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Theorem smufval 13986
Description: Define the addition of two bit sequences, using df-had 1431 and df-cad 1432 bit operations. (Contributed by Mario Carneiro, 9-Sep-2016.)
Hypotheses
Ref Expression
smuval.a  |-  ( ph  ->  A  C_  NN0 )
smuval.b  |-  ( ph  ->  B  C_  NN0 )
smuval.p  |-  P  =  seq 0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m
)  e.  B ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) )
Assertion
Ref Expression
smufval  |-  ( ph  ->  ( A smul  B )  =  { k  e. 
NN0  |  k  e.  ( P `  ( k  +  1 ) ) } )
Distinct variable groups:    k, m, n, p, A    ph, k, n    B, k, m, n, p    P, k
Allowed substitution hints:    ph( m, p)    P( m, n, p)

Proof of Theorem smufval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 smuval.a . . 3  |-  ( ph  ->  A  C_  NN0 )
2 nn0ex 10801 . . . 4  |-  NN0  e.  _V
32elpw2 4611 . . 3  |-  ( A  e.  ~P NN0  <->  A  C_  NN0 )
41, 3sylibr 212 . 2  |-  ( ph  ->  A  e.  ~P NN0 )
5 smuval.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 simp1l 1020 . . . . . . . . . . . . 13  |-  ( ( ( x  =  A  /\  y  =  B )  /\  p  e. 
~P NN0  /\  m  e.  NN0 )  ->  x  =  A )
98eleq2d 2537 . . . . . . . . . . . 12  |-  ( ( ( x  =  A  /\  y  =  B )  /\  p  e. 
~P NN0  /\  m  e.  NN0 )  ->  (
m  e.  x  <->  m  e.  A ) )
10 simp1r 1021 . . . . . . . . . . . . 13  |-  ( ( ( x  =  A  /\  y  =  B )  /\  p  e. 
~P NN0  /\  m  e.  NN0 )  ->  y  =  B )
1110eleq2d 2537 . . . . . . . . . . . 12  |-  ( ( ( x  =  A  /\  y  =  B )  /\  p  e. 
~P NN0  /\  m  e.  NN0 )  ->  (
( n  -  m
)  e.  y  <->  ( n  -  m )  e.  B
) )
129, 11anbi12d 710 . . . . . . . . . . 11  |-  ( ( ( x  =  A  /\  y  =  B )  /\  p  e. 
~P NN0  /\  m  e.  NN0 )  ->  (
( m  e.  x  /\  ( n  -  m
)  e.  y )  <-> 
( m  e.  A  /\  ( n  -  m
)  e.  B ) ) )
1312rabbidv 3105 . . . . . . . . . 10  |-  ( ( ( x  =  A  /\  y  =  B )  /\  p  e. 
~P NN0  /\  m  e.  NN0 )  ->  { n  e.  NN0  |  ( m  e.  x  /\  (
n  -  m )  e.  y ) }  =  { n  e. 
NN0  |  ( m  e.  A  /\  (
n  -  m )  e.  B ) } )
1413oveq2d 6300 . . . . . . . . 9  |-  ( ( ( x  =  A  /\  y  =  B )  /\  p  e. 
~P NN0  /\  m  e.  NN0 )  ->  (
p sadd  { n  e.  NN0  |  ( m  e.  x  /\  ( n  -  m
)  e.  y ) } )  =  ( p sadd  { n  e. 
NN0  |  ( m  e.  A  /\  (
n  -  m )  e.  B ) } ) )
1514mpt2eq3dva 6345 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B )  ->  ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e. 
NN0  |  ( m  e.  x  /\  (
n  -  m )  e.  y ) } ) )  =  ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m
)  e.  B ) } ) ) )
1615seqeq2d 12082 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  seq 0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  x  /\  ( n  -  m
)  e.  y ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) )  =  seq 0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m
)  e.  B ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) ) )
17 smuval.p . . . . . . 7  |-  P  =  seq 0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m
)  e.  B ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) )
1816, 17syl6eqr 2526 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  seq 0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  x  /\  ( n  -  m
)  e.  y ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) )  =  P )
1918fveq1d 5868 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  (  seq 0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e. 
NN0  |  ( m  e.  x  /\  (
n  -  m )  e.  y ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  - 
1 ) ) ) ) `  ( k  +  1 ) )  =  ( P `  ( k  +  1 ) ) )
2019eleq2d 2537 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( k  e.  (  seq 0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  x  /\  ( n  -  m
)  e.  y ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) ) `  (
k  +  1 ) )  <->  k  e.  ( P `  ( k  +  1 ) ) ) )
2120rabbidv 3105 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  { k  e.  NN0  |  k  e.  (  seq 0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  x  /\  ( n  -  m
)  e.  y ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) ) `  (
k  +  1 ) ) }  =  {
k  e.  NN0  | 
k  e.  ( P `
 ( k  +  1 ) ) } )
22 df-smu 13985 . . 3  |- smul  =  ( x  e.  ~P NN0 ,  y  e.  ~P NN0  |->  { k  e.  NN0  |  k  e.  (  seq 0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  x  /\  ( n  -  m
)  e.  y ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) ) `  (
k  +  1 ) ) } )
232rabex 4598 . . 3  |-  { k  e.  NN0  |  k  e.  ( P `  (
k  +  1 ) ) }  e.  _V
2421, 22, 23ovmpt2a 6417 . 2  |-  ( ( A  e.  ~P NN0  /\  B  e.  ~P NN0 )  ->  ( A smul  B
)  =  { k  e.  NN0  |  k  e.  ( P `  (
k  +  1 ) ) } )
254, 7, 24syl2anc 661 1  |-  ( ph  ->  ( A smul  B )  =  { k  e. 
NN0  |  k  e.  ( P `  ( k  +  1 ) ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {crab 2818    C_ wss 3476   (/)c0 3785   ifcif 3939   ~Pcpw 4010    |-> cmpt 4505   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   0cc0 9492   1c1 9493    + caddc 9495    - cmin 9805   NN0cn0 10795    seqcseq 12075   sadd csad 13929   smul csmu 13930
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 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-i2m1 9560  ax-1ne0 9561  ax-rrecex 9564  ax-cnre 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-recs 7042  df-rdg 7076  df-nn 10537  df-n0 10796  df-seq 12076  df-smu 13985
This theorem is referenced by:  smuval  13990  smupvallem  13992  smucl  13993
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