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

Theorem signsvtp 27121
Description: Adding a letter of the same sign as the highest coefficient does not change the sign. (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 ) )
signsvf.e  |-  ( ph  ->  E  e.  (Word  RR  \  { (/) } ) )
signsvf.0  |-  ( ph  ->  ( E `  0
)  =/=  0 )
signsvf.f  |-  ( ph  ->  F  =  ( E concat  <" A "> ) )
signsvf.a  |-  ( ph  ->  A  e.  RR )
signsvf.n  |-  N  =  ( # `  E
)
signsvt.b  |-  B  =  ( ( T `  E ) `  ( N  -  1 ) )
Assertion
Ref Expression
signsvtp  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( V `  F )  =  ( V `  E ) )
Distinct variable groups:    a, b,  .+^    f, i, n, F    f, W, i, n    f, a, i, j, b, n, A    E, a, b, f, i, j, n    T, a, b, f, j, n
Allowed substitution hints:    ph( f, i, j, n, a, b)    B( f, i, j, n, a, b)    .+^ ( f, i, j, n)    T( i)    F( j, a, b)    N( f, i, j, n, a, b)    V( f, i, j, n, a, b)    W( j, a, b)

Proof of Theorem signsvtp
StepHypRef Expression
1 signsvf.f . . . . 5  |-  ( ph  ->  F  =  ( E concat  <" A "> ) )
21fveq2d 5796 . . . 4  |-  ( ph  ->  ( V `  F
)  =  ( V `
 ( E concat  <" A "> ) ) )
3 signsvf.e . . . . 5  |-  ( ph  ->  E  e.  (Word  RR  \  { (/) } ) )
4 signsvf.0 . . . . 5  |-  ( ph  ->  ( E `  0
)  =/=  0 )
5 signsvf.a . . . . 5  |-  ( ph  ->  A  e.  RR )
6 signsv.p . . . . . 6  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
7 signsv.w . . . . . 6  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
8 signsv.t . . . . . 6  |-  T  =  ( f  e. Word  RR  |->  ( n  e.  (
0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) ) )
9 signsv.v . . . . . 6  |-  V  =  ( f  e. Word  RR  |->  sum_ j  e.  ( 1..^ ( # `  f
) ) if ( ( ( T `  f ) `  j
)  =/=  ( ( T `  f ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
106, 7, 8, 9signsvfn 27120 . . . . 5  |-  ( ( ( E  e.  (Word 
RR  \  { (/) } )  /\  ( E ` 
0 )  =/=  0
)  /\  A  e.  RR )  ->  ( V `
 ( E concat  <" A "> ) )  =  ( ( V `  E )  +  if ( ( ( ( T `  E ) `
 ( ( # `  E )  -  1 ) )  x.  A
)  <  0 , 
1 ,  0 ) ) )
113, 4, 5, 10syl21anc 1218 . . . 4  |-  ( ph  ->  ( V `  ( E concat  <" A "> ) )  =  ( ( V `  E
)  +  if ( ( ( ( T `
 E ) `  ( ( # `  E
)  -  1 ) )  x.  A )  <  0 ,  1 ,  0 ) ) )
122, 11eqtrd 2492 . . 3  |-  ( ph  ->  ( V `  F
)  =  ( ( V `  E )  +  if ( ( ( ( T `  E ) `  (
( # `  E )  -  1 ) )  x.  A )  <  0 ,  1 ,  0 ) ) )
1312adantr 465 . 2  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( V `  F )  =  ( ( V `  E
)  +  if ( ( ( ( T `
 E ) `  ( ( # `  E
)  -  1 ) )  x.  A )  <  0 ,  1 ,  0 ) ) )
14 0re 9490 . . . . . 6  |-  0  e.  RR
1514a1i 11 . . . . 5  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  0  e.  RR )
163adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  E  e.  (Word  RR  \  { (/) } ) )
1716eldifad 3441 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  E  e. Word  RR )
186, 7, 8, 9signstf 27104 . . . . . . . 8  |-  ( E  e. Word  RR  ->  ( T `
 E )  e. Word  RR )
19 wrdf 12351 . . . . . . . 8  |-  ( ( T `  E )  e. Word  RR  ->  ( T `
 E ) : ( 0..^ ( # `  ( T `  E
) ) ) --> RR )
2017, 18, 193syl 20 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( T `  E ) : ( 0..^ ( # `  ( T `  E )
) ) --> RR )
21 eldifsn 4101 . . . . . . . . . . . 12  |-  ( E  e.  (Word  RR  \  { (/) } )  <->  ( E  e. Word  RR  /\  E  =/=  (/) ) )
223, 21sylib 196 . . . . . . . . . . 11  |-  ( ph  ->  ( E  e. Word  RR  /\  E  =/=  (/) ) )
2322adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( E  e. Word  RR  /\  E  =/=  (/) ) )
24 lennncl 12361 . . . . . . . . . 10  |-  ( ( E  e. Word  RR  /\  E  =/=  (/) )  ->  ( # `
 E )  e.  NN )
2523, 24syl 16 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( # `  E
)  e.  NN )
26 fzo0end 11729 . . . . . . . . 9  |-  ( (
# `  E )  e.  NN  ->  ( ( # `
 E )  - 
1 )  e.  ( 0..^ ( # `  E
) ) )
2725, 26syl 16 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( ( # `
 E )  - 
1 )  e.  ( 0..^ ( # `  E
) ) )
286, 7, 8, 9signstlen 27105 . . . . . . . . . 10  |-  ( E  e. Word  RR  ->  ( # `  ( T `  E
) )  =  (
# `  E )
)
2917, 28syl 16 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( # `  ( T `  E )
)  =  ( # `  E ) )
3029oveq2d 6209 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( 0..^ ( # `  ( T `  E )
) )  =  ( 0..^ ( # `  E
) ) )
3127, 30eleqtrrd 2542 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( ( # `
 E )  - 
1 )  e.  ( 0..^ ( # `  ( T `  E )
) ) )
3220, 31ffvelrnd 5946 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( ( T `  E ) `  ( ( # `  E
)  -  1 ) )  e.  RR )
335adantr 465 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  A  e.  RR )
3432, 33remulcld 9518 . . . . 5  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( (
( T `  E
) `  ( ( # `
 E )  - 
1 ) )  x.  A )  e.  RR )
35 simpr 461 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  0  <  ( A  x.  B ) )
36 signsvt.b . . . . . . . . . . 11  |-  B  =  ( ( T `  E ) `  ( N  -  1 ) )
37 signsvf.n . . . . . . . . . . . . 13  |-  N  =  ( # `  E
)
3837oveq1i 6203 . . . . . . . . . . . 12  |-  ( N  -  1 )  =  ( ( # `  E
)  -  1 )
3938fveq2i 5795 . . . . . . . . . . 11  |-  ( ( T `  E ) `
 ( N  - 
1 ) )  =  ( ( T `  E ) `  (
( # `  E )  -  1 ) )
4036, 39eqtri 2480 . . . . . . . . . 10  |-  B  =  ( ( T `  E ) `  (
( # `  E )  -  1 ) )
4140, 32syl5eqel 2543 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  B  e.  RR )
4241recnd 9516 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  B  e.  CC )
4333recnd 9516 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  A  e.  CC )
4442, 43mulcomd 9511 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( B  x.  A )  =  ( A  x.  B ) )
4535, 44breqtrrd 4419 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  0  <  ( B  x.  A ) )
4640oveq1i 6203 . . . . . 6  |-  ( B  x.  A )  =  ( ( ( T `
 E ) `  ( ( # `  E
)  -  1 ) )  x.  A )
4745, 46syl6breq 4432 . . . . 5  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  0  <  ( ( ( T `  E ) `  (
( # `  E )  -  1 ) )  x.  A ) )
4815, 34, 47ltnsymd 9627 . . . 4  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  -.  (
( ( T `  E ) `  (
( # `  E )  -  1 ) )  x.  A )  <  0 )
49 iffalse 3900 . . . 4  |-  ( -.  ( ( ( T `
 E ) `  ( ( # `  E
)  -  1 ) )  x.  A )  <  0  ->  if ( ( ( ( T `  E ) `
 ( ( # `  E )  -  1 ) )  x.  A
)  <  0 , 
1 ,  0 )  =  0 )
5048, 49syl 16 . . 3  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  if (
( ( ( T `
 E ) `  ( ( # `  E
)  -  1 ) )  x.  A )  <  0 ,  1 ,  0 )  =  0 )
5150oveq2d 6209 . 2  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( ( V `  E )  +  if ( ( ( ( T `  E
) `  ( ( # `
 E )  - 
1 ) )  x.  A )  <  0 ,  1 ,  0 ) )  =  ( ( V `  E
)  +  0 ) )
526, 7, 8, 9signsvvf 27117 . . . . . 6  |-  V :Word  RR
--> NN0
5352a1i 11 . . . . 5  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  V :Word  RR
--> NN0 )
5453, 17ffvelrnd 5946 . . . 4  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( V `  E )  e.  NN0 )
5554nn0cnd 10742 . . 3  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( V `  E )  e.  CC )
5655addid1d 9673 . 2  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( ( V `  E )  +  0 )  =  ( V `  E
) )
5713, 51, 563eqtrd 2496 1  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( V `  F )  =  ( V `  E ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644    \ cdif 3426   (/)c0 3738   ifcif 3892   {csn 3978   {cpr 3980   {ctp 3982   <.cop 3984   class class class wbr 4393    |-> cmpt 4451   -->wf 5515   ` cfv 5519  (class class class)co 6193    |-> cmpt2 6195   RRcr 9385   0cc0 9386   1c1 9387    + caddc 9389    x. cmul 9391    < clt 9522    - cmin 9699   -ucneg 9700   NNcn 10426   NN0cn0 10683   ...cfz 11547  ..^cfzo 11658   #chash 12213  Word cword 12332   concat cconcat 12334   <"cs1 12335  sgncsgn 12686   sum_csu 13274   ndxcnx 14282   Basecbs 14285   +g cplusg 14349    gsumg cgsu 14490
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 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
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 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-oi 7828  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-n0 10684  df-z 10751  df-uz 10966  df-rp 11096  df-fz 11548  df-fzo 11659  df-seq 11917  df-exp 11976  df-hash 12214  df-word 12340  df-concat 12342  df-s1 12343  df-substr 12344  df-sgn 12687  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-clim 13077  df-sum 13275  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-plusg 14362  df-0g 14491  df-gsum 14492  df-mnd 15526  df-mulg 15659  df-cntz 15946
This theorem is referenced by:  signsvfpn  27123
  Copyright terms: Public domain W3C validator