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Theorem signsvtn 27121
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 5795 . . . . 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 27119 . . . . . 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 1218 . . . . 5  |-  ( ph  ->  ( V `  ( E concat  <" A "> ) )  =  ( ( V `  E
)  +  if ( ( ( ( T `
 E ) `  ( ( # `  E
)  -  1 ) )  x.  A )  <  0 ,  1 ,  0 ) ) )
122, 11eqtrd 2492 . . . 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 6202 . . . . . . . . 9  |-  ( N  -  1 )  =  ( ( # `  E
)  -  1 )
1716fveq2i 5794 . . . . . . . 8  |-  ( ( T `  E ) `
 ( N  - 
1 ) )  =  ( ( T `  E ) `  (
( # `  E )  -  1 ) )
1814, 17eqtri 2480 . . . . . . 7  |-  B  =  ( ( T `  E ) `  (
( # `  E )  -  1 ) )
1918oveq1i 6202 . . . . . 6  |-  ( B  x.  A )  =  ( ( ( T `
 E ) `  ( ( # `  E
)  -  1 ) )  x.  A )
203adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  E  e.  (Word  RR  \  { (/) } ) )
2120eldifad 3440 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  E  e. Word  RR )
226, 7, 8, 9signstf 27103 . . . . . . . . . . . 12  |-  ( E  e. Word  RR  ->  ( T `
 E )  e. Word  RR )
23 wrdf 12344 . . . . . . . . . . . 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 4100 . . . . . . . . . . . . . . . 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 12354 . . . . . . . . . . . . . 14  |-  ( ( E  e. Word  RR  /\  E  =/=  (/) )  ->  ( # `
 E )  e.  NN )
2927, 28syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( # `  E
)  e.  NN )
30 fzo0end 11722 . . . . . . . . . . . . 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 27104 . . . . . . . . . . . . . 14  |-  ( E  e. Word  RR  ->  ( # `  ( T `  E
) )  =  (
# `  E )
)
3321, 32syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( # `  ( T `  E )
)  =  ( # `  E ) )
3433oveq2d 6208 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( 0..^ ( # `  ( T `  E )
) )  =  ( 0..^ ( # `  E
) ) )
3531, 34eleqtrrd 2542 . . . . . . . . . . 11  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( ( # `
 E )  - 
1 )  e.  ( 0..^ ( # `  ( T `  E )
) ) )
3624, 35ffvelrnd 5945 . . . . . . . . . 10  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( ( T `  E ) `  ( ( # `  E
)  -  1 ) )  e.  RR )
3718, 36syl5eqel 2543 . . . . . . . . 9  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  B  e.  RR )
3837recnd 9515 . . . . . . . 8  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  B  e.  CC )
395adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  A  e.  RR )
4039recnd 9515 . . . . . . . 8  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  A  e.  CC )
4138, 40mulcomd 9510 . . . . . . 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 4412 . . . . . 6  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( B  x.  A )  <  0
)
4419, 43syl5eqbrr 4426 . . . . 5  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( (
( T `  E
) `  ( ( # `
 E )  - 
1 ) )  x.  A )  <  0
)
45 iftrue 3897 . . . . 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 6208 . . 3  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( ( V `  E )  +  if ( ( ( ( T `  E
) `  ( ( # `
 E )  - 
1 ) )  x.  A )  <  0 ,  1 ,  0 ) )  =  ( ( V `  E
)  +  1 ) )
4813, 47eqtr2d 2493 . 2  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( ( V `  E )  +  1 )  =  ( V `  F
) )
496, 7, 8, 9signsvvf 27116 . . . . . 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 12398 . . . . . . 7  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  <" A ">  e. Word  RR )
53 ccatcl 12378 . . . . . . 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 2539 . . . . 5  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  F  e. Word  RR )
5650, 55ffvelrnd 5945 . . . 4  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( V `  F )  e.  NN0 )
5756nn0cnd 10741 . . 3  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( V `  F )  e.  CC )
5850, 21ffvelrnd 5945 . . . 4  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( V `  E )  e.  NN0 )
5958nn0cnd 10741 . . 3  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( V `  E )  e.  CC )
60 ax-1cn 9443 . . . 4  |-  1  e.  CC
6160a1i 11 . . 3  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  1  e.  CC )
6257, 59, 61subaddd 9840 . 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 1370    e. wcel 1758    =/= wne 2644    \ cdif 3425   (/)c0 3737   ifcif 3891   {csn 3977   {cpr 3979   {ctp 3981   <.cop 3983   class class class wbr 4392    |-> cmpt 4450   -->wf 5514   ` cfv 5518  (class class class)co 6192    |-> cmpt2 6194   CCcc 9383   RRcr 9384   0cc0 9385   1c1 9386    + caddc 9388    x. cmul 9390    < clt 9521    - cmin 9698   -ucneg 9699   NNcn 10425   NN0cn0 10682   ...cfz 11540  ..^cfzo 11651   #chash 12206  Word cword 12325   concat cconcat 12327   <"cs1 12328  sgncsgn 12679   sum_csu 13267   ndxcnx 14275   Basecbs 14278   +g cplusg 14342    gsumg cgsu 14483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-inf2 7950  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462  ax-pre-sup 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-isom 5527  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-supp 6793  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-sup 7794  df-oi 7827  df-card 8212  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-div 10097  df-nn 10426  df-2 10483  df-3 10484  df-n0 10683  df-z 10750  df-uz 10965  df-rp 11095  df-fz 11541  df-fzo 11652  df-seq 11910  df-exp 11969  df-hash 12207  df-word 12333  df-concat 12335  df-s1 12336  df-substr 12337  df-sgn 12680  df-cj 12692  df-re 12693  df-im 12694  df-sqr 12828  df-abs 12829  df-clim 13070  df-sum 13268  df-struct 14280  df-ndx 14281  df-slot 14282  df-base 14283  df-plusg 14355  df-0g 14484  df-gsum 14485  df-mnd 15519  df-mulg 15652  df-cntz 15939
This theorem is referenced by:  signsvfnn  27123
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