Theorem sadadd2 14434

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 6298	. . . . . . . . . . 11 |- ( x = 0 -> ( 0..^ x ) = ( 0..^ 0 ) )
3		fzo0 11942	. . . . . . . . . . 11 |- ( 0..^ 0 ) = (/)
4	2, 3	syl6eq 2501	. . . . . . . . . 10 |- ( x = 0 -> ( 0..^ x ) = (/) )
5	4	ineq2d 3634	. . . . . . . . 9 |- ( x = 0 -> ( ( A sadd B ) i^i ( 0..^ x ) ) = ( ( A sadd B ) i^i (/) ) )
6		in0 3760	. . . . . . . . 9 |- ( ( A sadd B ) i^i (/) ) = (/)
7	5, 6	syl6eq 2501	. . . . . . . 8 |- ( x = 0 -> ( ( A sadd B ) i^i ( 0..^ x ) ) = (/) )
8	7	fveq2d 5869	. . . . . . 7 |- ( x = 0 -> ( K ` ( ( A sadd B ) i^i ( 0..^ x ) ) ) = ( K ` (/) ) )
9		sadcadd.k	. . . . . . . . 9 |- K = `' (bits |` NN0 )
10		0nn0 10884	. . . . . . . . . . 11 |- 0 e. NN0
11		fvres 5879	. . . . . . . . . . 11 |- ( 0 e. NN0 -> ( (bits |` NN0 ) ` 0 ) = (bits ` 0 ) )
12	10, 11	ax-mp 5	. . . . . . . . . 10 |- ( (bits |` NN0 ) ` 0 ) = (bits ` 0 )
13		0bits 14413	. . . . . . . . . 10 |- (bits ` 0 ) = (/)
14	12, 13	eqtr2i 2474	. . . . . . . . 9 |- (/) = ( (bits |` NN0 ) ` 0 )
15	9, 14	fveq12i 5870	. . . . . . . 8 |- ( K ` (/) ) = ( `' (bits |` NN0 ) ` ( (bits |` NN0 ) ` 0 ) )
16		bitsf1o 14419	. . . . . . . . 9 |- (bits |` NN0 ) : NN0 -1-1-onto-> ( ~P NN0 i^i Fin )
17		f1ocnvfv1 6175	. . . . . . . . 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 678	. . . . . . . 8 |- ( `' (bits |` NN0 ) ` ( (bits |` NN0 ) ` 0 ) ) = 0
19	15, 18	eqtri 2473	. . . . . . 7 |- ( K ` (/) ) = 0
20	8, 19	syl6eq 2501	. . . . . 6 |- ( x = 0 -> ( K ` ( ( A sadd B ) i^i ( 0..^ x ) ) ) = 0 )
21		fveq2 5865	. . . . . . . 8 |- ( x = 0 -> ( C ` x ) = ( C ` 0 ) )
22	21	eleq2d 2514	. . . . . . 7 |- ( x = 0 -> ( (/) e. ( C ` x ) <-> (/) e. ( C ` 0 ) ) )
23		oveq2 6298	. . . . . . 7 |- ( x = 0 -> ( 2 ^ x ) = ( 2 ^ 0 ) )
24	22, 23	ifbieq1d 3904	. . . . . 6 |- ( x = 0 -> if ( (/) e. ( C ` x ) , ( 2 ^ x ) , 0 ) = if ( (/) e. ( C ` 0 ) , ( 2 ^ 0 ) , 0 ) )
25	20, 24	oveq12d 6308	. . . . 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 3634	. . . . . . . . . 10 |- ( x = 0 -> ( A i^i ( 0..^ x ) ) = ( A i^i (/) ) )
27		in0 3760	. . . . . . . . . 10 |- ( A i^i (/) ) = (/)
28	26, 27	syl6eq 2501	. . . . . . . . 9 |- ( x = 0 -> ( A i^i ( 0..^ x ) ) = (/) )
29	28	fveq2d 5869	. . . . . . . 8 |- ( x = 0 -> ( K ` ( A i^i ( 0..^ x ) ) ) = ( K ` (/) ) )
30	29, 19	syl6eq 2501	. . . . . . 7 |- ( x = 0 -> ( K ` ( A i^i ( 0..^ x ) ) ) = 0 )
31	4	ineq2d 3634	. . . . . . . . . 10 |- ( x = 0 -> ( B i^i ( 0..^ x ) ) = ( B i^i (/) ) )
32		in0 3760	. . . . . . . . . 10 |- ( B i^i (/) ) = (/)
33	31, 32	syl6eq 2501	. . . . . . . . 9 |- ( x = 0 -> ( B i^i ( 0..^ x ) ) = (/) )
34	33	fveq2d 5869	. . . . . . . 8 |- ( x = 0 -> ( K ` ( B i^i ( 0..^ x ) ) ) = ( K ` (/) ) )
35	34, 19	syl6eq 2501	. . . . . . 7 |- ( x = 0 -> ( K ` ( B i^i ( 0..^ x ) ) ) = 0 )
36	30, 35	oveq12d 6308	. . . . . 6 |- ( x = 0 -> ( ( K ` ( A i^i ( 0..^ x ) ) ) + ( K ` ( B i^i ( 0..^ x ) ) ) ) = ( 0 + 0 ) )
37		00id 9808	. . . . . 6 |- ( 0 + 0 ) = 0
38	36, 37	syl6eq 2501	. . . . 5 |- ( x = 0 -> ( ( K ` ( A i^i ( 0..^ x ) ) ) + ( K ` ( B i^i ( 0..^ x ) ) ) ) = 0 )
39	25, 38	eqeq12d 2466	. . . 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 318	. . 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 6298	. . . . . . . 8 |- ( x = k -> ( 0..^ x ) = ( 0..^ k ) )
42	41	ineq2d 3634	. . . . . . 7 |- ( x = k -> ( ( A sadd B ) i^i ( 0..^ x ) ) = ( ( A sadd B ) i^i ( 0..^ k ) ) )
43	42	fveq2d 5869	. . . . . 6 |- ( x = k -> ( K ` ( ( A sadd B ) i^i ( 0..^ x ) ) ) = ( K ` ( ( A sadd B ) i^i ( 0..^ k ) ) ) )
44		fveq2 5865	. . . . . . . 8 |- ( x = k -> ( C ` x ) = ( C ` k ) )
45	44	eleq2d 2514	. . . . . . 7 |- ( x = k -> ( (/) e. ( C ` x ) <-> (/) e. ( C ` k ) ) )
46		oveq2 6298	. . . . . . 7 |- ( x = k -> ( 2 ^ x ) = ( 2 ^ k ) )
47	45, 46	ifbieq1d 3904	. . . . . 6 |- ( x = k -> if ( (/) e. ( C ` x ) , ( 2 ^ x ) , 0 ) = if ( (/) e. ( C ` k ) , ( 2 ^ k ) , 0 ) )
48	43, 47	oveq12d 6308	. . . . 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 3634	. . . . . . 7 |- ( x = k -> ( A i^i ( 0..^ x ) ) = ( A i^i ( 0..^ k ) ) )
50	49	fveq2d 5869	. . . . . 6 |- ( x = k -> ( K ` ( A i^i ( 0..^ x ) ) ) = ( K ` ( A i^i ( 0..^ k ) ) ) )
51	41	ineq2d 3634	. . . . . . 7 |- ( x = k -> ( B i^i ( 0..^ x ) ) = ( B i^i ( 0..^ k ) ) )
52	51	fveq2d 5869	. . . . . 6 |- ( x = k -> ( K ` ( B i^i ( 0..^ x ) ) ) = ( K ` ( B i^i ( 0..^ k ) ) ) )
53	50, 52	oveq12d 6308	. . . . 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 2466	. . . 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 318	. . 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 6298	. . . . . . . 8 |- ( x = ( k + 1 ) -> ( 0..^ x ) = ( 0..^ ( k + 1 ) ) )
57	56	ineq2d 3634	. . . . . . 7 |- ( x = ( k + 1 ) -> ( ( A sadd B ) i^i ( 0..^ x ) ) = ( ( A sadd B ) i^i ( 0..^ ( k + 1 ) ) ) )
58	57	fveq2d 5869	. . . . . 6 |- ( x = ( k + 1 ) -> ( K ` ( ( A sadd B ) i^i ( 0..^ x ) ) ) = ( K ` ( ( A sadd B ) i^i ( 0..^ ( k + 1 ) ) ) ) )
59		fveq2 5865	. . . . . . . 8 |- ( x = ( k + 1 ) -> ( C ` x ) = ( C ` ( k + 1 ) ) )
60	59	eleq2d 2514	. . . . . . 7 |- ( x = ( k + 1 ) -> ( (/) e. ( C ` x ) <-> (/) e. ( C ` ( k + 1 ) ) ) )
61		oveq2 6298	. . . . . . 7 |- ( x = ( k + 1 ) -> ( 2 ^ x ) = ( 2 ^ ( k + 1 ) ) )
62	60, 61	ifbieq1d 3904	. . . . . 6 |- ( x = ( k + 1 ) -> if ( (/) e. ( C ` x ) , ( 2 ^ x ) , 0 ) = if ( (/) e. ( C ` ( k + 1 ) ) , ( 2 ^ ( k + 1 ) ) , 0 ) )
63	58, 62	oveq12d 6308	. . . . 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 3634	. . . . . . 7 |- ( x = ( k + 1 ) -> ( A i^i ( 0..^ x ) ) = ( A i^i ( 0..^ ( k + 1 ) ) ) )
65	64	fveq2d 5869	. . . . . 6 |- ( x = ( k + 1 ) -> ( K ` ( A i^i ( 0..^ x ) ) ) = ( K ` ( A i^i ( 0..^ ( k + 1 ) ) ) ) )
66	56	ineq2d 3634	. . . . . . 7 |- ( x = ( k + 1 ) -> ( B i^i ( 0..^ x ) ) = ( B i^i ( 0..^ ( k + 1 ) ) ) )
67	66	fveq2d 5869	. . . . . 6 |- ( x = ( k + 1 ) -> ( K ` ( B i^i ( 0..^ x ) ) ) = ( K ` ( B i^i ( 0..^ ( k + 1 ) ) ) ) )
68	65, 67	oveq12d 6308	. . . . 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 2466	. . . 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 318	. . 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 6298	. . . . . . . 8 |- ( x = N -> ( 0..^ x ) = ( 0..^ N ) )
72	71	ineq2d 3634	. . . . . . 7 |- ( x = N -> ( ( A sadd B ) i^i ( 0..^ x ) ) = ( ( A sadd B ) i^i ( 0..^ N ) ) )
73	72	fveq2d 5869	. . . . . 6 |- ( x = N -> ( K ` ( ( A sadd B ) i^i ( 0..^ x ) ) ) = ( K ` ( ( A sadd B ) i^i ( 0..^ N ) ) ) )
74		fveq2 5865	. . . . . . . 8 |- ( x = N -> ( C ` x ) = ( C ` N ) )
75	74	eleq2d 2514	. . . . . . 7 |- ( x = N -> ( (/) e. ( C ` x ) <-> (/) e. ( C ` N ) ) )
76		oveq2 6298	. . . . . . 7 |- ( x = N -> ( 2 ^ x ) = ( 2 ^ N ) )
77	75, 76	ifbieq1d 3904	. . . . . 6 |- ( x = N -> if ( (/) e. ( C ` x ) , ( 2 ^ x ) , 0 ) = if ( (/) e. ( C ` N ) , ( 2 ^ N ) , 0 ) )
78	73, 77	oveq12d 6308	. . . . 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 3634	. . . . . . 7 |- ( x = N -> ( A i^i ( 0..^ x ) ) = ( A i^i ( 0..^ N ) ) )
80	79	fveq2d 5869	. . . . . 6 |- ( x = N -> ( K ` ( A i^i ( 0..^ x ) ) ) = ( K ` ( A i^i ( 0..^ N ) ) ) )
81	71	ineq2d 3634	. . . . . . 7 |- ( x = N -> ( B i^i ( 0..^ x ) ) = ( B i^i ( 0..^ N ) ) )
82	81	fveq2d 5869	. . . . . 6 |- ( x = N -> ( K ` ( B i^i ( 0..^ x ) ) ) = ( K ` ( B i^i ( 0..^ N ) ) ) )
83	80, 82	oveq12d 6308	. . . . 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 2466	. . . 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 318	. . 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 14428	. . . . . 6 |- ( ph -> -. (/) e. ( C ` 0 ) )
90	89	iffalsed 3892	. . . . 5 |- ( ph -> if ( (/) e. ( C ` 0 ) , ( 2 ^ 0 ) , 0 ) = 0 )
91	90	oveq2d 6306	. . . 4 |- ( ph -> ( 0 + if ( (/) e. ( C ` 0 ) , ( 2 ^ 0 ) , 0 ) ) = ( 0 + 0 ) )
92	91, 37	syl6eq 2501	. . 3 |- ( ph -> ( 0 + if ( (/) e. ( C ` 0 ) , ( 2 ^ 0 ) , 0 ) ) = 0 )
93	86	ad2antrr 732	. . . . . . 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 732	. . . . . . 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 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 ) ) ) ) ) -> k e. NN0 )
96		simpr 463	. . . . . . 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 14433	. . . . . 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 436	. . . . 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 437	. . . 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 11030	. 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 37	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 371   = wceq 1444  caddwcad 1509   e. wcel 1887   i^i cin 3403   C_ wss 3404   (/)c0 3731   ifcif 3881   ~Pcpw 3951   |-> cmpt 4461   `'ccnv 4833   |` cres 4836   -1-1-onto->wf1o 5581   ` cfv 5582  (class class class)co 6290   |-> cmpt2 6292   1oc1o 7175   2oc2o 7176   Fincfn 7569   0cc0 9539   1c1 9540   + caddc 9542   - cmin 9860   2c2 10659   NN0cn0 10869  ..^cfzo 11915   seqcseq 12213   ^cexp 12272  bitscbits 14392   sadd csad 14393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-xor 1406  df-tru 1447  df-fal 1450  df-had 1497  df-cad 1510  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-disj 4374  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-sum 13753  df-dvds 14306  df-bits 14395  df-sad 14425

This theorem is referenced by:  sadadd3  14435