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Theorem signstfval 28785
Description: Value of the zero-skipping sign word. (Contributed by Thierry Arnoux, 8-Oct-2018.)
Hypotheses
Ref Expression
signsv.p  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
signsv.w  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
signsv.t  |-  T  =  ( f  e. Word  RR  |->  ( n  e.  (
0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) ) )
signsv.v  |-  V  =  ( f  e. Word  RR  |->  sum_ j  e.  ( 1..^ ( # `  f
) ) if ( ( ( T `  f ) `  j
)  =/=  ( ( T `  f ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
Assertion
Ref Expression
signstfval  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( ( T `  F ) `  N
)  =  ( W 
gsumg  ( i  e.  ( 0 ... N ) 
|->  (sgn `  ( F `  i ) ) ) ) )
Distinct variable groups:    f, i, n, F    i, N, n   
f, W, n
Allowed substitution hints:    .+^ ( f, i,
j, n, a, b)    T( f, i, j, n, a, b)    F( j, a, b)    N( f, j, a, b)    V( f, i, j, n, a, b)    W( i, j, a, b)

Proof of Theorem signstfval
StepHypRef Expression
1 signsv.p . . . 4  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
2 signsv.w . . . 4  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
3 signsv.t . . . 4  |-  T  =  ( f  e. Word  RR  |->  ( n  e.  (
0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) ) )
4 signsv.v . . . 4  |-  V  =  ( f  e. Word  RR  |->  sum_ j  e.  ( 1..^ ( # `  f
) ) if ( ( ( T `  f ) `  j
)  =/=  ( ( T `  f ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
51, 2, 3, 4signstfv 28784 . . 3  |-  ( F  e. Word  RR  ->  ( T `
 F )  =  ( n  e.  ( 0..^ ( # `  F
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( F `  i ) ) ) ) ) )
65adantr 463 . 2  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( T `  F
)  =  ( n  e.  ( 0..^ (
# `  F )
)  |->  ( W  gsumg  ( i  e.  ( 0 ... n )  |->  (sgn `  ( F `  i ) ) ) ) ) )
7 simpr 459 . . . . 5  |-  ( ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  /\  n  =  N )  ->  n  =  N )
87oveq2d 6286 . . . 4  |-  ( ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  /\  n  =  N )  ->  ( 0 ... n
)  =  ( 0 ... N ) )
98mpteq1d 4520 . . 3  |-  ( ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  /\  n  =  N )  ->  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( F `  i ) ) )  =  ( i  e.  ( 0 ... N
)  |->  (sgn `  ( F `  i )
) ) )
109oveq2d 6286 . 2  |-  ( ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  /\  n  =  N )  ->  ( W  gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( F `  i ) ) ) )  =  ( W 
gsumg  ( i  e.  ( 0 ... N ) 
|->  (sgn `  ( F `  i ) ) ) ) )
11 simpr 459 . 2  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  ->  N  e.  ( 0..^ ( # `  F
) ) )
12 ovex 6298 . . 3  |-  ( W 
gsumg  ( i  e.  ( 0 ... N ) 
|->  (sgn `  ( F `  i ) ) ) )  e.  _V
1312a1i 11 . 2  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( W  gsumg  ( i  e.  ( 0 ... N ) 
|->  (sgn `  ( F `  i ) ) ) )  e.  _V )
146, 10, 11, 13fvmptd 5936 1  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( ( T `  F ) `  N
)  =  ( W 
gsumg  ( i  e.  ( 0 ... N ) 
|->  (sgn `  ( F `  i ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   _Vcvv 3106   ifcif 3929   {cpr 4018   {ctp 4020   <.cop 4022    |-> cmpt 4497   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   RRcr 9480   0cc0 9481   1c1 9482    - cmin 9796   -ucneg 9797   ...cfz 11675  ..^cfzo 11799   #chash 12387  Word cword 12518  sgncsgn 13001   sum_csu 13590   ndxcnx 14713   Basecbs 14716   +g cplusg 14784    gsumg cgsu 14930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273
This theorem is referenced by:  signstcl  28786  signstfvn  28790  signstfvp  28792
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