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Theorem signlem0 28727
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 ++  <" 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 9507 . . 3  |-  0  e.  RR
2 signsv.p . . . 4  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
3 signsv.w . . . 4  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
4 signsv.t . . . 4  |-  T  =  ( f  e. Word  RR  |->  ( n  e.  (
0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) ) )
5 signsv.v . . . 4  |-  V  =  ( f  e. Word  RR  |->  sum_ j  e.  ( 1..^ ( # `  f
) ) if ( ( ( T `  f ) `  j
)  =/=  ( ( T `  f ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
62, 3, 4, 5signsvfn 28722 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  0  e.  RR )  ->  ( V `
 ( F ++  <" 0 "> )
)  =  ( ( V `  F )  +  if ( ( ( ( T `  F ) `  (
( # `  F )  -  1 ) )  x.  0 )  <  0 ,  1 ,  0 ) ) )
71, 6mpan2 669 . 2  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( V `  ( F ++  <" 0 "> ) )  =  ( ( V `  F )  +  if ( ( ( ( T `  F ) `
 ( ( # `  F )  -  1 ) )  x.  0 )  <  0 ,  1 ,  0 ) ) )
81ltnri 9604 . . . . 5  |-  -.  0  <  0
9 neg1cn 10556 . . . . . . . . 9  |-  -u 1  e.  CC
10 ax-1cn 9461 . . . . . . . . 9  |-  1  e.  CC
11 prssi 4100 . . . . . . . . 9  |-  ( (
-u 1  e.  CC  /\  1  e.  CC )  ->  { -u 1 ,  1 }  C_  CC )
129, 10, 11mp2an 670 . . . . . . . 8  |-  { -u
1 ,  1 } 
C_  CC
13 simpl 455 . . . . . . . . . . 11  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  ->  F  e.  (Word  RR  \  { (/) } ) )
14 eldifsn 4069 . . . . . . . . . . 11  |-  ( F  e.  (Word  RR  \  { (/) } )  <->  ( F  e. Word  RR  /\  F  =/=  (/) ) )
1513, 14sylib 196 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( F  e. Word  RR  /\  F  =/=  (/) ) )
16 lennncl 12470 . . . . . . . . . 10  |-  ( ( F  e. Word  RR  /\  F  =/=  (/) )  ->  ( # `
 F )  e.  NN )
17 fzo0end 11803 . . . . . . . . . 10  |-  ( (
# `  F )  e.  NN  ->  ( ( # `
 F )  - 
1 )  e.  ( 0..^ ( # `  F
) ) )
1815, 16, 173syl 20 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( ( # `  F
)  -  1 )  e.  ( 0..^ (
# `  F )
) )
192, 3, 4, 5signstfvcl 28713 . . . . . . . . 9  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( ( # `
 F )  - 
1 )  e.  ( 0..^ ( # `  F
) ) )  -> 
( ( T `  F ) `  (
( # `  F )  -  1 ) )  e.  { -u 1 ,  1 } )
2018, 19mpdan 666 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( ( T `  F ) `  (
( # `  F )  -  1 ) )  e.  { -u 1 ,  1 } )
2112, 20sseldi 3415 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( ( T `  F ) `  (
( # `  F )  -  1 ) )  e.  CC )
2221mul01d 9690 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( ( ( T `
 F ) `  ( ( # `  F
)  -  1 ) )  x.  0 )  =  0 )
2322breq1d 4377 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( ( ( ( T `  F ) `
 ( ( # `  F )  -  1 ) )  x.  0 )  <  0  <->  0  <  0 ) )
248, 23mtbiri 301 . . . 4  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  ->  -.  ( ( ( T `
 F ) `  ( ( # `  F
)  -  1 ) )  x.  0 )  <  0 )
2524iffalsed 3868 . . 3  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  ->  if ( ( ( ( T `  F ) `
 ( ( # `  F )  -  1 ) )  x.  0 )  <  0 ,  1 ,  0 )  =  0 )
2625oveq2d 6212 . 2  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( ( V `  F )  +  if ( ( ( ( T `  F ) `
 ( ( # `  F )  -  1 ) )  x.  0 )  <  0 ,  1 ,  0 ) )  =  ( ( V `  F )  +  0 ) )
272, 3, 4, 5signsvvf 28719 . . . . . 6  |-  V :Word  RR
--> NN0
2827a1i 11 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  ->  V :Word  RR --> NN0 )
2913eldifad 3401 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  ->  F  e. Word  RR )
3028, 29ffvelrnd 5934 . . . 4  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( V `  F
)  e.  NN0 )
3130nn0cnd 10771 . . 3  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( V `  F
)  e.  CC )
3231addid1d 9691 . 2  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( ( V `  F )  +  0 )  =  ( V `
 F ) )
337, 26, 323eqtrd 2427 1  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  0 )  =/=  0 )  -> 
( V `  ( F ++  <" 0 "> ) )  =  ( V `  F
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826    =/= wne 2577    \ cdif 3386    C_ wss 3389   (/)c0 3711   ifcif 3857   {csn 3944   {cpr 3946   {ctp 3948   <.cop 3950   class class class wbr 4367    |-> cmpt 4425   -->wf 5492   ` cfv 5496  (class class class)co 6196    |-> cmpt2 6198   CCcc 9401   RRcr 9402   0cc0 9403   1c1 9404    + caddc 9406    x. cmul 9408    < clt 9539    - cmin 9718   -ucneg 9719   NNcn 10452   NN0cn0 10712   ...cfz 11593  ..^cfzo 11717   #chash 12307  Word cword 12438   ++ cconcat 12440   <"cs1 12441  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-8 1828  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-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  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-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  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-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  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-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-supp 6818  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-sup 7816  df-oi 7850  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-n0 10713  df-z 10782  df-uz 11002  df-rp 11140  df-fz 11594  df-fzo 11718  df-seq 12011  df-exp 12070  df-hash 12308  df-word 12446  df-lsw 12447  df-concat 12448  df-s1 12449  df-substr 12450  df-sgn 12922  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-clim 13313  df-sum 13511  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-plusg 14715  df-0g 14849  df-gsum 14850  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-mulg 16177  df-cntz 16472
This theorem is referenced by: (None)
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