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Theorem signsvfpn 26991
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 9417 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
3 signsvf.b . . . . . . . . 9  |-  B  =  ( E `  ( N  -  1 ) )
4 signsvf.e . . . . . . . . . . . . . 14  |-  ( ph  ->  E  e.  (Word  RR  \  { (/) } ) )
5 eldifsn 4005 . . . . . . . . . . . . . 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 12245 . . . . . . . . . . . 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 6106 . . . . . . . . . . . 12  |-  ( N  -  1 )  =  ( ( # `  E
)  -  1 )
12 lennncl 12255 . . . . . . . . . . . . 13  |-  ( ( E  e. Word  RR  /\  E  =/=  (/) )  ->  ( # `
 E )  e.  NN )
13 fzo0end 11624 . . . . . . . . . . . . 13  |-  ( (
# `  E )  e.  NN  ->  ( ( # `
 E )  - 
1 )  e.  ( 0..^ ( # `  E
) ) )
146, 12, 133syl 20 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( # `  E
)  -  1 )  e.  ( 0..^ (
# `  E )
) )
1511, 14syl5eqel 2527 . . . . . . . . . . 11  |-  ( ph  ->  ( N  -  1 )  e.  ( 0..^ ( # `  E
) ) )
169, 15ffvelrnd 5849 . . . . . . . . . 10  |-  ( ph  ->  ( E `  ( N  -  1 ) )  e.  RR )
1716recnd 9417 . . . . . . . . 9  |-  ( ph  ->  ( E `  ( N  -  1 ) )  e.  CC )
183, 17syl5eqel 2527 . . . . . . . 8  |-  ( ph  ->  B  e.  CC )
192, 18mulcomd 9412 . . . . . . 7  |-  ( ph  ->  ( A  x.  B
)  =  ( B  x.  A ) )
2019breq2d 4309 . . . . . 6  |-  ( ph  ->  ( 0  <  ( A  x.  B )  <->  0  <  ( B  x.  A ) ) )
213, 16syl5eqel 2527 . . . . . . 7  |-  ( ph  ->  B  e.  RR )
22 sgnmulsgp 26938 . . . . . . 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 9391 . . . . . . . . . . . . 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 9515 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  0  =/=  ( B  x.  A
) )
3332necomd 2700 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  ( B  x.  A )  =/=  0
)
3427, 28, 33mulne0bad 9996 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  B  =/=  0 )
353, 34syl5eqner 2638 . . . . . . . 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 26976 . . . . . . . . 9  |-  ( ( E  e.  (Word  RR  \  { (/) } )  /\  ( E `  ( N  -  1 ) )  =/=  0 )  -> 
( ( T `  E ) `  ( N  -  1 ) )  =  (sgn `  ( E `  ( N  -  1 ) ) ) )
413fveq2i 5699 . . . . . . . . 9  |-  (sgn `  B )  =  (sgn
`  ( E `  ( N  -  1
) ) )
4240, 41syl6eqr 2493 . . . . . . . 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 5700 . . . . . 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 9438 . . . . . . 7  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  B  e.  RR* )
47 sgnsgn 26936 . . . . . . 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 2475 . . . . 5  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  (sgn `  (
( T `  E
) `  ( N  -  1 ) ) )  =  (sgn `  B ) )
5049oveq2d 6112 . . . 4  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  ( (sgn `  A )  x.  (sgn `  ( ( T `  E ) `  ( N  -  1 ) ) ) )  =  ( (sgn `  A
)  x.  (sgn `  B ) ) )
5125, 50breqtrrd 4323 . . 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 26927 . . . . . 6  |-  ( B  e.  RR  ->  (sgn `  B )  e.  RR )
5445, 53syl 16 . . . . 5  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  (sgn `  B
)  e.  RR )
5543, 54eqeltrd 2517 . . . 4  |-  ( (
ph  /\  0  <  ( B  x.  A ) )  ->  ( ( T `  E ) `  ( N  -  1 ) )  e.  RR )
56 sgnmulsgp 26938 . . . 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 2443 . . 3  |-  ( ( T `  E ) `
 ( N  - 
1 ) )  =  ( ( T `  E ) `  ( N  -  1 ) )
6236, 37, 38, 39, 4, 59, 60, 1, 10, 61signsvtp 26989 . 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 1369    e. wcel 1756    =/= wne 2611    \ cdif 3330   (/)c0 3642   ifcif 3796   {csn 3882   {cpr 3884   {ctp 3886   <.cop 3888   class class class wbr 4297    e. cmpt 4355   -->wf 5419   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   CCcc 9285   RRcr 9286   0cc0 9287   1c1 9288    x. cmul 9292   RR*cxr 9422    < clt 9423    - cmin 9600   -ucneg 9601   NNcn 10327   ...cfz 11442  ..^cfzo 11553   #chash 12108  Word cword 12226   concat cconcat 12228   <"cs1 12229  sgncsgn 12580   sum_csu 13168   ndxcnx 14176   Basecbs 14179   +g cplusg 14243    gsumg cgsu 14384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-oi 7729  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-fz 11443  df-fzo 11554  df-seq 11812  df-exp 11871  df-hash 12109  df-word 12234  df-concat 12236  df-s1 12237  df-substr 12238  df-sgn 12581  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-clim 12971  df-sum 13169  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-plusg 14256  df-0g 14385  df-gsum 14386  df-mnd 15420  df-mulg 15553  df-cntz 15840
This theorem is referenced by: (None)
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