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Theorem signstfval 27102
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 27101 . . 3  |-  ( F  e. Word  RR  ->  ( T `
 F )  =  ( n  e.  ( 0..^ ( # `  F
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( F `  i ) ) ) ) ) )
65adantr 465 . 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 461 . . . . 5  |-  ( ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  /\  n  =  N )  ->  n  =  N )
87oveq2d 6209 . . . 4  |-  ( ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  /\  n  =  N )  ->  ( 0 ... n
)  =  ( 0 ... N ) )
98mpteq1d 4474 . . 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 6209 . 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 461 . 2  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  ->  N  e.  ( 0..^ ( # `  F
) ) )
12 ovex 6218 . . 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 5881 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 369    = wceq 1370    e. wcel 1758    =/= wne 2644   _Vcvv 3071   ifcif 3892   {cpr 3980   {ctp 3982   <.cop 3984    |-> cmpt 4451   ` cfv 5519  (class class class)co 6193    |-> cmpt2 6195   RRcr 9385   0cc0 9386   1c1 9387    - cmin 9699   -ucneg 9700   ...cfz 11547  ..^cfzo 11658   #chash 12213  Word cword 12332  sgncsgn 12686   sum_csu 13274   ndxcnx 14282   Basecbs 14285   +g cplusg 14349    gsumg cgsu 14490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196
This theorem is referenced by:  signstcl  27103  signstfvn  27107  signstfvp  27109
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