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Theorem smuval 12688
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 ) ) ) )
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 12684 . . 3  |-  ( ph  ->  ( A smul  B )  =  { k  e. 
NN0  |  k  e.  ( P `  ( k  +  1 ) ) } )
54eleq2d 2363 . 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 19 . . . . . 6  |-  ( k  =  N  ->  k  =  N )
8 oveq1 5881 . . . . . . 7  |-  ( k  =  N  ->  (
k  +  1 )  =  ( N  + 
1 ) )
98fveq2d 5545 . . . . . 6  |-  ( k  =  N  ->  ( P `  ( k  +  1 ) )  =  ( P `  ( N  +  1
) ) )
107, 9eleq12d 2364 . . . . 5  |-  ( k  =  N  ->  (
k  e.  ( P `
 ( k  +  1 ) )  <->  N  e.  ( P `  ( N  +  1 ) ) ) )
1110elrab 2936 . . . 4  |-  ( N  e.  { k  e. 
NN0  |  k  e.  ( P `  ( k  +  1 ) ) }  <->  ( N  e. 
NN0  /\  N  e.  ( P `  ( N  +  1 ) ) ) )
1211baib 871 . . 3  |-  ( N  e.  NN0  ->  ( N  e.  { k  e. 
NN0  |  k  e.  ( P `  ( k  +  1 ) ) }  <->  N  e.  ( P `  ( N  +  1 ) ) ) )
136, 12syl 15 . 2  |-  ( ph  ->  ( N  e.  {
k  e.  NN0  | 
k  e.  ( P `
 ( k  +  1 ) ) }  <-> 
N  e.  ( P `
 ( N  + 
1 ) ) ) )
145, 13bitrd 244 1  |-  ( ph  ->  ( N  e.  ( A smul  B )  <->  N  e.  ( P `  ( N  +  1 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = 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   0cc0 8753   1c1 8754    + caddc 8756    - cmin 9053   NN0cn0 9981    seq cseq 11062   sadd csad 12627   smul csmu 12628
This theorem is referenced by:  smuval2  12689  smupvallem  12690  smu01lem  12692
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-tru 1310  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-smu 12683
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