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Theorem sadval 12647
Description: The full adder sequence is the half adder function applied to the inputs and the carry sequence. (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
sadval  |-  ( ph  ->  ( N  e.  ( A sadd  B )  <-> hadd ( 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 sadval
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 sadval.a . . . 4  |-  ( ph  ->  A  C_  NN0 )
2 sadval.b . . . 4  |-  ( ph  ->  B  C_  NN0 )
3 sadval.c . . . 4  |-  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 ) ) ) )
41, 2, 3sadfval 12643 . . 3  |-  ( ph  ->  ( A sadd  B )  =  { k  e. 
NN0  | hadd ( k  e.  A ,  k  e.  B ,  (/)  e.  ( C `  k ) ) } )
54eleq2d 2350 . 2  |-  ( ph  ->  ( N  e.  ( A sadd  B )  <->  N  e.  { k  e.  NN0  | hadd ( k  e.  A ,  k  e.  B ,  (/)  e.  ( C `
 k ) ) } ) )
6 sadcp1.n . . 3  |-  ( ph  ->  N  e.  NN0 )
7 eleq1 2343 . . . . . 6  |-  ( k  =  N  ->  (
k  e.  A  <->  N  e.  A ) )
8 eleq1 2343 . . . . . 6  |-  ( k  =  N  ->  (
k  e.  B  <->  N  e.  B ) )
9 fveq2 5525 . . . . . . 7  |-  ( k  =  N  ->  ( C `  k )  =  ( C `  N ) )
109eleq2d 2350 . . . . . 6  |-  ( k  =  N  ->  ( (/) 
e.  ( C `  k )  <->  (/)  e.  ( C `  N ) ) )
117, 8, 10hadbi123d 1372 . . . . 5  |-  ( k  =  N  ->  (hadd ( k  e.  A ,  k  e.  B ,  (/)  e.  ( C `
 k ) )  <-> hadd ( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `
 N ) ) ) )
1211elrab 2923 . . . 4  |-  ( N  e.  { k  e. 
NN0  | hadd ( k  e.  A ,  k  e.  B ,  (/)  e.  ( C `  k ) ) }  <->  ( N  e.  NN0  /\ hadd ( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `  N ) ) ) )
1312baib 871 . . 3  |-  ( N  e.  NN0  ->  ( N  e.  { k  e. 
NN0  | hadd ( k  e.  A ,  k  e.  B ,  (/)  e.  ( C `  k ) ) }  <-> hadd ( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `  N ) ) ) )
146, 13syl 15 . 2  |-  ( ph  ->  ( N  e.  {
k  e.  NN0  | hadd ( k  e.  A ,  k  e.  B ,  (/)  e.  ( C `
 k ) ) }  <-> hadd ( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `
 N ) ) ) )
155, 14bitrd 244 1  |-  ( ph  ->  ( N  e.  ( A sadd  B )  <-> hadd ( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `  N ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176  haddwhad 1368  caddwcad 1369    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152   (/)c0 3455   ifcif 3565    e. cmpt 4077   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1oc1o 6472   2oc2o 6473   0cc0 8737   1c1 8738    - cmin 9037   NN0cn0 9965    seq cseq 11046   sadd csad 12611
This theorem is referenced by:  sadadd2lem  12650  saddisjlem  12655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-xor 1296  df-tru 1310  df-had 1370  df-cad 1371  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-nn 9747  df-n0 9966  df-seq 11047  df-sad 12642
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