B ) i^i ( 0..^ x ) ) = ( ( A sadd B ) i^i ( 0..^ ( k + 1 ) ) ) )
58	57	fveq2d 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 ) ) )
60	59	eleq2d 2524	. . . . . . 7 |- ( x = ( k + 1 ) -> ( (/) e. ( C ` x ) <-> (/) e. ( C ` ( k + 1 ) ) ) )
61		oveq2 6209	. . . . . . 7 |- ( x = ( k + 1 ) -> ( 2 ^ x ) = ( 2 ^ ( k + 1 ) ) )
62	60, 61	ifbieq1d 3921	. . . . . 6 |- ( x = ( k + 1 ) -> if ( (/) e. ( C ` x ) , ( 2 ^ x ) , 0 ) = if ( (/) e. ( C ` ( k + 1 ) ) , ( 2 ^ ( k + 1 ) ) , 0 ) )
63	58, 62	oveq12d 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 ) ) )
64	56	ineq2d 3661	. . . . . . 7 |- ( x = ( k + 1 ) -> ( A i^i ( 0..^ x ) ) = ( A i^i ( 0..^ ( k + 1 ) ) ) )
65	64	fveq2d 5804	. . . . . 6 |- ( x = ( k + 1 ) -> ( K ` ( A i^i ( 0..^ x ) ) ) = ( K ` ( A i^i ( 0..^ ( k + 1 ) ) ) ) )
66	56	ineq2d 3661	. . . . . . 7 |- ( x = ( k + 1 ) -> ( B i^i ( 0..^ x ) ) = ( B i^i ( 0..^ ( k + 1 ) ) ) )
67	66	fveq2d 5804	. . . . . 6 |- ( x = ( k + 1 ) -> ( K ` ( B i^i ( 0..^ x ) ) ) = ( K ` ( B i^i ( 0..^ ( k + 1 ) ) ) ) )
68	65, 67	oveq12d 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 ) ) ) ) ) )
69	63, 68	eqeq12d 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 ) ) ) ) ) ) )
70	69	imbi2d 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 ) )
72	71	ineq2d 3661	. . . . . . 7 |- ( x = N -> ( ( A sadd B ) i^i ( 0..^ x ) ) = ( ( A sadd B ) i^i ( 0..^ N ) ) )
73	72	fveq2d 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 ) )
75	74	eleq2d 2524	. . . . . . 7 |- ( x = N -> ( (/) e. ( C ` x ) <-> (/) e. ( C ` N ) ) )
76		oveq2 6209	. . . . . . 7 |- ( x = N -> ( 2 ^ x ) = ( 2 ^ N ) )
77	75, 76	ifbieq1d 3921	. . . . . 6 |- ( x = N -> if ( (/) e. ( C ` x ) , ( 2 ^ x ) , 0 ) = if ( (/) e. ( C ` N ) , ( 2 ^ N ) , 0 ) )
78	73, 77	oveq12d 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 ) ) )
79	71	ineq2d 3661	. . . . . . 7 |- ( x = N -> ( A i^i ( 0..^ x ) ) = ( A i^i ( 0..^ N ) ) )
80	79	fveq2d 5804	. . . . . 6 |- ( x = N -> ( K ` ( A i^i ( 0..^ x ) ) ) = ( K ` ( A i^i ( 0..^ N ) ) ) )
81	71	ineq2d 3661	. . . . . . 7 |- ( x = N -> ( B i^i ( 0..^ x ) ) = ( B i^i ( 0..^ N ) ) )
82	81	fveq2d 5804	. . . . . 6 |- ( x = N -> ( K ` ( B i^i ( 0..^ x ) ) ) = ( K ` ( B i^i ( 0..^ N ) ) ) )
83	80, 82	oveq12d 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 ) ) ) ) )
84	78, 83	eqeq12d 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 ) ) ) ) ) )
85	84	imbi2d 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 ) ) ) )
89	86, 87, 88	sadc0 13769	. . . . . 6 |- ( ph -> -. (/) e. ( C ` 0 ) )
90		iffalse 3908	. . . . . 6 |- ( -. (/) e. ( C ` 0 ) -> if ( (/) e. ( C ` 0 ) , ( 2 ^ 0 ) , 0 ) = 0 )
91	89, 90	syl 16	. . . . 5 |- ( ph -> if ( (/) e. ( C ` 0 ) , ( 2 ^ 0 ) , 0 ) = 0 )
92	91	oveq2d 6217	. . . 4 |- ( ph -> ( 0 + if ( (/) e. ( C ` 0 ) , ( 2 ^ 0 ) , 0 ) ) = ( 0 + 0 ) )
93	92, 37	syl6eq 2511	. . 3 |- ( ph -> ( 0 + if ( (/) e. ( C ` 0 ) , ( 2 ^ 0 ) , 0 ) ) = 0 )
94	86	ad2antrr 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 )
95	87	ad2antrr 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 ) ) ) ) )
98	94, 95, 88, 96, 9, 97	sadadd2lem 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 ) ) ) ) ) )
99	98	ex 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 ) ) ) ) ) ) )
100	99	expcom 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 ) ) ) ) ) ) ) )
101	100	a2d 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 ) ) ) ) ) ) ) )
102	40, 55, 70, 85, 93, 101	nn0ind 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 ) ) ) ) ) )
103	1, 102	mpcom 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 ) ) ) ) )

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