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.

Step	Hyp	Ref	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 ) = (/)
4	2, 3	syl6eq 2517	. . . . . . . . . 10 |- ( x = 0 -> ( 0..^ x ) = (/) )
5	4	ineq2d 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 (/) ) = (/)
7	5, 6	syl6eq 2517	. . . . . . . 8 |- ( x = 0 -> ( ( A sadd B ) i^i ( 0..^ x ) ) = (/) )
8	7	fveq2d 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 ) )
12	10, 11	ax-mp 5	. . . . . . . . . 10 |- ( (bits |` NN0 ) ` 0 ) = (bits ` 0 )
13		0bits 13937	. . . . . . . . . 10 |- (bits ` 0 ) = (/)
14	12, 13	eqtr2i 2490	. . . . . . . . 9 |- (/) = ( (bits |` NN0 ) ` 0 )
15	9, 14	fveq12i 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 )
18	16, 10, 17	mp2an 672	. . . . . . . 8 |- ( `' (bits |` NN0 ) ` ( (bits |` NN0 ) ` 0 ) ) = 0
19	15, 18	eqtri 2489	. . . . . . 7 |- ( K ` (/) ) = 0
20	8, 19	syl6eq 2517	. . . . . 6 |- ( x = 0 -> ( K ` ( ( A sadd B ) i^i ( 0..^ x ) ) ) = 0 )
21		fveq2 5857	. . . . . . . 8 |- ( x = 0 -> ( C ` x ) = ( C ` 0 ) )
22	21	eleq2d 2530	. . . . . . 7 |- ( x = 0 -> ( (/) e. ( C ` x ) <-> (/) e. ( C ` 0 ) ) )
23		oveq2 6283	. . . . . . 7 |- ( x = 0 -> ( 2 ^ x ) = ( 2 ^ 0 ) )
24	22, 23	ifbieq1d 3955	. . . . . 6 |- ( x = 0 -> if ( (/) e. ( C ` x ) , ( 2 ^ x ) , 0 ) = if ( (/) e. ( C ` 0 ) , ( 2 ^ 0 ) , 0 ) )
25	20, 24	oveq12d 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 ) ) )
26	4	ineq2d 3693	. . . . . . . . . 10 |- ( x = 0 -> ( A i^i ( 0..^ x ) ) = ( A i^i (/) ) )
27		in0 3804	. . . . . . . . . 10 |- ( A i^i (/) ) = (/)
28	26, 27	syl6eq 2517	. . . . . . . . 9 |- ( x = 0 -> ( A i^i ( 0..^ x ) ) = (/) )
29	28	fveq2d 5861	. . . . . . . 8 |- ( x = 0 -> ( K ` ( A i^i ( 0..^ x ) ) ) = ( K ` (/) ) )
30	29, 19	syl6eq 2517	. . . . . . 7 |- ( x = 0 -> ( K ` ( A i^i ( 0..^ x ) ) ) = 0 )
31	4	ineq2d 3693	. . . . . . . . . 10 |- ( x = 0 -> ( B i^i ( 0..^ x ) ) = ( B i^i (/) ) )
32		in0 3804	. . . . . . . . . 10 |- ( B i^i (/) ) = (/)
33	31, 32	syl6eq 2517	. . . . . . . . 9 |- ( x = 0 -> ( B i^i ( 0..^ x ) ) = (/) )
34	33	fveq2d 5861	. . . . . . . 8 |- ( x = 0 -> ( K ` ( B i^i ( 0..^ x ) ) ) = ( K ` (/) ) )
35	34, 19	syl6eq 2517	. . . . . . 7 |- ( x = 0 -> ( K ` ( B i^i ( 0..^ x ) ) ) = 0 )
36	30, 35	oveq12d 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
38	36, 37	syl6eq 2517	. . . . 5 |- ( x = 0 -> ( ( K ` ( A i^i ( 0..^ x ) ) ) + ( K ` ( B i^i ( 0..^ x ) ) ) ) = 0 )
39	25, 38	eqeq12d 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 ) )
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 6283	. . . . . . . 8 |- ( x = k -> ( 0..^ x ) = ( 0..^ k ) )
42	41	ineq2d 3693	. . . . . . 7 |- ( x = k -> ( ( A sadd B ) i^i ( 0..^ x ) ) = ( ( A sadd B ) i^i ( 0..^ k ) ) )
43	42	fveq2d 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 ) )
45	44	eleq2d 2530	. . . . . . 7 |- ( x = k -> ( (/) e. ( C ` x ) <-> (/) e. ( C ` k ) ) )
46		oveq2 6283	. . . . . . 7 |- ( x = k -> ( 2 ^ x ) = ( 2 ^ k ) )
47	45, 46	ifbieq1d 3955	. . . . . 6 |- ( x = k -> if ( (/) e. ( C ` x ) , ( 2 ^ x ) , 0 ) = if ( (/) e. ( C ` k ) , ( 2 ^ k ) , 0 ) )
48	43, 47	oveq12d 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 ) ) )
49	41	ineq2d 3693	. . . . . . 7 |- ( x = k -> ( A i^i ( 0..^ x ) ) = ( A i^i ( 0..^ k ) ) )
50	49	fveq2d 5861	. . . . . 6 |- ( x = k -> ( K ` ( A i^i ( 0..^ x ) ) ) = ( K ` ( A i^i ( 0..^ k ) ) ) )
51	41	ineq2d 3693	. . . . . . 7 |- ( x = k -> ( B i^i ( 0..^ x ) ) = ( B i^i ( 0..^ k ) ) )
52	51	fveq2d 5861	. . . . . 6 |- ( x = k -> ( K ` ( B i^i ( 0..^ x ) ) ) = ( K ` ( B i^i ( 0..^ k ) ) ) )
53	50, 52	oveq12d 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 ) ) ) ) )
54	48, 53	eqeq12d 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 ) ) ) ) ) )
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 6283	. . . . . . . 8 |- ( x = ( k + 1 ) -> ( 0..^ x ) = ( 0..^ ( k + 1 ) ) )
57	56	ineq2d 3693	. . . . . . 7 |- ( x = ( k + 1 ) -> ( ( A sadd B ) i^i ( 0..^ x ) ) = ( ( A sadd B ) i^i ( 0..^ ( k + 1 ) ) ) )
58	57	fveq2d 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 ) ) )
60	59	eleq2d 2530	. . . . . . 7 |- ( x = ( k + 1 ) -> ( (/) e. ( C ` x ) <-> (/) e. ( C ` ( k + 1 ) ) ) )
61		oveq2 6283	. . . . . . 7 |- ( x = ( k + 1 ) -> ( 2 ^ x ) = ( 2 ^ ( k + 1 ) ) )
62	60, 61	ifbieq1d 3955	. . . . . 6 |- ( x = ( k + 1 ) -> if ( (/) e. ( C ` x ) , ( 2 ^ x ) , 0 ) = if ( (/) e. ( C ` ( k + 1 ) ) , ( 2 ^ ( k + 1 ) ) , 0 ) )
63	58, 62	oveq12d 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 ) ) )
64	56	ineq2d 3693	. . . . . . 7 |- ( x = ( k + 1 ) -> ( A i^i ( 0..^ x ) ) = ( A i^i ( 0..^ ( k + 1 ) ) ) )
65	64	fveq2d 5861	. . . . . 6 |- ( x = ( k + 1 ) -> ( K ` ( A i^i ( 0..^ x ) ) ) = ( K ` ( A i^i ( 0..^ ( k + 1 ) ) ) ) )
66	56	ineq2d 3693	. . . . . . 7 |- ( x = ( k + 1 ) -> ( B i^i ( 0..^ x ) ) = ( B i^i ( 0..^ ( k + 1 ) ) ) )
67	66	fveq2d 5861	. . . . . 6 |- ( x = ( k + 1 ) -> ( K ` ( B i^i ( 0..^ x ) ) ) = ( K ` ( B i^i ( 0..^ ( k + 1 ) ) ) ) )
68	65, 67	oveq12d 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 ) ) ) ) ) )
69	63, 68	eqeq12d 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 ) ) ) ) ) ) )
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 6283	. . . . . . . 8 |- ( x = N -> ( 0..^ x ) = ( 0..^ N ) )
72	71	ineq2d 3693	. . . . . . 7 |- ( x = N -> ( ( A sadd B ) i^i ( 0..^ x ) ) = ( ( A sadd B ) i^i ( 0..^ N ) ) )
73	72	fveq2d 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 ) )
75	74	eleq2d 2530	. . . . . . 7 |- ( x = N -> ( (/) e. ( C ` x ) <-> (/) e. ( C ` N ) ) )
76		oveq2 6283	. . . . . . 7 |- ( x = N -> ( 2 ^ x ) = ( 2 ^ N ) )
77	75, 76	ifbieq1d 3955	. . . . . 6 |- ( x = N -> if ( (/) e. ( C ` x ) , ( 2 ^ x ) , 0 ) = if ( (/) e. ( C ` N ) , ( 2 ^ N ) , 0 ) )
78	73, 77	oveq12d 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 ) ) )
79	71	ineq2d 3693	. . . . . . 7 |- ( x = N -> ( A i^i ( 0..^ x ) ) = ( A i^i ( 0..^ N ) ) )
80	79	fveq2d 5861	. . . . . 6 |- ( x = N -> ( K ` ( A i^i ( 0..^ x ) ) ) = ( K ` ( A i^i ( 0..^ N ) ) ) )
81	71	ineq2d 3693	. . . . . . 7 |- ( x = N -> ( B i^i ( 0..^ x ) ) = ( B i^i ( 0..^ N ) ) )
82	81	fveq2d 5861	. . . . . 6 |- ( x = N -> ( K ` ( B i^i ( 0..^ x ) ) ) = ( K ` ( B i^i ( 0..^ N ) ) ) )
83	80, 82	oveq12d 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 ) ) ) ) )
84	78, 83	eqeq12d 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 ) ) ) ) ) )
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 13952	. . . . . 6 |- ( ph -> -. (/) e. ( C ` 0 ) )
90		iffalse 3941	. . . . . 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 6291	. . . 4 |- ( ph -> ( 0 + if ( (/) e. ( C ` 0 ) , ( 2 ^ 0 ) , 0 ) ) = ( 0 + 0 ) )
93	92, 37	syl6eq 2517	. . 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 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 ) ) ) ) ) )
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 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 ) ) ) ) ) )
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 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