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Theorem signsvtn 29060
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 ++ 
<" 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 ++ 
<" A "> ) )
21fveq2d 5855 . . . . 5  |-  ( ph  ->  ( V `  F
)  =  ( V `
 ( E ++  <" 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 29058 . . . . . 6  |-  ( ( ( 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 1231 . . . . 5  |-  ( ph  ->  ( V `  ( E ++  <" A "> ) )  =  ( ( V `  E
)  +  if ( ( ( ( T `
 E ) `  ( ( # `  E
)  -  1 ) )  x.  A )  <  0 ,  1 ,  0 ) ) )
122, 11eqtrd 2445 . . . 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 6290 . . . . . . . . 9  |-  ( N  -  1 )  =  ( ( # `  E
)  -  1 )
1716fveq2i 5854 . . . . . . . 8  |-  ( ( T `  E ) `
 ( N  - 
1 ) )  =  ( ( T `  E ) `  (
( # `  E )  -  1 ) )
1814, 17eqtri 2433 . . . . . . 7  |-  B  =  ( ( T `  E ) `  (
( # `  E )  -  1 ) )
1918oveq1i 6290 . . . . . 6  |-  ( B  x.  A )  =  ( ( ( T `
 E ) `  ( ( # `  E
)  -  1 ) )  x.  A )
203adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  E  e.  (Word  RR  \  { (/) } ) )
2120eldifad 3428 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  E  e. Word  RR )
226, 7, 8, 9signstf 29042 . . . . . . . . . . . 12  |-  ( E  e. Word  RR  ->  ( T `
 E )  e. Word  RR )
23 wrdf 12605 . . . . . . . . . . . 12  |-  ( ( T `  E )  e. Word  RR  ->  ( T `
 E ) : ( 0..^ ( # `  ( T `  E
) ) ) --> RR )
2421, 22, 233syl 18 . . . . . . . . . . 11  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( T `  E ) : ( 0..^ ( # `  ( T `  E )
) ) --> RR )
25 eldifsn 4099 . . . . . . . . . . . . . . 15  |-  ( E  e.  (Word  RR  \  { (/) } )  <->  ( E  e. Word  RR  /\  E  =/=  (/) ) )
263, 25sylib 198 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( E  e. Word  RR  /\  E  =/=  (/) ) )
2726adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( E  e. Word  RR  /\  E  =/=  (/) ) )
28 lennncl 12617 . . . . . . . . . . . . 13  |-  ( ( E  e. Word  RR  /\  E  =/=  (/) )  ->  ( # `
 E )  e.  NN )
29 fzo0end 11943 . . . . . . . . . . . . 13  |-  ( (
# `  E )  e.  NN  ->  ( ( # `
 E )  - 
1 )  e.  ( 0..^ ( # `  E
) ) )
3027, 28, 293syl 18 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( ( # `
 E )  - 
1 )  e.  ( 0..^ ( # `  E
) ) )
316, 7, 8, 9signstlen 29043 . . . . . . . . . . . . . 14  |-  ( E  e. Word  RR  ->  ( # `  ( T `  E
) )  =  (
# `  E )
)
3221, 31syl 17 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( # `  ( T `  E )
)  =  ( # `  E ) )
3332oveq2d 6296 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( 0..^ ( # `  ( T `  E )
) )  =  ( 0..^ ( # `  E
) ) )
3430, 33eleqtrrd 2495 . . . . . . . . . . 11  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( ( # `
 E )  - 
1 )  e.  ( 0..^ ( # `  ( T `  E )
) ) )
3524, 34ffvelrnd 6012 . . . . . . . . . 10  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( ( T `  E ) `  ( ( # `  E
)  -  1 ) )  e.  RR )
3618, 35syl5eqel 2496 . . . . . . . . 9  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  B  e.  RR )
3736recnd 9654 . . . . . . . 8  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  B  e.  CC )
385adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  A  e.  RR )
3938recnd 9654 . . . . . . . 8  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  A  e.  CC )
4037, 39mulcomd 9649 . . . . . . 7  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( B  x.  A )  =  ( A  x.  B ) )
41 simpr 461 . . . . . . 7  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( A  x.  B )  <  0
)
4240, 41eqbrtrd 4417 . . . . . 6  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( B  x.  A )  <  0
)
4319, 42syl5eqbrr 4431 . . . . 5  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( (
( T `  E
) `  ( ( # `
 E )  - 
1 ) )  x.  A )  <  0
)
4443iftrued 3895 . . . 4  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  if (
( ( ( T `
 E ) `  ( ( # `  E
)  -  1 ) )  x.  A )  <  0 ,  1 ,  0 )  =  1 )
4544oveq2d 6296 . . 3  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( ( V `  E )  +  if ( ( ( ( T `  E
) `  ( ( # `
 E )  - 
1 ) )  x.  A )  <  0 ,  1 ,  0 ) )  =  ( ( V `  E
)  +  1 ) )
4613, 45eqtr2d 2446 . 2  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( ( V `  E )  +  1 )  =  ( V `  F
) )
476, 7, 8, 9signsvvf 29055 . . . . . 6  |-  V :Word  RR
--> NN0
4847a1i 11 . . . . 5  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  V :Word  RR
--> NN0 )
491adantr 465 . . . . . 6  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  F  =  ( E ++  <" A "> ) )
5038s1cld 12671 . . . . . . 7  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  <" A ">  e. Word  RR )
51 ccatcl 12649 . . . . . . 7  |-  ( ( E  e. Word  RR  /\  <" A ">  e. Word  RR )  ->  ( E ++  <" A "> )  e. Word  RR )
5221, 50, 51syl2anc 661 . . . . . 6  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( E ++  <" A "> )  e. Word  RR )
5349, 52eqeltrd 2492 . . . . 5  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  F  e. Word  RR )
5448, 53ffvelrnd 6012 . . . 4  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( V `  F )  e.  NN0 )
5554nn0cnd 10897 . . 3  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( V `  F )  e.  CC )
5648, 21ffvelrnd 6012 . . . 4  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( V `  E )  e.  NN0 )
5756nn0cnd 10897 . . 3  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( V `  E )  e.  CC )
58 1cnd 9644 . . 3  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  1  e.  CC )
5955, 57, 58subaddd 9987 . 2  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( (
( V `  F
)  -  ( V `
 E ) )  =  1  <->  ( ( V `  E )  +  1 )  =  ( V `  F
) ) )
6046, 59mpbird 234 1  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( ( V `  F )  -  ( V `  E ) )  =  1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1407    e. wcel 1844    =/= wne 2600    \ cdif 3413   (/)c0 3740   ifcif 3887   {csn 3974   {cpr 3976   {ctp 3978   <.cop 3980   class class class wbr 4397    |-> cmpt 4455   -->wf 5567   ` cfv 5571  (class class class)co 6280    |-> cmpt2 6282   RRcr 9523   0cc0 9524   1c1 9525    + caddc 9527    x. cmul 9529    < clt 9660    - cmin 9843   -ucneg 9844   NNcn 10578   NN0cn0 10838   ...cfz 11728  ..^cfzo 11856   #chash 12454  Word cword 12585   ++ cconcat 12587   <"cs1 12588  sgncsgn 13070   sum_csu 13659   ndxcnx 14840   Basecbs 14843   +g cplusg 14911    gsumg cgsu 15057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-inf2 8093  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-pre-sup 9602
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-fal 1413  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-se 4785  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-isom 5580  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-supp 6905  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-sup 7937  df-oi 7971  df-card 8354  df-cda 8582  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-div 10250  df-nn 10579  df-2 10637  df-3 10638  df-n0 10839  df-z 10908  df-uz 11130  df-rp 11268  df-fz 11729  df-fzo 11857  df-seq 12154  df-exp 12213  df-hash 12455  df-word 12593  df-lsw 12594  df-concat 12595  df-s1 12596  df-substr 12597  df-sgn 13071  df-cj 13083  df-re 13084  df-im 13085  df-sqrt 13219  df-abs 13220  df-clim 13462  df-sum 13660  df-struct 14845  df-ndx 14846  df-slot 14847  df-base 14848  df-plusg 14924  df-0g 15058  df-gsum 15059  df-mgm 16198  df-sgrp 16237  df-mnd 16247  df-mulg 16386  df-cntz 16681
This theorem is referenced by:  signsvfnn  29062
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