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Theorem signlem0 27119
Description: Adding a zero as the highest coefficient does not change the parity of the sign changes. (Contributed by Thierry Arnoux, 12-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
signlem0  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( V `  ( F concat  <" 0 "> ) )  =  ( V `  F
) )
Distinct variable groups:    a, b,  .+^    f, i, n, F    f, W, i, n    f, a, i, j, b, F, n    T, a    n, b, T, f, j
Allowed substitution hints:    .+^ ( f, i,
j, n)    T( i)    V( f, i, j, n, a, b)    W( j, a, b)

Proof of Theorem signlem0
StepHypRef Expression
1 0re 9484 . . . 4  |-  0  e.  RR
21a1i 11 . . 3  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
0  e.  RR )
3 signsv.p . . . 4  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
4 signsv.w . . . 4  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
5 signsv.t . . . 4  |-  T  =  ( f  e. Word  RR  |->  ( n  e.  (
0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) ) )
6 signsv.v . . . 4  |-  V  =  ( f  e. Word  RR  |->  sum_ j  e.  ( 1..^ ( # `  f
) ) if ( ( ( T `  f ) `  j
)  =/=  ( ( T `  f ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
73, 4, 5, 6signsvfn 27114 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  0  e.  RR )  ->  ( V `
 ( F concat  <" 0 "> ) )  =  ( ( V `  F )  +  if ( ( ( ( T `  F ) `
 ( ( # `  F )  -  1 ) )  x.  0 )  <  0 ,  1 ,  0 ) ) )
82, 7mpdan 668 . 2  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( V `  ( F concat  <" 0 "> ) )  =  ( ( V `  F )  +  if ( ( ( ( T `  F ) `
 ( ( # `  F )  -  1 ) )  x.  0 )  <  0 ,  1 ,  0 ) ) )
91ltnri 9581 . . . . 5  |-  -.  0  <  0
10 neg1cn 10523 . . . . . . . . 9  |-  -u 1  e.  CC
11 ax-1cn 9438 . . . . . . . . 9  |-  1  e.  CC
12 prssi 4124 . . . . . . . . 9  |-  ( (
-u 1  e.  CC  /\  1  e.  CC )  ->  { -u 1 ,  1 }  C_  CC )
1310, 11, 12mp2an 672 . . . . . . . 8  |-  { -u
1 ,  1 } 
C_  CC
14 simpl 457 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  ->  F  e.  (Word  RR  \  { (/) } ) )
15 simpr 461 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( F `  0
)  =/=  0 )
16 eldifsn 4095 . . . . . . . . . . . 12  |-  ( F  e.  (Word  RR  \  { (/) } )  <->  ( F  e. Word  RR  /\  F  =/=  (/) ) )
1714, 16sylib 196 . . . . . . . . . . 11  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( F  e. Word  RR  /\  F  =/=  (/) ) )
18 lennncl 12349 . . . . . . . . . . 11  |-  ( ( F  e. Word  RR  /\  F  =/=  (/) )  ->  ( # `
 F )  e.  NN )
1917, 18syl 16 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( # `  F )  e.  NN )
20 fzo0end 11717 . . . . . . . . . 10  |-  ( (
# `  F )  e.  NN  ->  ( ( # `
 F )  - 
1 )  e.  ( 0..^ ( # `  F
) ) )
2119, 20syl 16 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( ( # `  F
)  -  1 )  e.  ( 0..^ (
# `  F )
) )
223, 4, 5, 6signstfvcl 27105 . . . . . . . . 9  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( ( # `
 F )  - 
1 )  e.  ( 0..^ ( # `  F
) ) )  -> 
( ( T `  F ) `  (
( # `  F )  -  1 ) )  e.  { -u 1 ,  1 } )
2314, 15, 21, 22syl21anc 1218 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( ( T `  F ) `  (
( # `  F )  -  1 ) )  e.  { -u 1 ,  1 } )
2413, 23sseldi 3449 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( ( T `  F ) `  (
( # `  F )  -  1 ) )  e.  CC )
2524mul01d 9666 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( ( ( T `
 F ) `  ( ( # `  F
)  -  1 ) )  x.  0 )  =  0 )
2625breq1d 4397 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( ( ( ( T `  F ) `
 ( ( # `  F )  -  1 ) )  x.  0 )  <  0  <->  0  <  0 ) )
279, 26mtbiri 303 . . . 4  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  ->  -.  ( ( ( T `
 F ) `  ( ( # `  F
)  -  1 ) )  x.  0 )  <  0 )
28 iffalse 3894 . . . 4  |-  ( -.  ( ( ( T `
 F ) `  ( ( # `  F
)  -  1 ) )  x.  0 )  <  0  ->  if ( ( ( ( T `  F ) `
 ( ( # `  F )  -  1 ) )  x.  0 )  <  0 ,  1 ,  0 )  =  0 )
2927, 28syl 16 . . 3  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  ->  if ( ( ( ( T `  F ) `
 ( ( # `  F )  -  1 ) )  x.  0 )  <  0 ,  1 ,  0 )  =  0 )
3029oveq2d 6203 . 2  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( ( V `  F )  +  if ( ( ( ( T `  F ) `
 ( ( # `  F )  -  1 ) )  x.  0 )  <  0 ,  1 ,  0 ) )  =  ( ( V `  F )  +  0 ) )
313, 4, 5, 6signsvvf 27111 . . . . . 6  |-  V :Word  RR
--> NN0
3231a1i 11 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  ->  V :Word  RR --> NN0 )
3314eldifad 3435 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  ->  F  e. Word  RR )
3432, 33ffvelrnd 5940 . . . 4  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( V `  F
)  e.  NN0 )
3534nn0cnd 10736 . . 3  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( V `  F
)  e.  CC )
3635addid1d 9667 . 2  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( ( V `  F )  +  0 )  =  ( V `
 F ) )
378, 30, 363eqtrd 2495 1  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( V `  ( F concat  <" 0 "> ) )  =  ( V `  F
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2642    \ cdif 3420    C_ wss 3423   (/)c0 3732   ifcif 3886   {csn 3972   {cpr 3974   {ctp 3976   <.cop 3978   class class class wbr 4387    |-> cmpt 4445   -->wf 5509   ` cfv 5513  (class class class)co 6187    |-> cmpt2 6189   CCcc 9378   RRcr 9379   0cc0 9380   1c1 9381    + caddc 9383    x. cmul 9385    < clt 9516    - cmin 9693   -ucneg 9694   NNcn 10420   NN0cn0 10677   ...cfz 11535  ..^cfzo 11646   #chash 12201  Word cword 12320   concat cconcat 12322   <"cs1 12323  sgncsgn 12674   sum_csu 13262   ndxcnx 14270   Basecbs 14273   +g cplusg 14337    gsumg cgsu 14478
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-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-inf2 7945  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457  ax-pre-sup 9458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  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 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-se 4775  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-isom 5522  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-supp 6788  df-recs 6929  df-rdg 6963  df-1o 7017  df-oadd 7021  df-er 7198  df-en 7408  df-dom 7409  df-sdom 7410  df-fin 7411  df-sup 7789  df-oi 7822  df-card 8207  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-div 10092  df-nn 10421  df-2 10478  df-3 10479  df-n0 10678  df-z 10745  df-uz 10960  df-rp 11090  df-fz 11536  df-fzo 11647  df-seq 11905  df-exp 11964  df-hash 12202  df-word 12328  df-concat 12330  df-s1 12331  df-substr 12332  df-sgn 12675  df-cj 12687  df-re 12688  df-im 12689  df-sqr 12823  df-abs 12824  df-clim 13065  df-sum 13263  df-struct 14275  df-ndx 14276  df-slot 14277  df-base 14278  df-plusg 14350  df-0g 14479  df-gsum 14480  df-mnd 15514  df-mulg 15647  df-cntz 15934
This theorem is referenced by: (None)
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