Theorem sadcp1 15177

Description: The carry sequence (which is a sequence of wffs, encoded as 1o and (/)) is defined recursively as the carry operation applied to the previous carry and the two current inputs. (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 ) ) ) )
sadcp1.n	|- ( ph -> N e. NN0 )

Assertion

Ref	Expression
sadcp1	|- ( ph -> ( (/) e. ( C ` ( N + 1 ) ) <-> cadd ( N e. A , N e. B , (/) e. ( C ` 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)   N( m, c)

Proof of Theorem sadcp1

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

Step	Hyp	Ref	Expression
1		sadcp1.n	. . . . . . 7 |- ( ph -> N e. NN0 )
2		nn0uz 11722	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
3	1, 2	syl6eleq 2711	. . . . . 6 |- ( ph -> N e. ( ZZ>= ` 0 ) )
4		seqp1 12816	. . . . . 6 |- ( N e. ( ZZ>= ` 0 ) -> ( 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 ) ) ) ) ` ( 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 ) ) ) ) ` N ) ( 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 ) ) ) ` ( N + 1 ) ) ) )
5	3, 4	syl 17	. . . . 5 |- ( ph -> ( 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 ) ) ) ) ` ( 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 ) ) ) ) ` N ) ( 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 ) ) ) ` ( N + 1 ) ) ) )
6		sadval.c	. . . . . 6 |- 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 ) ) ) )
7	6	fveq1i 6192	. . . . 5 |- ( C ` ( 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 ) ) ) ) ` ( N + 1 ) )
8	6	fveq1i 6192	. . . . . 6 |- ( C ` N ) = ( 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 ) ) ) ) ` N )
9	8	oveq1i 6660	. . . . 5 |- ( ( C ` N ) ( 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 ) ) ) ` ( 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 ) ) ) ) ` N ) ( 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 ) ) ) ` ( N + 1 ) ) )
10	5, 7, 9	3eqtr4g 2681	. . . 4 |- ( ph -> ( C ` ( N + 1 ) ) = ( ( C ` N ) ( 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 ) ) ) ` ( N + 1 ) ) ) )
11		peano2nn0 11333	. . . . . . 7 |- ( N e. NN0 -> ( N + 1 ) e. NN0 )
12		eqeq1 2626	. . . . . . . . 9 |- ( n = ( N + 1 ) -> ( n = 0 <-> ( N + 1 ) = 0 ) )
13		oveq1 6657	. . . . . . . . 9 |- ( n = ( N + 1 ) -> ( n - 1 ) = ( ( N + 1 ) - 1 ) )
14	12, 13	ifbieq2d 4111	. . . . . . . 8 |- ( n = ( N + 1 ) -> if ( n = 0 , (/) , ( n - 1 ) ) = if ( ( N + 1 ) = 0 , (/) , ( ( N + 1 ) - 1 ) ) )
15		eqid 2622	. . . . . . . 8 |- ( n e. NN0 |-> if ( n = 0 , (/) , ( n - 1 ) ) ) = ( n e. NN0 |-> if ( n = 0 , (/) , ( n - 1 ) ) )
16		0ex 4790	. . . . . . . . 9 |- (/) e. _V
17		ovex 6678	. . . . . . . . 9 |- ( ( N + 1 ) - 1 ) e. _V
18	16, 17	ifex 4156	. . . . . . . 8 |- if ( ( N + 1 ) = 0 , (/) , ( ( N + 1 ) - 1 ) ) e. _V
19	14, 15, 18	fvmpt 6282	. . . . . . 7 |- ( ( N + 1 ) e. NN0 -> ( ( n e. NN0 |-> if ( n = 0 , (/) , ( n - 1 ) ) ) ` ( N + 1 ) ) = if ( ( N + 1 ) = 0 , (/) , ( ( N + 1 ) - 1 ) ) )
20	1, 11, 19	3syl 18	. . . . . 6 |- ( ph -> ( ( n e. NN0 |-> if ( n = 0 , (/) , ( n - 1 ) ) ) ` ( N + 1 ) ) = if ( ( N + 1 ) = 0 , (/) , ( ( N + 1 ) - 1 ) ) )
21		nn0p1nn 11332	. . . . . . . . 9 |- ( N e. NN0 -> ( N + 1 ) e. NN )
22	1, 21	syl 17	. . . . . . . 8 |- ( ph -> ( N + 1 ) e. NN )
23	22	nnne0d 11065	. . . . . . 7 |- ( ph -> ( N + 1 ) =/= 0 )
24		ifnefalse 4098	. . . . . . 7 |- ( ( N + 1 ) =/= 0 -> if ( ( N + 1 ) = 0 , (/) , ( ( N + 1 ) - 1 ) ) = ( ( N + 1 ) - 1 ) )
25	23, 24	syl 17	. . . . . 6 |- ( ph -> if ( ( N + 1 ) = 0 , (/) , ( ( N + 1 ) - 1 ) ) = ( ( N + 1 ) - 1 ) )
26	1	nn0cnd 11353	. . . . . . 7 |- ( ph -> N e. CC )
27		1cnd 10056	. . . . . . 7 |- ( ph -> 1 e. CC )
28	26, 27	pncand 10393	. . . . . 6 |- ( ph -> ( ( N + 1 ) - 1 ) = N )
29	20, 25, 28	3eqtrd 2660	. . . . 5 |- ( ph -> ( ( n e. NN0 |-> if ( n = 0 , (/) , ( n - 1 ) ) ) ` ( N + 1 ) ) = N )
30	29	oveq2d 6666	. . . 4 |- ( ph -> ( ( C ` N ) ( 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 ) ) ) ` ( N + 1 ) ) ) = ( ( C ` N ) ( c e. 2o , m e. NN0 |-> if (cadd ( m e. A , m e. B , (/) e. c ) , 1o , (/) ) ) N ) )
31		sadval.a	. . . . . . 7 |- ( ph -> A C_ NN0 )
32		sadval.b	. . . . . . 7 |- ( ph -> B C_ NN0 )
33	31, 32, 6	sadcf 15175	. . . . . 6 |- ( ph -> C : NN0 --> 2o )
34	33, 1	ffvelrnd 6360	. . . . 5 |- ( ph -> ( C ` N ) e. 2o )
35		simpr 477	. . . . . . . . 9 |- ( ( x = ( C ` N ) /\ y = N ) -> y = N )
36	35	eleq1d 2686	. . . . . . . 8 |- ( ( x = ( C ` N ) /\ y = N ) -> ( y e. A <-> N e. A ) )
37	35	eleq1d 2686	. . . . . . . 8 |- ( ( x = ( C ` N ) /\ y = N ) -> ( y e. B <-> N e. B ) )
38		simpl 473	. . . . . . . . 9 |- ( ( x = ( C ` N ) /\ y = N ) -> x = ( C ` N ) )
39	38	eleq2d 2687	. . . . . . . 8 |- ( ( x = ( C ` N ) /\ y = N ) -> ( (/) e. x <-> (/) e. ( C ` N ) ) )
40	36, 37, 39	cadbi123d 1549	. . . . . . 7 |- ( ( x = ( C ` N ) /\ y = N ) -> (cadd ( y e. A , y e. B , (/) e. x ) <-> cadd ( N e. A , N e. B , (/) e. ( C ` N ) ) ) )
41	40	ifbid 4108	. . . . . 6 |- ( ( x = ( C ` N ) /\ y = N ) -> if (cadd ( y e. A , y e. B , (/) e. x ) , 1o , (/) ) = if (cadd ( N e. A , N e. B , (/) e. ( C ` N ) ) , 1o , (/) ) )
42		biidd 252	. . . . . . . . 9 |- ( c = x -> ( m e. A <-> m e. A ) )
43		biidd 252	. . . . . . . . 9 |- ( c = x -> ( m e. B <-> m e. B ) )
44		eleq2 2690	. . . . . . . . 9 |- ( c = x -> ( (/) e. c <-> (/) e. x ) )
45	42, 43, 44	cadbi123d 1549	. . . . . . . 8 |- ( c = x -> (cadd ( m e. A , m e. B , (/) e. c ) <-> cadd ( m e. A , m e. B , (/) e. x ) ) )
46	45	ifbid 4108	. . . . . . 7 |- ( c = x -> if (cadd ( m e. A , m e. B , (/) e. c ) , 1o , (/) ) = if (cadd ( m e. A , m e. B , (/) e. x ) , 1o , (/) ) )
47		eleq1 2689	. . . . . . . . 9 |- ( m = y -> ( m e. A <-> y e. A ) )
48		eleq1 2689	. . . . . . . . 9 |- ( m = y -> ( m e. B <-> y e. B ) )
49		biidd 252	. . . . . . . . 9 |- ( m = y -> ( (/) e. x <-> (/) e. x ) )
50	47, 48, 49	cadbi123d 1549	. . . . . . . 8 |- ( m = y -> (cadd ( m e. A , m e. B , (/) e. x ) <-> cadd ( y e. A , y e. B , (/) e. x ) ) )
51	50	ifbid 4108	. . . . . . 7 |- ( m = y -> if (cadd ( m e. A , m e. B , (/) e. x ) , 1o , (/) ) = if (cadd ( y e. A , y e. B , (/) e. x ) , 1o , (/) ) )
52	46, 51	cbvmpt2v 6735	. . . . . 6 |- ( c e. 2o , m e. NN0 |-> if (cadd ( m e. A , m e. B , (/) e. c ) , 1o , (/) ) ) = ( x e. 2o , y e. NN0 |-> if (cadd ( y e. A , y e. B , (/) e. x ) , 1o , (/) ) )
53		1on 7567	. . . . . . . 8 |- 1o e. On
54	53	elexi 3213	. . . . . . 7 |- 1o e. _V
55	54, 16	ifex 4156	. . . . . 6 |- if (cadd ( N e. A , N e. B , (/) e. ( C ` N ) ) , 1o , (/) ) e. _V
56	41, 52, 55	ovmpt2a 6791	. . . . 5 |- ( ( ( C ` N ) e. 2o /\ N e. NN0 ) -> ( ( C ` N ) ( c e. 2o , m e. NN0 |-> if (cadd ( m e. A , m e. B , (/) e. c ) , 1o , (/) ) ) N ) = if (cadd ( N e. A , N e. B , (/) e. ( C ` N ) ) , 1o , (/) ) )
57	34, 1, 56	syl2anc 693	. . . 4 |- ( ph -> ( ( C ` N ) ( c e. 2o , m e. NN0 |-> if (cadd ( m e. A , m e. B , (/) e. c ) , 1o , (/) ) ) N ) = if (cadd ( N e. A , N e. B , (/) e. ( C ` N ) ) , 1o , (/) ) )
58	10, 30, 57	3eqtrd 2660	. . 3 |- ( ph -> ( C ` ( N + 1 ) ) = if (cadd ( N e. A , N e. B , (/) e. ( C ` N ) ) , 1o , (/) ) )
59	58	eleq2d 2687	. 2 |- ( ph -> ( (/) e. ( C ` ( N + 1 ) ) <-> (/) e. if (cadd ( N e. A , N e. B , (/) e. ( C ` N ) ) , 1o , (/) ) ) )
60		noel 3919	. . . . 5 |- -. (/) e. (/)
61		iffalse 4095	. . . . . 6 |- ( -. cadd ( N e. A , N e. B , (/) e. ( C ` N ) ) -> if (cadd ( N e. A , N e. B , (/) e. ( C ` N ) ) , 1o , (/) ) = (/) )
62	61	eleq2d 2687	. . . . 5 |- ( -. cadd ( N e. A , N e. B , (/) e. ( C ` N ) ) -> ( (/) e. if (cadd ( N e. A , N e. B , (/) e. ( C ` N ) ) , 1o , (/) ) <-> (/) e. (/) ) )
63	60, 62	mtbiri 317	. . . 4 |- ( -. cadd ( N e. A , N e. B , (/) e. ( C ` N ) ) -> -. (/) e. if (cadd ( N e. A , N e. B , (/) e. ( C ` N ) ) , 1o , (/) ) )
64	63	con4i 113	. . 3 |- ( (/) e. if (cadd ( N e. A , N e. B , (/) e. ( C ` N ) ) , 1o , (/) ) -> cadd ( N e. A , N e. B , (/) e. ( C ` N ) ) )
65		0lt1o 7584	. . . 4 |- (/) e. 1o
66		iftrue 4092	. . . 4 |- (cadd ( N e. A , N e. B , (/) e. ( C ` N ) ) -> if (cadd ( N e. A , N e. B , (/) e. ( C ` N ) ) , 1o , (/) ) = 1o )
67	65, 66	syl5eleqr 2708	. . 3 |- (cadd ( N e. A , N e. B , (/) e. ( C ` N ) ) -> (/) e. if (cadd ( N e. A , N e. B , (/) e. ( C ` N ) ) , 1o , (/) ) )
68	64, 67	impbii 199	. 2 |- ( (/) e. if (cadd ( N e. A , N e. B , (/) e. ( C ` N ) ) , 1o , (/) ) <-> cadd ( N e. A , N e. B , (/) e. ( C ` N ) ) )
69	59, 68	syl6bb 276	1 |- ( ph -> ( (/) e. ( C ` ( N + 1 ) ) <-> cadd ( N e. A , N e. B , (/) e. ( C ` N ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483  caddwcad 1545   e. wcel 1990   =/= wne 2794   C_ wss 3574   (/)c0 3915   ifcif 4086   |-> cmpt 4729   Oncon0 5723   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1oc1o 7553   2oc2o 7554   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZ>=cuz 11687   seqcseq 12801

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-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

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-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-nel 2898  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802

This theorem is referenced by:  sadcaddlem  15179  sadadd2lem  15181  saddisjlem  15186