Theorem sadfval 15174

Description: Define the addition of two bit sequences, using df-had 1533 and df-cad 1546 bit operations. (Contributed by Mario Carneiro, 5-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 ) ) ) )

Assertion

Ref	Expression
sadfval	|- ( ph -> ( A sadd B ) = { k e. NN0 | hadd ( k e. A , k e. B , (/) e. ( C ` k ) ) } )

Distinct variable groups:   k, c, m, n   A, c, k, m   B, c, k, m   C, k   ph, k

Allowed substitution hints:   ph( m, n, c)   A( n)   B( n)   C( m, n, c)

Proof of Theorem sadfval

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sadval.a	. . 3 |- ( ph -> A C_ NN0 )
2		nn0ex 11298	. . . 4 |- NN0 e. _V
3	2	elpw2 4828	. . 3 |- ( A e. ~P NN0 <-> A C_ NN0 )
4	1, 3	sylibr 224	. 2 |- ( ph -> A e. ~P NN0 )
5		sadval.b	. . 3 |- ( ph -> B C_ NN0 )
6	2	elpw2 4828	. . 3 |- ( B e. ~P NN0 <-> B C_ NN0 )
7	5, 6	sylibr 224	. 2 |- ( ph -> B e. ~P NN0 )
8		simpl 473	. . . . . 6 |- ( ( x = A /\ y = B ) -> x = A )
9	8	eleq2d 2687	. . . . 5 |- ( ( x = A /\ y = B ) -> ( k e. x <-> k e. A ) )
10		simpr 477	. . . . . 6 |- ( ( x = A /\ y = B ) -> y = B )
11	10	eleq2d 2687	. . . . 5 |- ( ( x = A /\ y = B ) -> ( k e. y <-> k e. B ) )
12		simp1l 1085	. . . . . . . . . . . . 13 |- ( ( ( x = A /\ y = B ) /\ c e. 2o /\ m e. NN0 ) -> x = A )
13	12	eleq2d 2687	. . . . . . . . . . . 12 |- ( ( ( x = A /\ y = B ) /\ c e. 2o /\ m e. NN0 ) -> ( m e. x <-> m e. A ) )
14		simp1r 1086	. . . . . . . . . . . . 13 |- ( ( ( x = A /\ y = B ) /\ c e. 2o /\ m e. NN0 ) -> y = B )
15	14	eleq2d 2687	. . . . . . . . . . . 12 |- ( ( ( x = A /\ y = B ) /\ c e. 2o /\ m e. NN0 ) -> ( m e. y <-> m e. B ) )
16		biidd 252	. . . . . . . . . . . 12 |- ( ( ( x = A /\ y = B ) /\ c e. 2o /\ m e. NN0 ) -> ( (/) e. c <-> (/) e. c ) )
17	13, 15, 16	cadbi123d 1549	. . . . . . . . . . 11 |- ( ( ( x = A /\ y = B ) /\ c e. 2o /\ m e. NN0 ) -> (cadd ( m e. x , m e. y , (/) e. c ) <-> cadd ( m e. A , m e. B , (/) e. c ) ) )
18	17	ifbid 4108	. . . . . . . . . 10 |- ( ( ( x = A /\ y = B ) /\ c e. 2o /\ m e. NN0 ) -> if (cadd ( m e. x , m e. y , (/) e. c ) , 1o , (/) ) = if (cadd ( m e. A , m e. B , (/) e. c ) , 1o , (/) ) )
19	18	mpt2eq3dva 6719	. . . . . . . . 9 |- ( ( x = A /\ y = B ) -> ( c e. 2o , m e. NN0 |-> if (cadd ( m e. x , m e. y , (/) e. c ) , 1o , (/) ) ) = ( c e. 2o , m e. NN0 |-> if (cadd ( m e. A , m e. B , (/) e. c ) , 1o , (/) ) ) )
20	19	seqeq2d 12808	. . . . . . . 8 |- ( ( x = A /\ y = B ) -> seq 0 ( ( c e. 2o , m e. NN0 |-> if (cadd ( m e. x , m e. y , (/) e. c ) , 1o , (/) ) ) , ( n e. NN0 |-> if ( n = 0 , (/) , ( n - 1 ) ) ) ) = 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 ) ) ) ) )
21		sadval.c	. . . . . . . 8 |- 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 ) ) ) )
22	20, 21	syl6eqr 2674	. . . . . . 7 |- ( ( x = A /\ y = B ) -> seq 0 ( ( c e. 2o , m e. NN0 |-> if (cadd ( m e. x , m e. y , (/) e. c ) , 1o , (/) ) ) , ( n e. NN0 |-> if ( n = 0 , (/) , ( n - 1 ) ) ) ) = C )
23	22	fveq1d 6193	. . . . . 6 |- ( ( x = A /\ y = B ) -> ( seq 0 ( ( c e. 2o , m e. NN0 |-> if (cadd ( m e. x , m e. y , (/) e. c ) , 1o , (/) ) ) , ( n e. NN0 |-> if ( n = 0 , (/) , ( n - 1 ) ) ) ) ` k ) = ( C ` k ) )
24	23	eleq2d 2687	. . . . 5 |- ( ( x = A /\ y = B ) -> ( (/) e. ( seq 0 ( ( c e. 2o , m e. NN0 |-> if (cadd ( m e. x , m e. y , (/) e. c ) , 1o , (/) ) ) , ( n e. NN0 |-> if ( n = 0 , (/) , ( n - 1 ) ) ) ) ` k ) <-> (/) e. ( C ` k ) ) )
25	9, 11, 24	hadbi123d 1534	. . . 4 |- ( ( x = A /\ y = B ) -> (hadd ( k e. x , k e. y , (/) e. ( seq 0 ( ( c e. 2o , m e. NN0 |-> if (cadd ( m e. x , m e. y , (/) e. c ) , 1o , (/) ) ) , ( n e. NN0 |-> if ( n = 0 , (/) , ( n - 1 ) ) ) ) ` k ) ) <-> hadd ( k e. A , k e. B , (/) e. ( C ` k ) ) ) )
26	25	rabbidv 3189	. . 3 |- ( ( x = A /\ y = B ) -> { k e. NN0 | hadd ( k e. x , k e. y , (/) e. ( seq 0 ( ( c e. 2o , m e. NN0 |-> if (cadd ( m e. x , m e. y , (/) e. c ) , 1o , (/) ) ) , ( n e. NN0 |-> if ( n = 0 , (/) , ( n - 1 ) ) ) ) ` k ) ) } = { k e. NN0 | hadd ( k e. A , k e. B , (/) e. ( C ` k ) ) } )
27		df-sad 15173	. . 3 |- sadd = ( x e. ~P NN0 , y e. ~P NN0 |-> { k e. NN0 | hadd ( k e. x , k e. y , (/) e. ( seq 0 ( ( c e. 2o , m e. NN0 |-> if (cadd ( m e. x , m e. y , (/) e. c ) , 1o , (/) ) ) , ( n e. NN0 |-> if ( n = 0 , (/) , ( n - 1 ) ) ) ) ` k ) ) } )
28	2	rabex 4813	. . 3 |- { k e. NN0 | hadd ( k e. A , k e. B , (/) e. ( C ` k ) ) } e. _V
29	26, 27, 28	ovmpt2a 6791	. 2 |- ( ( A e. ~P NN0 /\ B e. ~P NN0 ) -> ( A sadd B ) = { k e. NN0 | hadd ( k e. A , k e. B , (/) e. ( C ` k ) ) } )
30	4, 7, 29	syl2anc 693	1 |- ( ph -> ( A sadd B ) = { k e. NN0 | hadd ( k e. A , k e. B , (/) e. ( C ` k ) ) } )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483  haddwhad 1532  caddwcad 1545   e. wcel 1990   {crab 2916   C_ wss 3574   (/)c0 3915   ifcif 4086   ~Pcpw 4158   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1oc1o 7553   2oc2o 7554   0cc0 9936   1c1 9937   - cmin 10266   NN0cn0 11292   seqcseq 12801   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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009

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-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-ral 2917  df-rex 2918  df-reu 2919  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-iun 4522  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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-nn 11021  df-n0 11293  df-seq 12802  df-sad 15173

This theorem is referenced by:  sadval  15178  sadadd2lem  15181  sadadd3  15183  sadcl  15184  sadcom  15185