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Theorem sadfval 12659
Description: Define the addition of two bit sequences, using df-had 1370 and df-cad 1371 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 9987 . . . 4  |-  NN0  e.  _V
32elpw2 4191 . . 3  |-  ( A  e.  ~P NN0  <->  A  C_  NN0 )
41, 3sylibr 203 . 2  |-  ( ph  ->  A  e.  ~P NN0 )
5 sadval.b . . 3  |-  ( ph  ->  B  C_  NN0 )
62elpw2 4191 . . 3  |-  ( B  e.  ~P NN0  <->  B  C_  NN0 )
75, 6sylibr 203 . 2  |-  ( ph  ->  B  e.  ~P NN0 )
8 simpl 443 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  x  =  A )
98eleq2d 2363 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( k  e.  x  <->  k  e.  A ) )
10 simpr 447 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
1110eleq2d 2363 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( k  e.  y  <-> 
k  e.  B ) )
12 simp1l 979 . . . . . . . . . . . . 13  |-  ( ( ( x  =  A  /\  y  =  B )  /\  c  e.  2o  /\  m  e. 
NN0 )  ->  x  =  A )
1312eleq2d 2363 . . . . . . . . . . . 12  |-  ( ( ( x  =  A  /\  y  =  B )  /\  c  e.  2o  /\  m  e. 
NN0 )  ->  (
m  e.  x  <->  m  e.  A ) )
14 simp1r 980 . . . . . . . . . . . . 13  |-  ( ( ( x  =  A  /\  y  =  B )  /\  c  e.  2o  /\  m  e. 
NN0 )  ->  y  =  B )
1514eleq2d 2363 . . . . . . . . . . . 12  |-  ( ( ( x  =  A  /\  y  =  B )  /\  c  e.  2o  /\  m  e. 
NN0 )  ->  (
m  e.  y  <->  m  e.  B ) )
16 biidd 228 . . . . . . . . . . . 12  |-  ( ( ( x  =  A  /\  y  =  B )  /\  c  e.  2o  /\  m  e. 
NN0 )  ->  ( (/) 
e.  c  <->  (/)  e.  c ) )
1713, 15, 16cadbi123d 1373 . . . . . . . . . . 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 3596 . . . . . . . . . 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 5928 . . . . . . . . 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 11069 . . . . . . . 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 2346 . . . . . . 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 5543 . . . . . 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 2363 . . . . 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 1372 . . . 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 2793 . . 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 12658 . . 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 4181 . . 3  |-  { k  e.  NN0  | hadd (
k  e.  A , 
k  e.  B ,  (/) 
e.  ( C `  k ) ) }  e.  _V
2926, 27, 28ovmpt2a 5994 . 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 642 1  |-  ( ph  ->  ( A sadd  B )  =  { k  e. 
NN0  | hadd ( k  e.  A ,  k  e.  B ,  (/)  e.  ( C `  k ) ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934  haddwhad 1368  caddwcad 1369    = wceq 1632    e. wcel 1696   {crab 2560    C_ wss 3165   (/)c0 3468   ifcif 3578   ~Pcpw 3638    e. cmpt 4093   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1oc1o 6488   2oc2o 6489   0cc0 8753   1c1 8754    - cmin 9053   NN0cn0 9981    seq cseq 11062   sadd csad 12627
This theorem is referenced by:  sadval  12663  sadadd2lem  12666  sadadd3  12668  sadcl  12669  sadcom  12670
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-i2m1 8821  ax-1ne0 8822  ax-rrecex 8825  ax-cnre 8826
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-nn 9763  df-n0 9982  df-seq 11063  df-sad 12658
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