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Theorem signsvvfval 28172
Description: The value of  V, which represents the number of times the sign changes in a word. (Contributed by Thierry Arnoux, 7-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
signsvvfval  |-  ( F  e. Word  RR  ->  ( V `
 F )  = 
sum_ j  e.  ( 1..^ ( # `  F
) ) if ( ( ( T `  F ) `  j
)  =/=  ( ( T `  F ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
Distinct variable groups:    a, b,  .+^    f, i, n, F    f, W, i, n    f, j, F    T, f
Allowed substitution hints:    .+^ ( f, i,
j, n)    T( i,
j, n, a, b)    F( a, b)    V( f, i, j, n, a, b)    W( j, a, b)

Proof of Theorem signsvvfval
StepHypRef Expression
1 fveq2 5864 . . . 4  |-  ( f  =  F  ->  ( # `
 f )  =  ( # `  F
) )
21oveq2d 6298 . . 3  |-  ( f  =  F  ->  (
1..^ ( # `  f
) )  =  ( 1..^ ( # `  F
) ) )
3 fveq2 5864 . . . . . . . 8  |-  ( f  =  F  ->  ( T `  f )  =  ( T `  F ) )
4 eqidd 2468 . . . . . . . 8  |-  ( f  =  F  ->  j  =  j )
53, 4fveq12d 5870 . . . . . . 7  |-  ( f  =  F  ->  (
( T `  f
) `  j )  =  ( ( T `
 F ) `  j ) )
6 eqidd 2468 . . . . . . . 8  |-  ( f  =  F  ->  (
j  -  1 )  =  ( j  - 
1 ) )
73, 6fveq12d 5870 . . . . . . 7  |-  ( f  =  F  ->  (
( T `  f
) `  ( j  -  1 ) )  =  ( ( T `
 F ) `  ( j  -  1 ) ) )
85, 7neeq12d 2746 . . . . . 6  |-  ( f  =  F  ->  (
( ( T `  f ) `  j
)  =/=  ( ( T `  f ) `
 ( j  - 
1 ) )  <->  ( ( T `  F ) `  j )  =/=  (
( T `  F
) `  ( j  -  1 ) ) ) )
98ifbid 3961 . . . . 5  |-  ( f  =  F  ->  if ( ( ( T `
 f ) `  j )  =/=  (
( T `  f
) `  ( j  -  1 ) ) ,  1 ,  0 )  =  if ( ( ( T `  F ) `  j
)  =/=  ( ( T `  F ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
109ralrimivw 2879 . . . 4  |-  ( f  =  F  ->  A. j  e.  ( 1..^ ( # `  f ) ) if ( ( ( T `
 f ) `  j )  =/=  (
( T `  f
) `  ( j  -  1 ) ) ,  1 ,  0 )  =  if ( ( ( T `  F ) `  j
)  =/=  ( ( T `  F ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
1110r19.21bi 2833 . . 3  |-  ( ( f  =  F  /\  j  e.  ( 1..^ ( # `  f
) ) )  ->  if ( ( ( T `
 f ) `  j )  =/=  (
( T `  f
) `  ( j  -  1 ) ) ,  1 ,  0 )  =  if ( ( ( T `  F ) `  j
)  =/=  ( ( T `  F ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
122, 11sumeq12dv 13484 . 2  |-  ( f  =  F  ->  sum_ j  e.  ( 1..^ ( # `  f ) ) if ( ( ( T `
 f ) `  j )  =/=  (
( T `  f
) `  ( j  -  1 ) ) ,  1 ,  0 )  =  sum_ j  e.  ( 1..^ ( # `  F ) ) if ( ( ( T `
 F ) `  j )  =/=  (
( T `  F
) `  ( j  -  1 ) ) ,  1 ,  0 ) )
13 signsv.v . 2  |-  V  =  ( f  e. Word  RR  |->  sum_ j  e.  ( 1..^ ( # `  f
) ) if ( ( ( T `  f ) `  j
)  =/=  ( ( T `  f ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
14 sumex 13466 . 2  |-  sum_ j  e.  ( 1..^ ( # `  F ) ) if ( ( ( T `
 F ) `  j )  =/=  (
( T `  F
) `  ( j  -  1 ) ) ,  1 ,  0 )  e.  _V
1512, 13, 14fvmpt 5948 1  |-  ( F  e. Word  RR  ->  ( V `
 F )  = 
sum_ j  e.  ( 1..^ ( # `  F
) ) if ( ( ( T `  F ) `  j
)  =/=  ( ( T `  F ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    =/= wne 2662   ifcif 3939   {cpr 4029   {ctp 4031   <.cop 4033    |-> cmpt 4505   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   RRcr 9487   0cc0 9488   1c1 9489    - cmin 9801   -ucneg 9802   ...cfz 11668  ..^cfzo 11788   #chash 12367  Word cword 12494  sgncsgn 12876   sum_csu 13464   ndxcnx 14480   Basecbs 14483   +g cplusg 14548    gsumg cgsu 14689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-seq 12071  df-sum 13465
This theorem is referenced by:  signsvf0  28174  signsvf1  28175  signsvfn  28176
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