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Theorem sadfval 13960
Description: Define the addition of two bit sequences, using df-had 1431 and df-cad 1432 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.
StepHypRef Expression
1 sadval.a . . 3  |-  ( ph  ->  A  C_  NN0 )
2 nn0ex 10800 . . . 4  |-  NN0  e.  _V
32elpw2 4611 . . 3  |-  ( A  e.  ~P NN0  <->  A  C_  NN0 )
41, 3sylibr 212 . 2  |-  ( ph  ->  A  e.  ~P NN0 )
5 sadval.b . . 3  |-  ( ph  ->  B  C_  NN0 )
62elpw2 4611 . . 3  |-  ( B  e.  ~P NN0  <->  B  C_  NN0 )
75, 6sylibr 212 . 2  |-  ( ph  ->  B  e.  ~P NN0 )
8 simpl 457 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  x  =  A )
98eleq2d 2537 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( k  e.  x  <->  k  e.  A ) )
10 simpr 461 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
1110eleq2d 2537 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( k  e.  y  <-> 
k  e.  B ) )
12 simp1l 1020 . . . . . . . . . . . . 13  |-  ( ( ( x  =  A  /\  y  =  B )  /\  c  e.  2o  /\  m  e. 
NN0 )  ->  x  =  A )
1312eleq2d 2537 . . . . . . . . . . . 12  |-  ( ( ( x  =  A  /\  y  =  B )  /\  c  e.  2o  /\  m  e. 
NN0 )  ->  (
m  e.  x  <->  m  e.  A ) )
14 simp1r 1021 . . . . . . . . . . . . 13  |-  ( ( ( x  =  A  /\  y  =  B )  /\  c  e.  2o  /\  m  e. 
NN0 )  ->  y  =  B )
1514eleq2d 2537 . . . . . . . . . . . 12  |-  ( ( ( x  =  A  /\  y  =  B )  /\  c  e.  2o  /\  m  e. 
NN0 )  ->  (
m  e.  y  <->  m  e.  B ) )
16 biidd 237 . . . . . . . . . . . 12  |-  ( ( ( x  =  A  /\  y  =  B )  /\  c  e.  2o  /\  m  e. 
NN0 )  ->  ( (/) 
e.  c  <->  (/)  e.  c ) )
1713, 15, 16cadbi123d 1434 . . . . . . . . . . 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 ) ) )
1817ifbid 3961 . . . . . . . . . 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 ,  (/) ) )
1918mpt2eq3dva 6344 . . . . . . . . 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 ,  (/) ) ) )
2019seqeq2d 12081 . . . . . . . 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 ) ) ) )
2220, 21syl6eqr 2526 . . . . . . 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 )
2322fveq1d 5867 . . . . . 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 ) )
2423eleq2d 2537 . . . . 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 ) ) )
259, 11, 24hadbi123d 1433 . . . 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 ) ) ) )
2625rabbidv 3105 . . 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 13959 . . 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 ) ) } )
282rabex 4598 . . 3  |-  { k  e.  NN0  | hadd (
k  e.  A , 
k  e.  B ,  (/) 
e.  ( C `  k ) ) }  e.  _V
2926, 27, 28ovmpt2a 6416 . 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 ) ) } )
304, 7, 29syl2anc 661 1  |-  ( ph  ->  ( A sadd  B )  =  { k  e. 
NN0  | hadd ( k  e.  A ,  k  e.  B ,  (/)  e.  ( C `  k ) ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379  haddwhad 1429  caddwcad 1430    e. wcel 1767   {crab 2818    C_ wss 3476   (/)c0 3785   ifcif 3939   ~Pcpw 4010    |-> cmpt 4505   ` cfv 5587  (class class class)co 6283    |-> cmpt2 6285   1oc1o 7123   2oc2o 7124   0cc0 9491   1c1 9492    - cmin 9804   NN0cn0 10794    seqcseq 12074   sadd csad 13928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-i2m1 9559  ax-1ne0 9560  ax-rrecex 9563  ax-cnre 9564
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-xor 1361  df-tru 1382  df-had 1431  df-cad 1432  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-recs 7042  df-rdg 7076  df-nn 10536  df-n0 10795  df-seq 12075  df-sad 13959
This theorem is referenced by:  sadval  13964  sadadd2lem  13967  sadadd3  13969  sadcl  13970  sadcom  13971
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