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Theorem signstfv 27103
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 5794 . . . 4  |-  ( f  =  F  ->  ( # `
 f )  =  ( # `  F
) )
21oveq2d 6211 . . 3  |-  ( f  =  F  ->  (
0..^ ( # `  f
) )  =  ( 0..^ ( # `  F
) ) )
3 simpl 457 . . . . . . 7  |-  ( ( f  =  F  /\  i  e.  ( 0 ... n ) )  ->  f  =  F )
43fveq1d 5796 . . . . . 6  |-  ( ( f  =  F  /\  i  e.  ( 0 ... n ) )  ->  ( f `  i )  =  ( F `  i ) )
54fveq2d 5798 . . . . 5  |-  ( ( f  =  F  /\  i  e.  ( 0 ... n ) )  ->  (sgn `  (
f `  i )
)  =  (sgn `  ( F `  i ) ) )
65mpteq2dva 4481 . . . 4  |-  ( f  =  F  ->  (
i  e.  ( 0 ... n )  |->  (sgn
`  ( f `  i ) ) )  =  ( i  e.  ( 0 ... n
)  |->  (sgn `  ( F `  i )
) ) )
76oveq2d 6211 . . 3  |-  ( f  =  F  ->  ( W  gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) )  =  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( F `  i ) ) ) ) )
82, 7mpteq12dv 4473 . 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 6220 . . 3  |-  ( 0..^ ( # `  F
) )  e.  _V
1110mptex 6052 . 2  |-  ( n  e.  ( 0..^ (
# `  F )
)  |->  ( W  gsumg  ( i  e.  ( 0 ... n )  |->  (sgn `  ( F `  i ) ) ) ) )  e.  _V
128, 9, 11fvmpt 5878 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 369    = wceq 1370    e. wcel 1758    =/= wne 2645   ifcif 3894   {cpr 3982   {ctp 3984   <.cop 3986    |-> cmpt 4453   ` cfv 5521  (class class class)co 6195    |-> cmpt2 6197   RRcr 9387   0cc0 9388   1c1 9389    - cmin 9701   -ucneg 9702   ...cfz 11549  ..^cfzo 11660   #chash 12215  Word cword 12334  sgncsgn 12688   sum_csu 13276   ndxcnx 14284   Basecbs 14287   +g cplusg 14352    gsumg cgsu 14493
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 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pr 4634
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198
This theorem is referenced by:  signstfval  27104  signstf  27106  signstlen  27107  signstf0  27108
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