Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  signstfvp Structured version   Unicode version

Theorem signstfvp 28165
Description: Zero-skipping sign in a word compared to a shorter 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
signstfvp  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 ( F concat  <" K "> ) ) `  N )  =  ( ( T `  F
) `  N )
)
Distinct variable groups:    a, b,  .+^    f, i, n, F    f, K, i, n    f, W, i, n    i, N, n
Allowed substitution hints:    .+^ ( f, i,
j, n)    T( f,
i, j, n, a, b)    F( j, a, b)    K( j, a, b)    N( f, j, a, b)    V( f, i, j, n, a, b)    W( j, a, b)

Proof of Theorem signstfvp
StepHypRef Expression
1 simp1 996 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  F  e. Word  RR )
21adantr 465 . . . . . 6  |-  ( ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  /\  i  e.  ( 0 ... N ) )  ->  F  e. Word  RR )
3 s1cl 12571 . . . . . . . 8  |-  ( K  e.  RR  ->  <" K ">  e. Word  RR )
433ad2ant2 1018 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  <" K ">  e. Word  RR )
54adantr 465 . . . . . 6  |-  ( ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  /\  i  e.  ( 0 ... N ) )  ->  <" K ">  e. Word  RR )
6 simp3 998 . . . . . . . 8  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  N  e.  ( 0..^ ( # `  F
) ) )
7 fzssfzo 28127 . . . . . . . 8  |-  ( N  e.  ( 0..^ (
# `  F )
)  ->  ( 0 ... N )  C_  ( 0..^ ( # `  F
) ) )
86, 7syl 16 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( 0 ... N )  C_  (
0..^ ( # `  F
) ) )
98sselda 3504 . . . . . 6  |-  ( ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  /\  i  e.  ( 0 ... N ) )  ->  i  e.  ( 0..^ ( # `  F
) ) )
10 ccatval1 12554 . . . . . 6  |-  ( ( F  e. Word  RR  /\  <" K ">  e. Word  RR  /\  i  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F concat  <" K "> ) `  i )  =  ( F `  i ) )
112, 5, 9, 10syl3anc 1228 . . . . 5  |-  ( ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  /\  i  e.  ( 0 ... N ) )  ->  ( ( F concat  <" K "> ) `  i )  =  ( F `  i ) )
1211fveq2d 5868 . . . 4  |-  ( ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  /\  i  e.  ( 0 ... N ) )  ->  (sgn `  (
( F concat  <" K "> ) `  i
) )  =  (sgn
`  ( F `  i ) ) )
1312mpteq2dva 4533 . . 3  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( i  e.  ( 0 ... N
)  |->  (sgn `  (
( F concat  <" K "> ) `  i
) ) )  =  ( i  e.  ( 0 ... N ) 
|->  (sgn `  ( F `  i ) ) ) )
1413oveq2d 6298 . 2  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( W  gsumg  ( i  e.  ( 0 ... N )  |->  (sgn `  ( ( F concat  <" K "> ) `  i
) ) ) )  =  ( W  gsumg  ( i  e.  ( 0 ... N )  |->  (sgn `  ( F `  i ) ) ) ) )
15 ccatcl 12552 . . . 4  |-  ( ( F  e. Word  RR  /\  <" K ">  e. Word  RR )  ->  ( F concat  <" K "> )  e. Word  RR )
161, 4, 15syl2anc 661 . . 3  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( F concat  <" K "> )  e. Word  RR )
17 lencl 12522 . . . . . . . 8  |-  ( F  e. Word  RR  ->  ( # `  F )  e.  NN0 )
1817nn0zd 10960 . . . . . . 7  |-  ( F  e. Word  RR  ->  ( # `  F )  e.  ZZ )
19 uzid 11092 . . . . . . 7  |-  ( (
# `  F )  e.  ZZ  ->  ( # `  F
)  e.  ( ZZ>= `  ( # `  F ) ) )
20 peano2uz 11130 . . . . . . 7  |-  ( (
# `  F )  e.  ( ZZ>= `  ( # `  F
) )  ->  (
( # `  F )  +  1 )  e.  ( ZZ>= `  ( # `  F
) ) )
21 fzoss2 11817 . . . . . . 7  |-  ( ( ( # `  F
)  +  1 )  e.  ( ZZ>= `  ( # `
 F ) )  ->  ( 0..^ (
# `  F )
)  C_  ( 0..^ ( ( # `  F
)  +  1 ) ) )
2218, 19, 20, 214syl 21 . . . . . 6  |-  ( F  e. Word  RR  ->  ( 0..^ ( # `  F
) )  C_  (
0..^ ( ( # `  F )  +  1 ) ) )
23223ad2ant1 1017 . . . . 5  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( 0..^ (
# `  F )
)  C_  ( 0..^ ( ( # `  F
)  +  1 ) ) )
2423, 6sseldd 3505 . . . 4  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  N  e.  ( 0..^ ( ( # `  F )  +  1 ) ) )
25 ccatlen 12553 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  <" K ">  e. Word  RR )  ->  ( # `
 ( F concat  <" K "> ) )  =  ( ( # `  F
)  +  ( # `  <" K "> ) ) )
261, 4, 25syl2anc 661 . . . . . 6  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( # `  ( F concat  <" K "> ) )  =  ( ( # `  F
)  +  ( # `  <" K "> ) ) )
27 s1len 12574 . . . . . . 7  |-  ( # `  <" K "> )  =  1
2827oveq2i 6293 . . . . . 6  |-  ( (
# `  F )  +  ( # `  <" K "> )
)  =  ( (
# `  F )  +  1 )
2926, 28syl6eq 2524 . . . . 5  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( # `  ( F concat  <" K "> ) )  =  ( ( # `  F
)  +  1 ) )
3029oveq2d 6298 . . . 4  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( 0..^ (
# `  ( F concat  <" K "> ) ) )  =  ( 0..^ ( (
# `  F )  +  1 ) ) )
3124, 30eleqtrrd 2558 . . 3  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  N  e.  ( 0..^ ( # `  ( F concat  <" K "> ) ) ) )
32 signsv.p . . . 4  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
33 signsv.w . . . 4  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
34 signsv.t . . . 4  |-  T  =  ( f  e. Word  RR  |->  ( n  e.  (
0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) ) )
35 signsv.v . . . 4  |-  V  =  ( f  e. Word  RR  |->  sum_ j  e.  ( 1..^ ( # `  f
) ) if ( ( ( T `  f ) `  j
)  =/=  ( ( T `  f ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
3632, 33, 34, 35signstfval 28158 . . 3  |-  ( ( ( F concat  <" K "> )  e. Word  RR  /\  N  e.  ( 0..^ ( # `  ( F concat  <" K "> ) ) ) )  ->  ( ( T `
 ( F concat  <" K "> ) ) `  N )  =  ( W  gsumg  ( i  e.  ( 0 ... N ) 
|->  (sgn `  ( ( F concat  <" K "> ) `  i ) ) ) ) )
3716, 31, 36syl2anc 661 . 2  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 ( F concat  <" K "> ) ) `  N )  =  ( W  gsumg  ( i  e.  ( 0 ... N ) 
|->  (sgn `  ( ( F concat  <" K "> ) `  i ) ) ) ) )
3832, 33, 34, 35signstfval 28158 . . 3  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( ( T `  F ) `  N
)  =  ( W 
gsumg  ( i  e.  ( 0 ... N ) 
|->  (sgn `  ( F `  i ) ) ) ) )
391, 6, 38syl2anc 661 . 2  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 F ) `  N )  =  ( W  gsumg  ( i  e.  ( 0 ... N ) 
|->  (sgn `  ( F `  i ) ) ) ) )
4014, 37, 393eqtr4d 2518 1  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 ( F concat  <" K "> ) ) `  N )  =  ( ( T `  F
) `  N )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662    C_ wss 3476   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    + caddc 9491    - cmin 9801   -ucneg 9802   ZZcz 10860   ZZ>=cuz 11078   ...cfz 11668  ..^cfzo 11788   #chash 12367  Word cword 12494   concat cconcat 12496   <"cs1 12497  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-rep 4558  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-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-int 4283  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-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  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-fzo 11789  df-hash 12368  df-word 12502  df-concat 12504  df-s1 12505
This theorem is referenced by:  signstfvneq0  28166  signstfvc  28168  signstfveq0  28171  signsvfn  28176
  Copyright terms: Public domain W3C validator