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Theorem smuval 12948
Description: Define the addition of two bit sequences, using df-had 1386 and df-cad 1387 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 ) ) ) )
smuval.n  |-  ( ph  ->  N  e.  NN0 )
Assertion
Ref Expression
smuval  |-  ( ph  ->  ( N  e.  ( A smul  B )  <->  N  e.  ( P `  ( N  +  1 ) ) ) )
Distinct variable groups:    m, n, p, A    n, N    ph, n    B, m, n, p
Allowed substitution hints:    ph( m, p)    P( m, n, p)    N( m, p)

Proof of Theorem smuval
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 smuval.a . . . 4  |-  ( ph  ->  A  C_  NN0 )
2 smuval.b . . . 4  |-  ( ph  ->  B  C_  NN0 )
3 smuval.p . . . 4  |-  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 ) ) ) )
41, 2, 3smufval 12944 . . 3  |-  ( ph  ->  ( A smul  B )  =  { k  e. 
NN0  |  k  e.  ( P `  ( k  +  1 ) ) } )
54eleq2d 2471 . 2  |-  ( ph  ->  ( N  e.  ( A smul  B )  <->  N  e.  { k  e.  NN0  | 
k  e.  ( P `
 ( k  +  1 ) ) } ) )
6 smuval.n . . 3  |-  ( ph  ->  N  e.  NN0 )
7 id 20 . . . . 5  |-  ( k  =  N  ->  k  =  N )
8 oveq1 6047 . . . . . 6  |-  ( k  =  N  ->  (
k  +  1 )  =  ( N  + 
1 ) )
98fveq2d 5691 . . . . 5  |-  ( k  =  N  ->  ( P `  ( k  +  1 ) )  =  ( P `  ( N  +  1
) ) )
107, 9eleq12d 2472 . . . 4  |-  ( k  =  N  ->  (
k  e.  ( P `
 ( k  +  1 ) )  <->  N  e.  ( P `  ( N  +  1 ) ) ) )
1110elrab3 3053 . . 3  |-  ( N  e.  NN0  ->  ( N  e.  { k  e. 
NN0  |  k  e.  ( P `  ( k  +  1 ) ) }  <->  N  e.  ( P `  ( N  +  1 ) ) ) )
126, 11syl 16 . 2  |-  ( ph  ->  ( N  e.  {
k  e.  NN0  | 
k  e.  ( P `
 ( k  +  1 ) ) }  <-> 
N  e.  ( P `
 ( N  + 
1 ) ) ) )
135, 12bitrd 245 1  |-  ( ph  ->  ( N  e.  ( A smul  B )  <->  N  e.  ( P `  ( N  +  1 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   {crab 2670    C_ wss 3280   (/)c0 3588   ifcif 3699   ~Pcpw 3759    e. cmpt 4226   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   0cc0 8946   1c1 8947    + caddc 8949    - cmin 9247   NN0cn0 10177    seq cseq 11278   sadd csad 12887   smul csmu 12888
This theorem is referenced by:  smuval2  12949  smupvallem  12950  smu01lem  12952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-i2m1 9014  ax-1ne0 9015  ax-rrecex 9018  ax-cnre 9019
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-recs 6592  df-rdg 6627  df-nn 9957  df-n0 10178  df-seq 11279  df-smu 12943
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