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Theorem signsvtn 28181
Description: Adding a letter of a different sign as the highest coefficient changes 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
signsvtn  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( ( V `  F )  -  ( V `  E ) )  =  1 )
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 signsvtn
StepHypRef Expression
1 signsvf.f . . . . . 6  |-  ( ph  ->  F  =  ( E concat  <" A "> ) )
21fveq2d 5868 . . . . 5  |-  ( ph  ->  ( V `  F
)  =  ( V `
 ( E concat  <" A "> ) ) )
3 signsvf.e . . . . . 6  |-  ( ph  ->  E  e.  (Word  RR  \  { (/) } ) )
4 signsvf.0 . . . . . 6  |-  ( ph  ->  ( E `  0
)  =/=  0 )
5 signsvf.a . . . . . 6  |-  ( ph  ->  A  e.  RR )
6 signsv.p . . . . . . 7  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
7 signsv.w . . . . . . 7  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
8 signsv.t . . . . . . 7  |-  T  =  ( f  e. Word  RR  |->  ( n  e.  (
0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) ) )
9 signsv.v . . . . . . 7  |-  V  =  ( f  e. Word  RR  |->  sum_ j  e.  ( 1..^ ( # `  f
) ) if ( ( ( T `  f ) `  j
)  =/=  ( ( T `  f ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
106, 7, 8, 9signsvfn 28179 . . . . . 6  |-  ( ( ( 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 1227 . . . . 5  |-  ( ph  ->  ( V `  ( E concat  <" A "> ) )  =  ( ( V `  E
)  +  if ( ( ( ( T `
 E ) `  ( ( # `  E
)  -  1 ) )  x.  A )  <  0 ,  1 ,  0 ) ) )
122, 11eqtrd 2508 . . . 4  |-  ( ph  ->  ( V `  F
)  =  ( ( V `  E )  +  if ( ( ( ( T `  E ) `  (
( # `  E )  -  1 ) )  x.  A )  <  0 ,  1 ,  0 ) ) )
1312adantr 465 . . 3  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( V `  F )  =  ( ( V `  E
)  +  if ( ( ( ( T `
 E ) `  ( ( # `  E
)  -  1 ) )  x.  A )  <  0 ,  1 ,  0 ) ) )
14 signsvt.b . . . . . . . 8  |-  B  =  ( ( T `  E ) `  ( N  -  1 ) )
15 signsvf.n . . . . . . . . . 10  |-  N  =  ( # `  E
)
1615oveq1i 6292 . . . . . . . . 9  |-  ( N  -  1 )  =  ( ( # `  E
)  -  1 )
1716fveq2i 5867 . . . . . . . 8  |-  ( ( T `  E ) `
 ( N  - 
1 ) )  =  ( ( T `  E ) `  (
( # `  E )  -  1 ) )
1814, 17eqtri 2496 . . . . . . 7  |-  B  =  ( ( T `  E ) `  (
( # `  E )  -  1 ) )
1918oveq1i 6292 . . . . . 6  |-  ( B  x.  A )  =  ( ( ( T `
 E ) `  ( ( # `  E
)  -  1 ) )  x.  A )
203adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  E  e.  (Word  RR  \  { (/) } ) )
2120eldifad 3488 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  E  e. Word  RR )
226, 7, 8, 9signstf 28163 . . . . . . . . . . . 12  |-  ( E  e. Word  RR  ->  ( T `
 E )  e. Word  RR )
23 wrdf 12515 . . . . . . . . . . . 12  |-  ( ( T `  E )  e. Word  RR  ->  ( T `
 E ) : ( 0..^ ( # `  ( T `  E
) ) ) --> RR )
2421, 22, 233syl 20 . . . . . . . . . . 11  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( T `  E ) : ( 0..^ ( # `  ( T `  E )
) ) --> RR )
25 eldifsn 4152 . . . . . . . . . . . . . . . 16  |-  ( E  e.  (Word  RR  \  { (/) } )  <->  ( E  e. Word  RR  /\  E  =/=  (/) ) )
263, 25sylib 196 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( E  e. Word  RR  /\  E  =/=  (/) ) )
2726adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( E  e. Word  RR  /\  E  =/=  (/) ) )
28 lennncl 12525 . . . . . . . . . . . . . 14  |-  ( ( E  e. Word  RR  /\  E  =/=  (/) )  ->  ( # `
 E )  e.  NN )
2927, 28syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( # `  E
)  e.  NN )
30 fzo0end 11868 . . . . . . . . . . . . 13  |-  ( (
# `  E )  e.  NN  ->  ( ( # `
 E )  - 
1 )  e.  ( 0..^ ( # `  E
) ) )
3129, 30syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( ( # `
 E )  - 
1 )  e.  ( 0..^ ( # `  E
) ) )
326, 7, 8, 9signstlen 28164 . . . . . . . . . . . . . 14  |-  ( E  e. Word  RR  ->  ( # `  ( T `  E
) )  =  (
# `  E )
)
3321, 32syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( # `  ( T `  E )
)  =  ( # `  E ) )
3433oveq2d 6298 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( 0..^ ( # `  ( T `  E )
) )  =  ( 0..^ ( # `  E
) ) )
3531, 34eleqtrrd 2558 . . . . . . . . . . 11  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( ( # `
 E )  - 
1 )  e.  ( 0..^ ( # `  ( T `  E )
) ) )
3624, 35ffvelrnd 6020 . . . . . . . . . 10  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( ( T `  E ) `  ( ( # `  E
)  -  1 ) )  e.  RR )
3718, 36syl5eqel 2559 . . . . . . . . 9  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  B  e.  RR )
3837recnd 9618 . . . . . . . 8  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  B  e.  CC )
395adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  A  e.  RR )
4039recnd 9618 . . . . . . . 8  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  A  e.  CC )
4138, 40mulcomd 9613 . . . . . . 7  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( B  x.  A )  =  ( A  x.  B ) )
42 simpr 461 . . . . . . 7  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( A  x.  B )  <  0
)
4341, 42eqbrtrd 4467 . . . . . 6  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( B  x.  A )  <  0
)
4419, 43syl5eqbrr 4481 . . . . 5  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( (
( T `  E
) `  ( ( # `
 E )  - 
1 ) )  x.  A )  <  0
)
45 iftrue 3945 . . . . 5  |-  ( ( ( ( T `  E ) `  (
( # `  E )  -  1 ) )  x.  A )  <  0  ->  if (
( ( ( T `
 E ) `  ( ( # `  E
)  -  1 ) )  x.  A )  <  0 ,  1 ,  0 )  =  1 )
4644, 45syl 16 . . . 4  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  if (
( ( ( T `
 E ) `  ( ( # `  E
)  -  1 ) )  x.  A )  <  0 ,  1 ,  0 )  =  1 )
4746oveq2d 6298 . . 3  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( ( V `  E )  +  if ( ( ( ( T `  E
) `  ( ( # `
 E )  - 
1 ) )  x.  A )  <  0 ,  1 ,  0 ) )  =  ( ( V `  E
)  +  1 ) )
4813, 47eqtr2d 2509 . 2  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( ( V `  E )  +  1 )  =  ( V `  F
) )
496, 7, 8, 9signsvvf 28176 . . . . . 6  |-  V :Word  RR
--> NN0
5049a1i 11 . . . . 5  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  V :Word  RR
--> NN0 )
511adantr 465 . . . . . 6  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  F  =  ( E concat  <" A "> ) )
5239s1cld 12574 . . . . . . 7  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  <" A ">  e. Word  RR )
53 ccatcl 12554 . . . . . . 7  |-  ( ( E  e. Word  RR  /\  <" A ">  e. Word  RR )  ->  ( E concat  <" A "> )  e. Word  RR )
5421, 52, 53syl2anc 661 . . . . . 6  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( E concat  <" A "> )  e. Word  RR )
5551, 54eqeltrd 2555 . . . . 5  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  F  e. Word  RR )
5650, 55ffvelrnd 6020 . . . 4  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( V `  F )  e.  NN0 )
5756nn0cnd 10850 . . 3  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( V `  F )  e.  CC )
5850, 21ffvelrnd 6020 . . . 4  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( V `  E )  e.  NN0 )
5958nn0cnd 10850 . . 3  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( V `  E )  e.  CC )
60 ax-1cn 9546 . . . 4  |-  1  e.  CC
6160a1i 11 . . 3  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  1  e.  CC )
6257, 59, 61subaddd 9944 . 2  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( (
( V `  F
)  -  ( V `
 E ) )  =  1  <->  ( ( V `  E )  +  1 )  =  ( V `  F
) ) )
6348, 62mpbird 232 1  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( ( V `  F )  -  ( V `  E ) )  =  1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3473   (/)c0 3785   ifcif 3939   {csn 4027   {cpr 4029   {ctp 4031   <.cop 4033   class class class wbr 4447    |-> cmpt 4505   -->wf 5582   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493    < clt 9624    - cmin 9801   -ucneg 9802   NNcn 10532   NN0cn0 10791   ...cfz 11668  ..^cfzo 11788   #chash 12369  Word cword 12496   concat cconcat 12498   <"cs1 12499  sgncsgn 12878   sum_csu 13467   ndxcnx 14483   Basecbs 14486   +g cplusg 14551    gsumg cgsu 14692
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fz 11669  df-fzo 11789  df-seq 12072  df-exp 12131  df-hash 12370  df-word 12504  df-concat 12506  df-s1 12507  df-substr 12508  df-sgn 12879  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-sum 13468  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-plusg 14564  df-0g 14693  df-gsum 14694  df-mnd 15728  df-mulg 15861  df-cntz 16150
This theorem is referenced by:  signsvfnn  28183
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