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Theorem signsvfnn 26987
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
)
signsvf.b  |-  B  =  ( E `  ( N  -  1 ) )
Assertion
Ref Expression
signsvfnn  |-  ( (
ph  /\  ( B  x.  A )  <  0
)  ->  ( ( V `  F )  -  ( V `  E ) )  =  1 )
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 signsvfnn
StepHypRef Expression
1 signsvf.e . . . . . . . . 9  |-  ( ph  ->  E  e.  (Word  RR  \  { (/) } ) )
21adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( B  x.  A )  <  0
)  ->  E  e.  (Word  RR  \  { (/) } ) )
3 signsvf.b . . . . . . . . 9  |-  B  =  ( E `  ( N  -  1 ) )
4 eldifsn 4000 . . . . . . . . . . . . . . . . 17  |-  ( E  e.  (Word  RR  \  { (/) } )  <->  ( E  e. Word  RR  /\  E  =/=  (/) ) )
51, 4sylib 196 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( E  e. Word  RR  /\  E  =/=  (/) ) )
65simpld 459 . . . . . . . . . . . . . . 15  |-  ( ph  ->  E  e. Word  RR )
7 wrdf 12240 . . . . . . . . . . . . . . 15  |-  ( E  e. Word  RR  ->  E :
( 0..^ ( # `  E ) ) --> RR )
86, 7syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  E : ( 0..^ ( # `  E
) ) --> RR )
9 signsvf.n . . . . . . . . . . . . . . . 16  |-  N  =  ( # `  E
)
109oveq1i 6101 . . . . . . . . . . . . . . 15  |-  ( N  -  1 )  =  ( ( # `  E
)  -  1 )
11 lennncl 12250 . . . . . . . . . . . . . . . 16  |-  ( ( E  e. Word  RR  /\  E  =/=  (/) )  ->  ( # `
 E )  e.  NN )
12 fzo0end 11619 . . . . . . . . . . . . . . . 16  |-  ( (
# `  E )  e.  NN  ->  ( ( # `
 E )  - 
1 )  e.  ( 0..^ ( # `  E
) ) )
135, 11, 123syl 20 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( # `  E
)  -  1 )  e.  ( 0..^ (
# `  E )
) )
1410, 13syl5eqel 2527 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( N  -  1 )  e.  ( 0..^ ( # `  E
) ) )
158, 14ffvelrnd 5844 . . . . . . . . . . . . 13  |-  ( ph  ->  ( E `  ( N  -  1 ) )  e.  RR )
1615recnd 9412 . . . . . . . . . . . 12  |-  ( ph  ->  ( E `  ( N  -  1 ) )  e.  CC )
173, 16syl5eqel 2527 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  CC )
1817adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( B  x.  A )  <  0
)  ->  B  e.  CC )
19 signsvf.a . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  RR )
2019recnd 9412 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  CC )
2120adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( B  x.  A )  <  0
)  ->  A  e.  CC )
223, 15syl5eqel 2527 . . . . . . . . . . . . 13  |-  ( ph  ->  B  e.  RR )
2322adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( B  x.  A )  <  0
)  ->  B  e.  RR )
2419adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( B  x.  A )  <  0
)  ->  A  e.  RR )
2523, 24remulcld 9414 . . . . . . . . . . 11  |-  ( (
ph  /\  ( B  x.  A )  <  0
)  ->  ( B  x.  A )  e.  RR )
26 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  ( B  x.  A )  <  0
)  ->  ( B  x.  A )  <  0
)
2725, 26ltned 9510 . . . . . . . . . 10  |-  ( (
ph  /\  ( B  x.  A )  <  0
)  ->  ( B  x.  A )  =/=  0
)
2818, 21, 27mulne0bad 9991 . . . . . . . . 9  |-  ( (
ph  /\  ( B  x.  A )  <  0
)  ->  B  =/=  0 )
293, 28syl5eqner 2633 . . . . . . . 8  |-  ( (
ph  /\  ( B  x.  A )  <  0
)  ->  ( E `  ( N  -  1 ) )  =/=  0
)
30 signsv.p . . . . . . . . . 10  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
31 signsv.w . . . . . . . . . 10  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
32 signsv.t . . . . . . . . . 10  |-  T  =  ( f  e. Word  RR  |->  ( n  e.  (
0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) ) )
33 signsv.v . . . . . . . . . 10  |-  V  =  ( f  e. Word  RR  |->  sum_ j  e.  ( 1..^ ( # `  f
) ) if ( ( ( T `  f ) `  j
)  =/=  ( ( T `  f ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
3430, 31, 32, 33, 9signsvtn0 26971 . . . . . . . . 9  |-  ( ( E  e.  (Word  RR  \  { (/) } )  /\  ( E `  ( N  -  1 ) )  =/=  0 )  -> 
( ( T `  E ) `  ( N  -  1 ) )  =  (sgn `  ( E `  ( N  -  1 ) ) ) )
353fveq2i 5694 . . . . . . . . 9  |-  (sgn `  B )  =  (sgn
`  ( E `  ( N  -  1
) ) )
3634, 35syl6eqr 2493 . . . . . . . 8  |-  ( ( E  e.  (Word  RR  \  { (/) } )  /\  ( E `  ( N  -  1 ) )  =/=  0 )  -> 
( ( T `  E ) `  ( N  -  1 ) )  =  (sgn `  B ) )
372, 29, 36syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  ( B  x.  A )  <  0
)  ->  ( ( T `  E ) `  ( N  -  1 ) )  =  (sgn
`  B ) )
3837fveq2d 5695 . . . . . 6  |-  ( (
ph  /\  ( B  x.  A )  <  0
)  ->  (sgn `  (
( T `  E
) `  ( N  -  1 ) ) )  =  (sgn `  (sgn `  B ) ) )
3923rexrd 9433 . . . . . . 7  |-  ( (
ph  /\  ( B  x.  A )  <  0
)  ->  B  e.  RR* )
40 sgnsgn 26931 . . . . . . 7  |-  ( B  e.  RR*  ->  (sgn `  (sgn `  B ) )  =  (sgn `  B
) )
4139, 40syl 16 . . . . . 6  |-  ( (
ph  /\  ( B  x.  A )  <  0
)  ->  (sgn `  (sgn `  B ) )  =  (sgn `  B )
)
4238, 41eqtrd 2475 . . . . 5  |-  ( (
ph  /\  ( B  x.  A )  <  0
)  ->  (sgn `  (
( T `  E
) `  ( N  -  1 ) ) )  =  (sgn `  B ) )
4342oveq2d 6107 . . . 4  |-  ( (
ph  /\  ( B  x.  A )  <  0
)  ->  ( (sgn `  A )  x.  (sgn `  ( ( T `  E ) `  ( N  -  1 ) ) ) )  =  ( (sgn `  A
)  x.  (sgn `  B ) ) )
4420, 17mulcomd 9407 . . . . . . 7  |-  ( ph  ->  ( A  x.  B
)  =  ( B  x.  A ) )
4544breq1d 4302 . . . . . 6  |-  ( ph  ->  ( ( A  x.  B )  <  0  <->  ( B  x.  A )  <  0 ) )
46 sgnmulsgn 26932 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  B )  <  0  <->  ( (sgn `  A )  x.  (sgn `  B )
)  <  0 ) )
4719, 22, 46syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ( A  x.  B )  <  0  <->  ( (sgn `  A )  x.  (sgn `  B )
)  <  0 ) )
4845, 47bitr3d 255 . . . . 5  |-  ( ph  ->  ( ( B  x.  A )  <  0  <->  ( (sgn `  A )  x.  (sgn `  B )
)  <  0 ) )
4948biimpa 484 . . . 4  |-  ( (
ph  /\  ( B  x.  A )  <  0
)  ->  ( (sgn `  A )  x.  (sgn `  B ) )  <  0 )
5043, 49eqbrtrd 4312 . . 3  |-  ( (
ph  /\  ( B  x.  A )  <  0
)  ->  ( (sgn `  A )  x.  (sgn `  ( ( T `  E ) `  ( N  -  1 ) ) ) )  <  0 )
51 sgnclre 26922 . . . . . 6  |-  ( B  e.  RR  ->  (sgn `  B )  e.  RR )
5223, 51syl 16 . . . . 5  |-  ( (
ph  /\  ( B  x.  A )  <  0
)  ->  (sgn `  B
)  e.  RR )
5337, 52eqeltrd 2517 . . . 4  |-  ( (
ph  /\  ( B  x.  A )  <  0
)  ->  ( ( T `  E ) `  ( N  -  1 ) )  e.  RR )
54 sgnmulsgn 26932 . . . 4  |-  ( ( A  e.  RR  /\  ( ( T `  E ) `  ( N  -  1 ) )  e.  RR )  ->  ( ( A  x.  ( ( T `
 E ) `  ( N  -  1
) ) )  <  0  <->  ( (sgn `  A )  x.  (sgn `  ( ( T `  E ) `  ( N  -  1 ) ) ) )  <  0 ) )
5524, 53, 54syl2anc 661 . . 3  |-  ( (
ph  /\  ( B  x.  A )  <  0
)  ->  ( ( A  x.  ( ( T `  E ) `  ( N  -  1 ) ) )  <  0  <->  ( (sgn `  A )  x.  (sgn `  ( ( T `  E ) `  ( N  -  1 ) ) ) )  <  0 ) )
5650, 55mpbird 232 . 2  |-  ( (
ph  /\  ( B  x.  A )  <  0
)  ->  ( A  x.  ( ( T `  E ) `  ( N  -  1 ) ) )  <  0
)
57 signsvf.0 . . 3  |-  ( ph  ->  ( E `  0
)  =/=  0 )
58 signsvf.f . . 3  |-  ( ph  ->  F  =  ( E concat  <" A "> ) )
59 eqid 2443 . . 3  |-  ( ( T `  E ) `
 ( N  - 
1 ) )  =  ( ( T `  E ) `  ( N  -  1 ) )
6030, 31, 32, 33, 1, 57, 58, 19, 9, 59signsvtn 26985 . 2  |-  ( (
ph  /\  ( A  x.  ( ( T `  E ) `  ( N  -  1 ) ) )  <  0
)  ->  ( ( V `  F )  -  ( V `  E ) )  =  1 )
6156, 60syldan 470 1  |-  ( (
ph  /\  ( B  x.  A )  <  0
)  ->  ( ( V `  F )  -  ( V `  E ) )  =  1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606    \ cdif 3325   (/)c0 3637   ifcif 3791   {csn 3877   {cpr 3879   {ctp 3881   <.cop 3883   class class class wbr 4292    e. cmpt 4350   -->wf 5414   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    x. cmul 9287   RR*cxr 9417    < clt 9418    - cmin 9595   -ucneg 9596   NNcn 10322   ...cfz 11437  ..^cfzo 11548   #chash 12103  Word cword 12221   concat cconcat 12223   <"cs1 12224  sgncsgn 12575   sum_csu 13163   ndxcnx 14171   Basecbs 14174   +g cplusg 14238    gsumg cgsu 14379
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fz 11438  df-fzo 11549  df-seq 11807  df-exp 11866  df-hash 12104  df-word 12229  df-concat 12231  df-s1 12232  df-substr 12233  df-sgn 12576  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-sum 13164  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-plusg 14251  df-0g 14380  df-gsum 14381  df-mnd 15415  df-mulg 15548  df-cntz 15835
This theorem is referenced by: (None)
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