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Theorem signsvfpn 28335
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
)
signsvf.b  |-  B  =  ( E `  ( N  -  1 ) )
Assertion
Ref Expression
signsvfpn  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  ( V `  F )  =  ( V `  E ) )
Distinct variable groups:    a, b,  .+^    f, i, n, F    f, W, i, n    f, a, i, j, n, A, b    E, a, b, f, i, j, n    N, a, b, f, i, 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( j)    V( f, i, j, n, a, b)    W( j, a, b)

Proof of Theorem signsvfpn
StepHypRef Expression
1 signsvf.a . . . . . . . . 9  |-  ( ph  ->  A  e.  RR )
21recnd 9632 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
3 signsvf.b . . . . . . . . 9  |-  B  =  ( E `  ( N  -  1 ) )
4 signsvf.e . . . . . . . . . . . . . 14  |-  ( ph  ->  E  e.  (Word  RR  \  { (/) } ) )
5 eldifsn 4157 . . . . . . . . . . . . . 14  |-  ( E  e.  (Word  RR  \  { (/) } )  <->  ( E  e. Word  RR  /\  E  =/=  (/) ) )
64, 5sylib 196 . . . . . . . . . . . . 13  |-  ( ph  ->  ( E  e. Word  RR  /\  E  =/=  (/) ) )
76simpld 459 . . . . . . . . . . . 12  |-  ( ph  ->  E  e. Word  RR )
8 wrdf 12529 . . . . . . . . . . . 12  |-  ( E  e. Word  RR  ->  E :
( 0..^ ( # `  E ) ) --> RR )
97, 8syl 16 . . . . . . . . . . 11  |-  ( ph  ->  E : ( 0..^ ( # `  E
) ) --> RR )
10 signsvf.n . . . . . . . . . . . . 13  |-  N  =  ( # `  E
)
1110oveq1i 6304 . . . . . . . . . . . 12  |-  ( N  -  1 )  =  ( ( # `  E
)  -  1 )
12 lennncl 12539 . . . . . . . . . . . . 13  |-  ( ( E  e. Word  RR  /\  E  =/=  (/) )  ->  ( # `
 E )  e.  NN )
13 fzo0end 11882 . . . . . . . . . . . . 13  |-  ( (
# `  E )  e.  NN  ->  ( ( # `
 E )  - 
1 )  e.  ( 0..^ ( # `  E
) ) )
146, 12, 133syl 20 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( # `  E
)  -  1 )  e.  ( 0..^ (
# `  E )
) )
1511, 14syl5eqel 2559 . . . . . . . . . . 11  |-  ( ph  ->  ( N  -  1 )  e.  ( 0..^ ( # `  E
) ) )
169, 15ffvelrnd 6032 . . . . . . . . . 10  |-  ( ph  ->  ( E `  ( N  -  1 ) )  e.  RR )
1716recnd 9632 . . . . . . . . 9  |-  ( ph  ->  ( E `  ( N  -  1 ) )  e.  CC )
183, 17syl5eqel 2559 . . . . . . . 8  |-  ( ph  ->  B  e.  CC )
192, 18mulcomd 9627 . . . . . . 7  |-  ( ph  ->  ( A  x.  B
)  =  ( B  x.  A ) )
2019breq2d 4464 . . . . . 6  |-  ( ph  ->  ( 0  <  ( A  x.  B )  <->  0  <  ( B  x.  A ) ) )
213, 16syl5eqel 2559 . . . . . . 7  |-  ( ph  ->  B  e.  RR )
22 sgnmulsgp 28282 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  ( A  x.  B )  <->  0  <  ( (sgn `  A )  x.  (sgn `  B ) ) ) )
231, 21, 22syl2anc 661 . . . . . 6  |-  ( ph  ->  ( 0  <  ( A  x.  B )  <->  0  <  ( (sgn `  A )  x.  (sgn `  B ) ) ) )
2420, 23bitr3d 255 . . . . 5  |-  ( ph  ->  ( 0  <  ( B  x.  A )  <->  0  <  ( (sgn `  A )  x.  (sgn `  B ) ) ) )
2524biimpa 484 . . . 4  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  0  <  ( (sgn `  A )  x.  (sgn `  B )
) )
264adantr 465 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  E  e.  (Word  RR  \  { (/) } ) )
2718adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  B  e.  CC )
282adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  A  e.  CC )
29 0re 9606 . . . . . . . . . . . . 13  |-  0  e.  RR
3029a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  0  e.  RR )
31 simpr 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  0  <  ( B  x.  A ) )
3230, 31ltned 9730 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  0  =/=  ( B  x.  A
) )
3332necomd 2738 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  ( B  x.  A )  =/=  0
)
3427, 28, 33mulne0bad 10214 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  B  =/=  0 )
353, 34syl5eqner 2768 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  ( E `  ( N  -  1 ) )  =/=  0
)
36 signsv.p . . . . . . . . . 10  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
37 signsv.w . . . . . . . . . 10  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
38 signsv.t . . . . . . . . . 10  |-  T  =  ( f  e. Word  RR  |->  ( n  e.  (
0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) ) )
39 signsv.v . . . . . . . . . 10  |-  V  =  ( f  e. Word  RR  |->  sum_ j  e.  ( 1..^ ( # `  f
) ) if ( ( ( T `  f ) `  j
)  =/=  ( ( T `  f ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
4036, 37, 38, 39, 10signsvtn0 28320 . . . . . . . . 9  |-  ( ( E  e.  (Word  RR  \  { (/) } )  /\  ( E `  ( N  -  1 ) )  =/=  0 )  -> 
( ( T `  E ) `  ( N  -  1 ) )  =  (sgn `  ( E `  ( N  -  1 ) ) ) )
413fveq2i 5874 . . . . . . . . 9  |-  (sgn `  B )  =  (sgn
`  ( E `  ( N  -  1
) ) )
4240, 41syl6eqr 2526 . . . . . . . 8  |-  ( ( E  e.  (Word  RR  \  { (/) } )  /\  ( E `  ( N  -  1 ) )  =/=  0 )  -> 
( ( T `  E ) `  ( N  -  1 ) )  =  (sgn `  B ) )
4326, 35, 42syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  ( ( T `  E ) `  ( N  -  1 ) )  =  (sgn
`  B ) )
4443fveq2d 5875 . . . . . 6  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  (sgn `  (
( T `  E
) `  ( N  -  1 ) ) )  =  (sgn `  (sgn `  B ) ) )
4521adantr 465 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  B  e.  RR )
4645rexrd 9653 . . . . . . 7  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  B  e.  RR* )
47 sgnsgn 28280 . . . . . . 7  |-  ( B  e.  RR*  ->  (sgn `  (sgn `  B ) )  =  (sgn `  B
) )
4846, 47syl 16 . . . . . 6  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  (sgn `  (sgn `  B ) )  =  (sgn `  B )
)
4944, 48eqtrd 2508 . . . . 5  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  (sgn `  (
( T `  E
) `  ( N  -  1 ) ) )  =  (sgn `  B ) )
5049oveq2d 6310 . . . 4  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  ( (sgn `  A )  x.  (sgn `  ( ( T `  E ) `  ( N  -  1 ) ) ) )  =  ( (sgn `  A
)  x.  (sgn `  B ) ) )
5125, 50breqtrrd 4478 . . 3  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  0  <  ( (sgn `  A )  x.  (sgn `  ( ( T `  E ) `  ( N  -  1 ) ) ) ) )
521adantr 465 . . . 4  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  A  e.  RR )
53 sgnclre 28271 . . . . . 6  |-  ( B  e.  RR  ->  (sgn `  B )  e.  RR )
5445, 53syl 16 . . . . 5  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  (sgn `  B
)  e.  RR )
5543, 54eqeltrd 2555 . . . 4  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  ( ( T `  E ) `  ( N  -  1 ) )  e.  RR )
56 sgnmulsgp 28282 . . . 4  |-  ( ( A  e.  RR  /\  ( ( T `  E ) `  ( N  -  1 ) )  e.  RR )  ->  ( 0  < 
( A  x.  (
( T `  E
) `  ( N  -  1 ) ) )  <->  0  <  (
(sgn `  A )  x.  (sgn `  ( ( T `  E ) `  ( N  -  1 ) ) ) ) ) )
5752, 55, 56syl2anc 661 . . 3  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  ( 0  <  ( A  x.  ( ( T `  E ) `  ( N  -  1 ) ) )  <->  0  <  ( (sgn `  A )  x.  (sgn `  ( ( T `  E ) `  ( N  -  1 ) ) ) ) ) )
5851, 57mpbird 232 . 2  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  0  <  ( A  x.  ( ( T `  E ) `
 ( N  - 
1 ) ) ) )
59 signsvf.0 . . 3  |-  ( ph  ->  ( E `  0
)  =/=  0 )
60 signsvf.f . . 3  |-  ( ph  ->  F  =  ( E concat  <" A "> ) )
61 eqid 2467 . . 3  |-  ( ( T `  E ) `
 ( N  - 
1 ) )  =  ( ( T `  E ) `  ( N  -  1 ) )
6236, 37, 38, 39, 4, 59, 60, 1, 10, 61signsvtp 28333 . 2  |-  ( (
ph  /\  0  <  ( A  x.  ( ( T `  E ) `
 ( N  - 
1 ) ) ) )  ->  ( V `  F )  =  ( V `  E ) )
6358, 62syldan 470 1  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  ( V `  F )  =  ( V `  E ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3478   (/)c0 3790   ifcif 3944   {csn 4032   {cpr 4034   {ctp 4036   <.cop 4038   class class class wbr 4452    |-> cmpt 4510   -->wf 5589   ` cfv 5593  (class class class)co 6294    |-> cmpt2 6296   CCcc 9500   RRcr 9501   0cc0 9502   1c1 9503    x. cmul 9507   RR*cxr 9637    < clt 9638    - cmin 9815   -ucneg 9816   NNcn 10546   ...cfz 11682  ..^cfzo 11802   #chash 12383  Word cword 12510   concat cconcat 12512   <"cs1 12513  sgncsgn 12894   sum_csu 13483   ndxcnx 14499   Basecbs 14502   +g cplusg 14567    gsumg cgsu 14708
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 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-inf2 8068  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579  ax-pre-sup 9580
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-se 4844  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-supp 6912  df-recs 7052  df-rdg 7086  df-1o 7140  df-oadd 7144  df-er 7321  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-sup 7911  df-oi 7945  df-card 8330  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-2 10604  df-3 10605  df-n0 10806  df-z 10875  df-uz 11093  df-rp 11231  df-fz 11683  df-fzo 11803  df-seq 12086  df-exp 12145  df-hash 12384  df-word 12518  df-concat 12520  df-s1 12521  df-substr 12522  df-sgn 12895  df-cj 12907  df-re 12908  df-im 12909  df-sqrt 13043  df-abs 13044  df-clim 13286  df-sum 13484  df-struct 14504  df-ndx 14505  df-slot 14506  df-base 14507  df-plusg 14580  df-0g 14709  df-gsum 14710  df-mgm 15741  df-sgrp 15764  df-mnd 15774  df-mulg 15909  df-cntz 16204
This theorem is referenced by: (None)
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