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Theorem signsvtp 28804
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 ++ 
<" 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 ++ 
<" A "> ) )
21fveq2d 5852 . . . 4  |-  ( ph  ->  ( V `  F
)  =  ( V `
 ( E ++  <" 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 28803 . . . . 5  |-  ( ( ( E  e.  (Word 
RR  \  { (/) } )  /\  ( E ` 
0 )  =/=  0
)  /\  A  e.  RR )  ->  ( V `
 ( E ++  <" A "> )
)  =  ( ( V `  E )  +  if ( ( ( ( T `  E ) `  (
( # `  E )  -  1 ) )  x.  A )  <  0 ,  1 ,  0 ) ) )
113, 4, 5, 10syl21anc 1225 . . . 4  |-  ( ph  ->  ( V `  ( E ++  <" A "> ) )  =  ( ( V `  E
)  +  if ( ( ( ( T `
 E ) `  ( ( # `  E
)  -  1 ) )  x.  A )  <  0 ,  1 ,  0 ) ) )
122, 11eqtrd 2495 . . 3  |-  ( ph  ->  ( V `  F
)  =  ( ( V `  E )  +  if ( ( ( ( T `  E ) `  (
( # `  E )  -  1 ) )  x.  A )  <  0 ,  1 ,  0 ) ) )
1312adantr 463 . 2  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( V `  F )  =  ( ( V `  E
)  +  if ( ( ( ( T `
 E ) `  ( ( # `  E
)  -  1 ) )  x.  A )  <  0 ,  1 ,  0 ) ) )
14 0red 9586 . . . . 5  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  0  e.  RR )
153adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  E  e.  (Word  RR  \  { (/) } ) )
1615eldifad 3473 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  E  e. Word  RR )
176, 7, 8, 9signstf 28787 . . . . . . . 8  |-  ( E  e. Word  RR  ->  ( T `
 E )  e. Word  RR )
18 wrdf 12538 . . . . . . . 8  |-  ( ( T `  E )  e. Word  RR  ->  ( T `
 E ) : ( 0..^ ( # `  ( T `  E
) ) ) --> RR )
1916, 17, 183syl 20 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( T `  E ) : ( 0..^ ( # `  ( T `  E )
) ) --> RR )
20 eldifsn 4141 . . . . . . . . . . 11  |-  ( E  e.  (Word  RR  \  { (/) } )  <->  ( E  e. Word  RR  /\  E  =/=  (/) ) )
213, 20sylib 196 . . . . . . . . . 10  |-  ( ph  ->  ( E  e. Word  RR  /\  E  =/=  (/) ) )
2221adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( E  e. Word  RR  /\  E  =/=  (/) ) )
23 lennncl 12550 . . . . . . . . 9  |-  ( ( E  e. Word  RR  /\  E  =/=  (/) )  ->  ( # `
 E )  e.  NN )
24 fzo0end 11885 . . . . . . . . 9  |-  ( (
# `  E )  e.  NN  ->  ( ( # `
 E )  - 
1 )  e.  ( 0..^ ( # `  E
) ) )
2522, 23, 243syl 20 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( ( # `
 E )  - 
1 )  e.  ( 0..^ ( # `  E
) ) )
266, 7, 8, 9signstlen 28788 . . . . . . . . . 10  |-  ( E  e. Word  RR  ->  ( # `  ( T `  E
) )  =  (
# `  E )
)
2716, 26syl 16 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( # `  ( T `  E )
)  =  ( # `  E ) )
2827oveq2d 6286 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( 0..^ ( # `  ( T `  E )
) )  =  ( 0..^ ( # `  E
) ) )
2925, 28eleqtrrd 2545 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( ( # `
 E )  - 
1 )  e.  ( 0..^ ( # `  ( T `  E )
) ) )
3019, 29ffvelrnd 6008 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( ( T `  E ) `  ( ( # `  E
)  -  1 ) )  e.  RR )
315adantr 463 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  A  e.  RR )
3230, 31remulcld 9613 . . . . 5  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( (
( T `  E
) `  ( ( # `
 E )  - 
1 ) )  x.  A )  e.  RR )
33 simpr 459 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  0  <  ( A  x.  B ) )
34 signsvt.b . . . . . . . . . . 11  |-  B  =  ( ( T `  E ) `  ( N  -  1 ) )
35 signsvf.n . . . . . . . . . . . . 13  |-  N  =  ( # `  E
)
3635oveq1i 6280 . . . . . . . . . . . 12  |-  ( N  -  1 )  =  ( ( # `  E
)  -  1 )
3736fveq2i 5851 . . . . . . . . . . 11  |-  ( ( T `  E ) `
 ( N  - 
1 ) )  =  ( ( T `  E ) `  (
( # `  E )  -  1 ) )
3834, 37eqtri 2483 . . . . . . . . . 10  |-  B  =  ( ( T `  E ) `  (
( # `  E )  -  1 ) )
3938, 30syl5eqel 2546 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  B  e.  RR )
4039recnd 9611 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  B  e.  CC )
4131recnd 9611 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  A  e.  CC )
4240, 41mulcomd 9606 . . . . . . 7  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( B  x.  A )  =  ( A  x.  B ) )
4333, 42breqtrrd 4465 . . . . . 6  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  0  <  ( B  x.  A ) )
4438oveq1i 6280 . . . . . 6  |-  ( B  x.  A )  =  ( ( ( T `
 E ) `  ( ( # `  E
)  -  1 ) )  x.  A )
4543, 44syl6breq 4478 . . . . 5  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  0  <  ( ( ( T `  E ) `  (
( # `  E )  -  1 ) )  x.  A ) )
4614, 32, 45ltnsymd 9723 . . . 4  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  -.  (
( ( T `  E ) `  (
( # `  E )  -  1 ) )  x.  A )  <  0 )
4746iffalsed 3940 . . 3  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  if (
( ( ( T `
 E ) `  ( ( # `  E
)  -  1 ) )  x.  A )  <  0 ,  1 ,  0 )  =  0 )
4847oveq2d 6286 . 2  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( ( V `  E )  +  if ( ( ( ( T `  E
) `  ( ( # `
 E )  - 
1 ) )  x.  A )  <  0 ,  1 ,  0 ) )  =  ( ( V `  E
)  +  0 ) )
496, 7, 8, 9signsvvf 28800 . . . . . 6  |-  V :Word  RR
--> NN0
5049a1i 11 . . . . 5  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  V :Word  RR
--> NN0 )
5150, 16ffvelrnd 6008 . . . 4  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( V `  E )  e.  NN0 )
5251nn0cnd 10850 . . 3  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( V `  E )  e.  CC )
5352addid1d 9769 . 2  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( ( V `  E )  +  0 )  =  ( V `  E
) )
5413, 48, 533eqtrd 2499 1  |-  ( (
ph  /\  0  <  ( A  x.  B ) )  ->  ( V `  F )  =  ( V `  E ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649    \ cdif 3458   (/)c0 3783   ifcif 3929   {csn 4016   {cpr 4018   {ctp 4020   <.cop 4022   class class class wbr 4439    |-> cmpt 4497   -->wf 5566   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    < clt 9617    - cmin 9796   -ucneg 9797   NNcn 10531   NN0cn0 10791   ...cfz 11675  ..^cfzo 11799   #chash 12387  Word cword 12518   ++ cconcat 12520   <"cs1 12521  sgncsgn 13001   sum_csu 13590   ndxcnx 14713   Basecbs 14716   +g cplusg 14784    gsumg cgsu 14930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fz 11676  df-fzo 11800  df-seq 12090  df-exp 12149  df-hash 12388  df-word 12526  df-lsw 12527  df-concat 12528  df-s1 12529  df-substr 12530  df-sgn 13002  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-sum 13591  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-plusg 14797  df-0g 14931  df-gsum 14932  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-mulg 16259  df-cntz 16554
This theorem is referenced by:  signsvfpn  28806
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