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Theorem sadcp1 14117
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.
StepHypRef Expression
1 sadcp1.n . . . . . . 7  |-  ( ph  ->  N  e.  NN0 )
2 nn0uz 11140 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
31, 2syl6eleq 2555 . . . . . 6  |-  ( ph  ->  N  e.  ( ZZ>= ` 
0 ) )
4 seqp1 12125 . . . . . 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 ) ) ) )
53, 4syl 16 . . . . 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 ) ) ) )
76fveq1i 5873 . . . . 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 ) )
86fveq1i 5873 . . . . . 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 )
98oveq1i 6306 . . . . 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 ) ) )
105, 7, 93eqtr4g 2523 . . . 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 10857 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
12 eqeq1 2461 . . . . . . . . 9  |-  ( n  =  ( N  + 
1 )  ->  (
n  =  0  <->  ( N  +  1 )  =  0 ) )
13 oveq1 6303 . . . . . . . . 9  |-  ( n  =  ( N  + 
1 )  ->  (
n  -  1 )  =  ( ( N  +  1 )  - 
1 ) )
1412, 13ifbieq2d 3969 . . . . . . . 8  |-  ( n  =  ( N  + 
1 )  ->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) )  =  if ( ( N  +  1 )  =  0 ,  (/) ,  ( ( N  + 
1 )  -  1 ) ) )
15 eqid 2457 . . . . . . . 8  |-  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  - 
1 ) ) )  =  ( n  e. 
NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) )
16 0ex 4587 . . . . . . . . 9  |-  (/)  e.  _V
17 ovex 6324 . . . . . . . . 9  |-  ( ( N  +  1 )  -  1 )  e. 
_V
1816, 17ifex 4013 . . . . . . . 8  |-  if ( ( N  +  1 )  =  0 ,  (/) ,  ( ( N  +  1 )  - 
1 ) )  e. 
_V
1914, 15, 18fvmpt 5956 . . . . . . 7  |-  ( ( N  +  1 )  e.  NN0  ->  ( ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  - 
1 ) ) ) `
 ( N  + 
1 ) )  =  if ( ( N  +  1 )  =  0 ,  (/) ,  ( ( N  +  1 )  -  1 ) ) )
201, 11, 193syl 20 . . . . . 6  |-  ( ph  ->  ( ( n  e. 
NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) `  ( N  +  1
) )  =  if ( ( N  + 
1 )  =  0 ,  (/) ,  ( ( N  +  1 )  -  1 ) ) )
21 nn0p1nn 10856 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN )
221, 21syl 16 . . . . . . . 8  |-  ( ph  ->  ( N  +  1 )  e.  NN )
2322nnne0d 10601 . . . . . . 7  |-  ( ph  ->  ( N  +  1 )  =/=  0 )
24 ifnefalse 3956 . . . . . . 7  |-  ( ( N  +  1 )  =/=  0  ->  if ( ( N  + 
1 )  =  0 ,  (/) ,  ( ( N  +  1 )  -  1 ) )  =  ( ( N  +  1 )  - 
1 ) )
2523, 24syl 16 . . . . . 6  |-  ( ph  ->  if ( ( N  +  1 )  =  0 ,  (/) ,  ( ( N  +  1 )  -  1 ) )  =  ( ( N  +  1 )  -  1 ) )
261nn0cnd 10875 . . . . . . 7  |-  ( ph  ->  N  e.  CC )
27 1cnd 9629 . . . . . . 7  |-  ( ph  ->  1  e.  CC )
2826, 27pncand 9951 . . . . . 6  |-  ( ph  ->  ( ( N  + 
1 )  -  1 )  =  N )
2920, 25, 283eqtrd 2502 . . . . 5  |-  ( ph  ->  ( ( n  e. 
NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) `  ( N  +  1
) )  =  N )
3029oveq2d 6312 . . . 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 )
3331, 32, 6sadcf 14115 . . . . . 6  |-  ( ph  ->  C : NN0 --> 2o )
3433, 1ffvelrnd 6033 . . . . 5  |-  ( ph  ->  ( C `  N
)  e.  2o )
35 simpr 461 . . . . . . . . 9  |-  ( ( x  =  ( C `
 N )  /\  y  =  N )  ->  y  =  N )
3635eleq1d 2526 . . . . . . . 8  |-  ( ( x  =  ( C `
 N )  /\  y  =  N )  ->  ( y  e.  A  <->  N  e.  A ) )
3735eleq1d 2526 . . . . . . . 8  |-  ( ( x  =  ( C `
 N )  /\  y  =  N )  ->  ( y  e.  B  <->  N  e.  B ) )
38 simpl 457 . . . . . . . . 9  |-  ( ( x  =  ( C `
 N )  /\  y  =  N )  ->  x  =  ( C `
 N ) )
3938eleq2d 2527 . . . . . . . 8  |-  ( ( x  =  ( C `
 N )  /\  y  =  N )  ->  ( (/)  e.  x  <->  (/)  e.  ( C `  N
) ) )
4036, 37, 39cadbi123d 1450 . . . . . . 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 ) ) ) )
4140ifbid 3966 . . . . . 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 237 . . . . . . . . 9  |-  ( c  =  x  ->  (
m  e.  A  <->  m  e.  A ) )
43 biidd 237 . . . . . . . . 9  |-  ( c  =  x  ->  (
m  e.  B  <->  m  e.  B ) )
44 eleq2 2530 . . . . . . . . 9  |-  ( c  =  x  ->  ( (/) 
e.  c  <->  (/)  e.  x
) )
4542, 43, 44cadbi123d 1450 . . . . . . . 8  |-  ( c  =  x  ->  (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c )  <-> cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  x ) ) )
4645ifbid 3966 . . . . . . 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 2529 . . . . . . . . 9  |-  ( m  =  y  ->  (
m  e.  A  <->  y  e.  A ) )
48 eleq1 2529 . . . . . . . . 9  |-  ( m  =  y  ->  (
m  e.  B  <->  y  e.  B ) )
49 biidd 237 . . . . . . . . 9  |-  ( m  =  y  ->  ( (/) 
e.  x  <->  (/)  e.  x
) )
5047, 48, 49cadbi123d 1450 . . . . . . . 8  |-  ( m  =  y  ->  (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  x )  <-> cadd ( y  e.  A ,  y  e.  B ,  (/)  e.  x ) ) )
5150ifbid 3966 . . . . . . 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 ,  (/) ) )
5246, 51cbvmpt2v 6376 . . . . . 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 7155 . . . . . . . 8  |-  1o  e.  On
5453elexi 3119 . . . . . . 7  |-  1o  e.  _V
5554, 16ifex 4013 . . . . . 6  |-  if (cadd ( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `
 N ) ) ,  1o ,  (/) )  e.  _V
5641, 52, 55ovmpt2a 6432 . . . . 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 ,  (/) ) )
5734, 1, 56syl2anc 661 . . . 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 ,  (/) ) )
5810, 30, 573eqtrd 2502 . . 3  |-  ( ph  ->  ( C `  ( N  +  1 ) )  =  if (cadd ( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `
 N ) ) ,  1o ,  (/) ) )
5958eleq2d 2527 . 2  |-  ( ph  ->  ( (/)  e.  ( C `  ( N  +  1 ) )  <->  (/) 
e.  if (cadd ( N  e.  A ,  N  e.  B ,  (/) 
e.  ( C `  N ) ) ,  1o ,  (/) ) ) )
60 noel 3797 . . . . 5  |-  -.  (/)  e.  (/)
61 iffalse 3953 . . . . . 6  |-  ( -. cadd
( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `
 N ) )  ->  if (cadd ( N  e.  A ,  N  e.  B ,  (/) 
e.  ( C `  N ) ) ,  1o ,  (/) )  =  (/) )
6261eleq2d 2527 . . . . 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.  (/) ) )
6360, 62mtbiri 303 . . . 4  |-  ( -. cadd
( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `
 N ) )  ->  -.  (/)  e.  if (cadd ( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `
 N ) ) ,  1o ,  (/) ) )
6463con4i 130 . . 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 7172 . . . 4  |-  (/)  e.  1o
66 iftrue 3950 . . . 4  |-  (cadd ( N  e.  A ,  N  e.  B ,  (/) 
e.  ( C `  N ) )  ->  if (cadd ( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `  N ) ) ,  1o ,  (/) )  =  1o )
6765, 66syl5eleqr 2552 . . 3  |-  (cadd ( N  e.  A ,  N  e.  B ,  (/) 
e.  ( C `  N ) )  ->  (/) 
e.  if (cadd ( N  e.  A ,  N  e.  B ,  (/) 
e.  ( C `  N ) ) ,  1o ,  (/) ) )
6864, 67impbii 188 . 2  |-  ( (/)  e.  if (cadd ( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `  N ) ) ,  1o ,  (/) )  <-> cadd ( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `
 N ) ) )
6959, 68syl6bb 261 1  |-  ( ph  ->  ( (/)  e.  ( C `  ( N  +  1 ) )  <-> cadd ( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `
 N ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395  caddwcad 1446    e. wcel 1819    =/= wne 2652    C_ wss 3471   (/)c0 3793   ifcif 3944    |-> cmpt 4515   Oncon0 4887   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   1oc1o 7141   2oc2o 7142   0cc0 9509   1c1 9510    + caddc 9512    - cmin 9824   NNcn 10556   NN0cn0 10816   ZZ>=cuz 11106    seqcseq 12110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-xor 1364  df-tru 1398  df-cad 1448  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-seq 12111
This theorem is referenced by:  sadcaddlem  14119  sadadd2lem  14121  saddisjlem  14126
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