Theorem sadadd2 14211

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 6242	. . . . . . . . . . 11 |- ( x = 0 -> ( 0..^ x ) = ( 0..^ 0 ) )
3		fzo0 11794	. . . . . . . . . . 11 |- ( 0..^ 0 ) = (/)
4	2, 3	syl6eq 2459	. . . . . . . . . 10 |- ( x = 0 -> ( 0..^ x ) = (/) )
5	4	ineq2d 3640	. . . . . . . . 9 |- ( x = 0 -> ( ( A sadd B ) i^i ( 0..^ x ) ) = ( ( A sadd B ) i^i (/) ) )
6		in0 3764	. . . . . . . . 9 |- ( ( A sadd B ) i^i (/) ) = (/)
7	5, 6	syl6eq 2459	. . . . . . . 8 |- ( x = 0 -> ( ( A sadd B ) i^i ( 0..^ x ) ) = (/) )
8	7	fveq2d 5809	. . . . . . 7 |- ( x = 0 -> ( K ` ( ( A sadd B ) i^i ( 0..^ x ) ) ) = ( K ` (/) ) )
9		sadcadd.k	. . . . . . . . 9 |- K = `' (bits |` NN0 )
10		0nn0 10771	. . . . . . . . . . 11 |- 0 e. NN0
11		fvres 5819	. . . . . . . . . . 11 |- ( 0 e. NN0 -> ( (bits |` NN0 ) ` 0 ) = (bits ` 0 ) )
12	10, 11	ax-mp 5	. . . . . . . . . 10 |- ( (bits |` NN0 ) ` 0 ) = (bits ` 0 )
13		0bits 14190	. . . . . . . . . 10 |- (bits ` 0 ) = (/)
14	12, 13	eqtr2i 2432	. . . . . . . . 9 |- (/) = ( (bits |` NN0 ) ` 0 )
15	9, 14	fveq12i 5810	. . . . . . . 8 |- ( K ` (/) ) = ( `' (bits |` NN0 ) ` ( (bits |` NN0 ) ` 0 ) )
16		bitsf1o 14196	. . . . . . . . 9 |- (bits |` NN0 ) : NN0 -1-1-onto-> ( ~P NN0 i^i Fin )
17		f1ocnvfv1 6119	. . . . . . . . 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 670	. . . . . . . 8 |- ( `' (bits |` NN0 ) ` ( (bits |` NN0 ) ` 0 ) ) = 0
19	15, 18	eqtri 2431	. . . . . . 7 |- ( K ` (/) ) = 0
20	8, 19	syl6eq 2459	. . . . . 6 |- ( x = 0 -> ( K ` ( ( A sadd B ) i^i ( 0..^ x ) ) ) = 0 )
21		fveq2 5805	. . . . . . . 8 |- ( x = 0 -> ( C ` x ) = ( C ` 0 ) )
22	21	eleq2d 2472	. . . . . . 7 |- ( x = 0 -> ( (/) e. ( C ` x ) <-> (/) e. ( C ` 0 ) ) )
23		oveq2 6242	. . . . . . 7 |- ( x = 0 -> ( 2 ^ x ) = ( 2 ^ 0 ) )
24	22, 23	ifbieq1d 3907	. . . . . 6 |- ( x = 0 -> if ( (/) e. ( C ` x ) , ( 2 ^ x ) , 0 ) = if ( (/) e. ( C ` 0 ) , ( 2 ^ 0 ) , 0 ) )
25	20, 24	oveq12d 6252	. . . . 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 3640	. . . . . . . . . 10 |- ( x = 0 -> ( A i^i ( 0..^ x ) ) = ( A i^i (/) ) )
27		in0 3764	. . . . . . . . . 10 |- ( A i^i (/) ) = (/)
28	26, 27	syl6eq 2459	. . . . . . . . 9 |- ( x = 0 -> ( A i^i ( 0..^ x ) ) = (/) )
29	28	fveq2d 5809	. . . . . . . 8 |- ( x = 0 -> ( K ` ( A i^i ( 0..^ x ) ) ) = ( K ` (/) ) )
30	29, 19	syl6eq 2459	. . . . . . 7 |- ( x = 0 -> ( K ` ( A i^i ( 0..^ x ) ) ) = 0 )
31	4	ineq2d 3640	. . . . . . . . . 10 |- ( x = 0 -> ( B i^i ( 0..^ x ) ) = ( B i^i (/) ) )
32		in0 3764	. . . . . . . . . 10 |- ( B i^i (/) ) = (/)
33	31, 32	syl6eq 2459	. . . . . . . . 9 |- ( x = 0 -> ( B i^i ( 0..^ x ) ) = (/) )
34	33	fveq2d 5809	. . . . . . . 8 |- ( x = 0 -> ( K ` ( B i^i ( 0..^ x ) ) ) = ( K ` (/) ) )
35	34, 19	syl6eq 2459	. . . . . . 7 |- ( x = 0 -> ( K ` ( B i^i ( 0..^ x ) ) ) = 0 )
36	30, 35	oveq12d 6252	. . . . . 6 |- ( x = 0 -> ( ( K ` ( A i^i ( 0..^ x ) ) ) + ( K ` ( B i^i ( 0..^ x ) ) ) ) = ( 0 + 0 ) )
37		00id 9709	. . . . . 6 |- ( 0 + 0 ) = 0
38	36, 37	syl6eq 2459	. . . . 5 |- ( x = 0 -> ( ( K ` ( A i^i ( 0..^ x ) ) ) + ( K ` ( B i^i ( 0..^ x ) ) ) ) = 0 )
39	25, 38	eqeq12d 2424	. . . 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 314	. . 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 6242	. . . . . . . 8 |- ( x = k -> ( 0..^ x ) = ( 0..^ k ) )
42	41	ineq2d 3640	. . . . . . 7 |- ( x = k -> ( ( A sadd B ) i^i ( 0..^ x ) ) = ( ( A sadd B ) i^i ( 0..^ k ) ) )
43	42	fveq2d 5809	. . . . . 6 |- ( x = k -> ( K ` ( ( A sadd B ) i^i ( 0..^ x ) ) ) = ( K ` ( ( A sadd B ) i^i ( 0..^ k ) ) ) )
44		fveq2 5805	. . . . . . . 8 |- ( x = k -> ( C ` x ) = ( C ` k ) )
45	44	eleq2d 2472	. . . . . . 7 |- ( x = k -> ( (/) e. ( C ` x ) <-> (/) e. ( C ` k ) ) )
46		oveq2 6242	. . . . . . 7 |- ( x = k -> ( 2 ^ x ) = ( 2 ^ k ) )
47	45, 46	ifbieq1d 3907	. . . . . 6 |- ( x = k -> if ( (/) e. ( C ` x ) , ( 2 ^ x ) , 0 ) = if ( (/) e. ( C ` k ) , ( 2 ^ k ) , 0 ) )
48	43, 47	oveq12d 6252	. . . . 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 3640	. . . . . . 7 |- ( x = k -> ( A i^i ( 0..^ x ) ) = ( A i^i ( 0..^ k ) ) )
50	49	fveq2d 5809	. . . . . 6 |- ( x = k -> ( K ` ( A i^i ( 0..^ x ) ) ) = ( K ` ( A i^i ( 0..^ k ) ) ) )
51	41	ineq2d 3640	. . . . . . 7 |- ( x = k -> ( B i^i ( 0..^ x ) ) = ( B i^i ( 0..^ k ) ) )
52	51	fveq2d 5809	. . . . . 6 |- ( x = k -> ( K ` ( B i^i ( 0..^ x ) ) ) = ( K ` ( B i^i ( 0..^ k ) ) ) )
53	50, 52	oveq12d 6252	. . . . 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 2424	. . . 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 314	. . 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 6242	. . . . . . . 8 |- ( x = ( k + 1 ) -> ( 0..^ x ) = ( 0..^ ( k + 1 ) ) )
57	56	ineq2d 3640	. . . . . . 7 |- ( x = ( k + 1 ) -> ( ( A sadd B ) i^i ( 0..^ x ) ) = ( ( A sadd B ) i^i ( 0..^ ( k + 1 ) ) ) )
58	57	fveq2d 5809	. . . . . 6 |- ( x = ( k + 1 ) -> ( K ` ( ( A sadd B ) i^i ( 0..^ x ) ) ) = ( K ` ( ( A sadd B ) i^i ( 0..^ ( k + 1 ) ) ) ) )
59		fveq2 5805	. . . . . . . 8 |- ( x = ( k + 1 ) -> ( C ` x ) = ( C ` ( k + 1 ) ) )
60	59	eleq2d 2472	. . . . . . 7 |- ( x = ( k + 1 ) -> ( (/) e. ( C ` x ) <-> (/) e. ( C ` ( k + 1 ) ) ) )
61		oveq2 6242	. . . . . . 7 |- ( x = ( k + 1 ) -> ( 2 ^ x ) = ( 2 ^ ( k + 1 ) ) )
62	60, 61	ifbieq1d 3907	. . . . . 6 |- ( x = ( k + 1 ) -> if ( (/) e. ( C ` x ) , ( 2 ^ x ) , 0 ) = if ( (/) e. ( C ` ( k + 1 ) ) , ( 2 ^ ( k + 1 ) ) , 0 ) )
63	58, 62	oveq12d 6252	. . . . 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 3640	. . . . . . 7 |- ( x = ( k + 1 ) -> ( A i^i ( 0..^ x ) ) = ( A i^i ( 0..^ ( k + 1 ) ) ) )
65	64	fveq2d 5809	. . . . . 6 |- ( x = ( k + 1 ) -> ( K ` ( A i^i ( 0..^ x ) ) ) = ( K ` ( A i^i ( 0..^ ( k + 1 ) ) ) ) )
66	56	ineq2d 3640	. . . . . . 7 |- ( x = ( k + 1 ) -> ( B i^i ( 0..^ x ) ) = ( B i^i ( 0..^ ( k + 1 ) ) ) )
67	66	fveq2d 5809	. . . . . 6 |- ( x = ( k + 1 ) -> ( K ` ( B i^i ( 0..^ x ) ) ) = ( K ` ( B i^i ( 0..^ ( k + 1 ) ) ) ) )
68	65, 67	oveq12d 6252	. . . . 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 2424	. . . 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 314	. . 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 6242	. . . . . . . 8 |- ( x = N -> ( 0..^ x ) = ( 0..^ N ) )
72	71	ineq2d 3640	. . . . . . 7 |- ( x = N -> ( ( A sadd B ) i^i ( 0..^ x ) ) = ( ( A sadd B ) i^i ( 0..^ N ) ) )
73	72	fveq2d 5809	. . . . . 6 |- ( x = N -> ( K ` ( ( A sadd B ) i^i ( 0..^ x ) ) ) = ( K ` ( ( A sadd B ) i^i ( 0..^ N ) ) ) )
74		fveq2 5805	. . . . . . . 8 |- ( x = N -> ( C ` x ) = ( C ` N ) )
75	74	eleq2d 2472	. . . . . . 7 |- ( x = N -> ( (/) e. ( C ` x ) <-> (/) e. ( C ` N ) ) )
76		oveq2 6242	. . . . . . 7 |- ( x = N -> ( 2 ^ x ) = ( 2 ^ N ) )
77	75, 76	ifbieq1d 3907	. . . . . 6 |- ( x = N -> if ( (/) e. ( C ` x ) , ( 2 ^ x ) , 0 ) = if ( (/) e. ( C ` N ) , ( 2 ^ N ) , 0 ) )
78	73, 77	oveq12d 6252	. . . . 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 3640	. . . . . . 7 |- ( x = N -> ( A i^i ( 0..^ x ) ) = ( A i^i ( 0..^ N ) ) )
80	79	fveq2d 5809	. . . . . 6 |- ( x = N -> ( K ` ( A i^i ( 0..^ x ) ) ) = ( K ` ( A i^i ( 0..^ N ) ) ) )
81	71	ineq2d 3640	. . . . . . 7 |- ( x = N -> ( B i^i ( 0..^ x ) ) = ( B i^i ( 0..^ N ) ) )
82	81	fveq2d 5809	. . . . . 6 |- ( x = N -> ( K ` ( B i^i ( 0..^ x ) ) ) = ( K ` ( B i^i ( 0..^ N ) ) ) )
83	80, 82	oveq12d 6252	. . . . 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 2424	. . . 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 314	. . 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 14205	. . . . . 6 |- ( ph -> -. (/) e. ( C ` 0 ) )
90	89	iffalsed 3895	. . . . 5 |- ( ph -> if ( (/) e. ( C ` 0 ) , ( 2 ^ 0 ) , 0 ) = 0 )
91	90	oveq2d 6250	. . . 4 |- ( ph -> ( 0 + if ( (/) e. ( C ` 0 ) , ( 2 ^ 0 ) , 0 ) ) = ( 0 + 0 ) )
92	91, 37	syl6eq 2459	. . 3 |- ( ph -> ( 0 + if ( (/) e. ( C ` 0 ) , ( 2 ^ 0 ) , 0 ) ) = 0 )
93	86	ad2antrr 724	. . . . . . 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 724	. . . . . . 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 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 )
96		simpr 459	. . . . . . 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 14210	. . . . . 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 432	. . . . 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 433	. . . 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 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 ) ) ) ) ) ) ) )
101	40, 55, 70, 85, 92, 100	nn0ind 10918	. 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 34	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 367   = wceq 1405  caddwcad 1460   e. wcel 1842   i^i cin 3412   C_ wss 3413   (/)c0 3737   ifcif 3884   ~Pcpw 3954   |-> cmpt 4452   `'ccnv 4941   |` cres 4944   -1-1-onto->wf1o 5524   ` cfv 5525  (class class class)co 6234   |-> cmpt2 6236   1oc1o 7080   2oc2o 7081   Fincfn 7474   0cc0 9442   1c1 9443   + caddc 9445   - cmin 9761   2c2 10546   NN0cn0 10756  ..^cfzo 11767   seqcseq 12061   ^cexp 12120  bitscbits 14170   sadd csad 14171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-inf2 8011  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519  ax-pre-sup 9520

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-xor 1367  df-tru 1408  df-fal 1411  df-had 1461  df-cad 1462  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-disj 4366  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-1o 7087  df-2o 7088  df-oadd 7091  df-er 7268  df-map 7379  df-pm 7380  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-sup 7855  df-oi 7889  df-card 8272  df-cda 8500  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-div 10168  df-nn 10497  df-2 10555  df-3 10556  df-n0 10757  df-z 10826  df-uz 11046  df-rp 11184  df-fz 11644  df-fzo 11768  df-fl 11879  df-mod 11948  df-seq 12062  df-exp 12121  df-hash 12360  df-cj 12988  df-re 12989  df-im 12990  df-sqrt 13124  df-abs 13125  df-clim 13367  df-sum 13565  df-dvds 14088  df-bits 14173  df-sad 14202

This theorem is referenced by:  sadadd3  14212