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Theorem smufval 12668
Description: Define the addition of two bit sequences, using df-had 1370 and df-cad 1371 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 9971 . . . 4  |-  NN0  e.  _V
32elpw2 4175 . . 3  |-  ( A  e.  ~P NN0  <->  A  C_  NN0 )
41, 3sylibr 203 . 2  |-  ( ph  ->  A  e.  ~P NN0 )
5 smuval.b . . 3  |-  ( ph  ->  B  C_  NN0 )
62elpw2 4175 . . 3  |-  ( B  e.  ~P NN0  <->  B  C_  NN0 )
75, 6sylibr 203 . 2  |-  ( ph  ->  B  e.  ~P NN0 )
8 simp1l 979 . . . . . . . . . . . . 13  |-  ( ( ( x  =  A  /\  y  =  B )  /\  p  e. 
~P NN0  /\  m  e.  NN0 )  ->  x  =  A )
98eleq2d 2350 . . . . . . . . . . . 12  |-  ( ( ( x  =  A  /\  y  =  B )  /\  p  e. 
~P NN0  /\  m  e.  NN0 )  ->  (
m  e.  x  <->  m  e.  A ) )
10 simp1r 980 . . . . . . . . . . . . 13  |-  ( ( ( x  =  A  /\  y  =  B )  /\  p  e. 
~P NN0  /\  m  e.  NN0 )  ->  y  =  B )
1110eleq2d 2350 . . . . . . . . . . . 12  |-  ( ( ( x  =  A  /\  y  =  B )  /\  p  e. 
~P NN0  /\  m  e.  NN0 )  ->  (
( n  -  m
)  e.  y  <->  ( n  -  m )  e.  B
) )
129, 11anbi12d 691 . . . . . . . . . . 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 2780 . . . . . . . . . 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 5874 . . . . . . . . 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 5912 . . . . . . . 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 11053 . . . . . . 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 2333 . . . . . 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 5527 . . . . 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 2350 . . . 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 2780 . . 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 12667 . . 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 4165 . . 3  |-  { k  e.  NN0  |  k  e.  ( P `  (
k  +  1 ) ) }  e.  _V
2421, 22, 23ovmpt2a 5978 . 2  |-  ( ( A  e.  ~P NN0  /\  B  e.  ~P NN0 )  ->  ( A smul  B
)  =  { k  e.  NN0  |  k  e.  ( P `  (
k  +  1 ) ) } )
254, 7, 24syl2anc 642 1  |-  ( ph  ->  ( A smul  B )  =  { k  e. 
NN0  |  k  e.  ( P `  ( k  +  1 ) ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152   (/)c0 3455   ifcif 3565   ~Pcpw 3625    e. cmpt 4077   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   0cc0 8737   1c1 8738    + caddc 8740    - cmin 9037   NN0cn0 9965    seq cseq 11046   sadd csad 12611   smul csmu 12612
This theorem is referenced by:  smuval  12672  smupvallem  12674  smucl  12675
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-tru 1310  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-smu 12667
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