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Theorem signstfv 28703
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
signstfv  |-  ( F  e. Word  RR  ->  ( T `
 F )  =  ( n  e.  ( 0..^ ( # `  F
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( F `  i ) ) ) ) ) )
Distinct variable groups:    f, i, n, F    f, W
Allowed substitution hints:    .+^ ( f, i,
j, n, a, b)    T( f, i, j, n, a, b)    F( j, a, b)    V( f, i, j, n, a, b)    W( i, j, n, a, b)

Proof of Theorem signstfv
StepHypRef Expression
1 fveq2 5774 . . . 4  |-  ( f  =  F  ->  ( # `
 f )  =  ( # `  F
) )
21oveq2d 6212 . . 3  |-  ( f  =  F  ->  (
0..^ ( # `  f
) )  =  ( 0..^ ( # `  F
) ) )
3 simpl 455 . . . . . . 7  |-  ( ( f  =  F  /\  i  e.  ( 0 ... n ) )  ->  f  =  F )
43fveq1d 5776 . . . . . 6  |-  ( ( f  =  F  /\  i  e.  ( 0 ... n ) )  ->  ( f `  i )  =  ( F `  i ) )
54fveq2d 5778 . . . . 5  |-  ( ( f  =  F  /\  i  e.  ( 0 ... n ) )  ->  (sgn `  (
f `  i )
)  =  (sgn `  ( F `  i ) ) )
65mpteq2dva 4453 . . . 4  |-  ( f  =  F  ->  (
i  e.  ( 0 ... n )  |->  (sgn
`  ( f `  i ) ) )  =  ( i  e.  ( 0 ... n
)  |->  (sgn `  ( F `  i )
) ) )
76oveq2d 6212 . . 3  |-  ( f  =  F  ->  ( W  gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) )  =  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( F `  i ) ) ) ) )
82, 7mpteq12dv 4445 . 2  |-  ( f  =  F  ->  (
n  e.  ( 0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) )  =  ( n  e.  ( 0..^ ( # `  F
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( F `  i ) ) ) ) ) )
9 signsv.t . 2  |-  T  =  ( f  e. Word  RR  |->  ( n  e.  (
0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) ) )
10 ovex 6224 . . 3  |-  ( 0..^ ( # `  F
) )  e.  _V
1110mptex 6044 . 2  |-  ( n  e.  ( 0..^ (
# `  F )
)  |->  ( W  gsumg  ( i  e.  ( 0 ... n )  |->  (sgn `  ( F `  i ) ) ) ) )  e.  _V
128, 9, 11fvmpt 5857 1  |-  ( F  e. Word  RR  ->  ( T `
 F )  =  ( n  e.  ( 0..^ ( # `  F
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( F `  i ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826    =/= wne 2577   ifcif 3857   {cpr 3946   {ctp 3948   <.cop 3950    |-> cmpt 4425   ` cfv 5496  (class class class)co 6196    |-> cmpt2 6198   RRcr 9402   0cc0 9403   1c1 9404    - cmin 9718   -ucneg 9719   ...cfz 11593  ..^cfzo 11717   #chash 12307  Word cword 12438  sgncsgn 12921   sum_csu 13510   ndxcnx 14631   Basecbs 14634   +g cplusg 14702    gsumg cgsu 14848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199
This theorem is referenced by:  signstfval  28704  signstf  28706  signstlen  28707  signstf0  28708
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