MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  smufval Structured version   Unicode version

Theorem smufval 14425
Description: The multiplication of two bit sequences as repeated sequence addition. (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 10875 . . . 4  |-  NN0  e.  _V
32elpw2 4589 . . 3  |-  ( A  e.  ~P NN0  <->  A  C_  NN0 )
41, 3sylibr 215 . 2  |-  ( ph  ->  A  e.  ~P NN0 )
5 smuval.b . . 3  |-  ( ph  ->  B  C_  NN0 )
62elpw2 4589 . . 3  |-  ( B  e.  ~P NN0  <->  B  C_  NN0 )
75, 6sylibr 215 . 2  |-  ( ph  ->  B  e.  ~P NN0 )
8 simp1l 1029 . . . . . . . . . . . . 13  |-  ( ( ( x  =  A  /\  y  =  B )  /\  p  e. 
~P NN0  /\  m  e.  NN0 )  ->  x  =  A )
98eleq2d 2499 . . . . . . . . . . . 12  |-  ( ( ( x  =  A  /\  y  =  B )  /\  p  e. 
~P NN0  /\  m  e.  NN0 )  ->  (
m  e.  x  <->  m  e.  A ) )
10 simp1r 1030 . . . . . . . . . . . . 13  |-  ( ( ( x  =  A  /\  y  =  B )  /\  p  e. 
~P NN0  /\  m  e.  NN0 )  ->  y  =  B )
1110eleq2d 2499 . . . . . . . . . . . 12  |-  ( ( ( x  =  A  /\  y  =  B )  /\  p  e. 
~P NN0  /\  m  e.  NN0 )  ->  (
( n  -  m
)  e.  y  <->  ( n  -  m )  e.  B
) )
129, 11anbi12d 715 . . . . . . . . . . 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 3079 . . . . . . . . . 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 6321 . . . . . . . . 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 6369 . . . . . . . 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 12217 . . . . . . 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 2488 . . . . . 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 5883 . . . . 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 2499 . . . 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 3079 . . 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 14424 . . 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 4576 . . 3  |-  { k  e.  NN0  |  k  e.  ( P `  (
k  +  1 ) ) }  e.  _V
2421, 22, 23ovmpt2a 6441 . 2  |-  ( ( A  e.  ~P NN0  /\  B  e.  ~P NN0 )  ->  ( A smul  B
)  =  { k  e.  NN0  |  k  e.  ( P `  (
k  +  1 ) ) } )
254, 7, 24syl2anc 665 1  |-  ( ph  ->  ( A smul  B )  =  { k  e. 
NN0  |  k  e.  ( P `  ( k  +  1 ) ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   {crab 2786    C_ wss 3442   (/)c0 3767   ifcif 3915   ~Pcpw 3985    |-> cmpt 4484   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   0cc0 9538   1c1 9539    + caddc 9541    - cmin 9859   NN0cn0 10869    seqcseq 12210   sadd csad 14368   smul csmu 14369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-i2m1 9606  ax-1ne0 9607  ax-rrecex 9610  ax-cnre 9611
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-nn 10610  df-n0 10870  df-seq 12211  df-smu 14424
This theorem is referenced by:  smuval  14429  smupvallem  14431  smucl  14432
  Copyright terms: Public domain W3C validator