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Theorem signstf0 28151
Description: Sign of a single letter word. (Contributed by Thierry Arnoux, 8-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 ) )
Assertion
Ref Expression
signstf0  |-  ( K  e.  RR  ->  ( T `  <" K "> )  =  <" (sgn `  K ) "> )
Distinct variable groups:    a, b,  .+^    f, i, n, W    f, K, i, n
Allowed substitution hints:    .+^ ( f, i,
j, n)    T( f,
i, j, n, a, b)    K( j, a, b)    V( f, i, j, n, a, b)    W( j, a, b)

Proof of Theorem signstf0
StepHypRef Expression
1 s1len 12567 . . . . . 6  |-  ( # `  <" K "> )  =  1
21oveq2i 6286 . . . . 5  |-  ( 0..^ ( # `  <" K "> )
)  =  ( 0..^ 1 )
3 fzo01 11854 . . . . 5  |-  ( 0..^ 1 )  =  {
0 }
42, 3eqtri 2489 . . . 4  |-  ( 0..^ ( # `  <" K "> )
)  =  { 0 }
54a1i 11 . . 3  |-  ( K  e.  RR  ->  (
0..^ ( # `  <" K "> )
)  =  { 0 } )
6 simpr 461 . . . . . 6  |-  ( ( K  e.  RR  /\  n  e.  ( 0..^ ( # `  <" K "> )
) )  ->  n  e.  ( 0..^ ( # `  <" K "> ) ) )
76, 4syl6eleq 2558 . . . . 5  |-  ( ( K  e.  RR  /\  n  e.  ( 0..^ ( # `  <" K "> )
) )  ->  n  e.  { 0 } )
8 elsn 4034 . . . . 5  |-  ( n  e.  { 0 }  <-> 
n  =  0 )
97, 8sylib 196 . . . 4  |-  ( ( K  e.  RR  /\  n  e.  ( 0..^ ( # `  <" K "> )
) )  ->  n  =  0 )
10 oveq2 6283 . . . . . . . . 9  |-  ( n  =  0  ->  (
0 ... n )  =  ( 0 ... 0
) )
11 0z 10864 . . . . . . . . . 10  |-  0  e.  ZZ
12 fzsn 11714 . . . . . . . . . 10  |-  ( 0  e.  ZZ  ->  (
0 ... 0 )  =  { 0 } )
1311, 12ax-mp 5 . . . . . . . . 9  |-  ( 0 ... 0 )  =  { 0 }
1410, 13syl6eq 2517 . . . . . . . 8  |-  ( n  =  0  ->  (
0 ... n )  =  { 0 } )
1514mpteq1d 4521 . . . . . . 7  |-  ( n  =  0  ->  (
i  e.  ( 0 ... n )  |->  (sgn
`  ( <" K "> `  i )
) )  =  ( i  e.  { 0 }  |->  (sgn `  ( <" K "> `  i ) ) ) )
1615oveq2d 6291 . . . . . 6  |-  ( n  =  0  ->  ( W  gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( <" K "> `  i
) ) ) )  =  ( W  gsumg  ( i  e.  { 0 } 
|->  (sgn `  ( <" K "> `  i
) ) ) ) )
1716adantl 466 . . . . 5  |-  ( ( K  e.  RR  /\  n  =  0 )  ->  ( W  gsumg  ( i  e.  ( 0 ... n )  |->  (sgn `  ( <" K "> `  i ) ) ) )  =  ( W  gsumg  ( i  e.  {
0 }  |->  (sgn `  ( <" K "> `  i ) ) ) ) )
18 signsv.p . . . . . . . . 9  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
19 signsv.w . . . . . . . . 9  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
2018, 19signswmnd 28140 . . . . . . . 8  |-  W  e. 
Mnd
2120a1i 11 . . . . . . 7  |-  ( K  e.  RR  ->  W  e.  Mnd )
22 0re 9585 . . . . . . . 8  |-  0  e.  RR
2322a1i 11 . . . . . . 7  |-  ( K  e.  RR  ->  0  e.  RR )
24 s1fv 12569 . . . . . . . . . 10  |-  ( K  e.  RR  ->  ( <" K "> `  0 )  =  K )
25 id 22 . . . . . . . . . 10  |-  ( K  e.  RR  ->  K  e.  RR )
2624, 25eqeltrd 2548 . . . . . . . . 9  |-  ( K  e.  RR  ->  ( <" K "> `  0 )  e.  RR )
2726rexrd 9632 . . . . . . . 8  |-  ( K  e.  RR  ->  ( <" K "> `  0 )  e.  RR* )
28 sgncl 28103 . . . . . . . 8  |-  ( (
<" K "> `  0 )  e.  RR*  ->  (sgn `  ( <" K "> `  0
) )  e.  { -u 1 ,  0 ,  1 } )
2927, 28syl 16 . . . . . . 7  |-  ( K  e.  RR  ->  (sgn `  ( <" K "> `  0 )
)  e.  { -u
1 ,  0 ,  1 } )
3018, 19signswbase 28137 . . . . . . . 8  |-  { -u
1 ,  0 ,  1 }  =  (
Base `  W )
31 fveq2 5857 . . . . . . . . 9  |-  ( i  =  0  ->  ( <" K "> `  i )  =  (
<" K "> `  0 ) )
3231fveq2d 5861 . . . . . . . 8  |-  ( i  =  0  ->  (sgn `  ( <" K "> `  i )
)  =  (sgn `  ( <" K "> `  0 ) ) )
3330, 32gsumsn 16765 . . . . . . 7  |-  ( ( W  e.  Mnd  /\  0  e.  RR  /\  (sgn `  ( <" K "> `  0 )
)  e.  { -u
1 ,  0 ,  1 } )  -> 
( W  gsumg  ( i  e.  {
0 }  |->  (sgn `  ( <" K "> `  i ) ) ) )  =  (sgn
`  ( <" K "> `  0 )
) )
3421, 23, 29, 33syl3anc 1223 . . . . . 6  |-  ( K  e.  RR  ->  ( W  gsumg  ( i  e.  {
0 }  |->  (sgn `  ( <" K "> `  i ) ) ) )  =  (sgn
`  ( <" K "> `  0 )
) )
3534adantr 465 . . . . 5  |-  ( ( K  e.  RR  /\  n  =  0 )  ->  ( W  gsumg  ( i  e.  { 0 } 
|->  (sgn `  ( <" K "> `  i
) ) ) )  =  (sgn `  ( <" K "> `  0 ) ) )
3624fveq2d 5861 . . . . . 6  |-  ( K  e.  RR  ->  (sgn `  ( <" K "> `  0 )
)  =  (sgn `  K ) )
3736adantr 465 . . . . 5  |-  ( ( K  e.  RR  /\  n  =  0 )  ->  (sgn `  ( <" K "> `  0 ) )  =  (sgn `  K )
)
3817, 35, 373eqtrd 2505 . . . 4  |-  ( ( K  e.  RR  /\  n  =  0 )  ->  ( W  gsumg  ( i  e.  ( 0 ... n )  |->  (sgn `  ( <" K "> `  i ) ) ) )  =  (sgn
`  K ) )
399, 38syldan 470 . . 3  |-  ( ( K  e.  RR  /\  n  e.  ( 0..^ ( # `  <" K "> )
) )  ->  ( W  gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( <" K "> `  i
) ) ) )  =  (sgn `  K
) )
405, 39mpteq12dva 4517 . 2  |-  ( K  e.  RR  ->  (
n  e.  ( 0..^ ( # `  <" K "> )
)  |->  ( W  gsumg  ( i  e.  ( 0 ... n )  |->  (sgn `  ( <" K "> `  i ) ) ) ) )  =  ( n  e.  {
0 }  |->  (sgn `  K ) ) )
41 s1cl 12564 . . 3  |-  ( K  e.  RR  ->  <" K ">  e. Word  RR )
42 signsv.t . . . 4  |-  T  =  ( f  e. Word  RR  |->  ( n  e.  (
0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) ) )
43 signsv.v . . . 4  |-  V  =  ( f  e. Word  RR  |->  sum_ j  e.  ( 1..^ ( # `  f
) ) if ( ( ( T `  f ) `  j
)  =/=  ( ( T `  f ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
4418, 19, 42, 43signstfv 28146 . . 3  |-  ( <" K ">  e. Word  RR  ->  ( T `  <" K "> )  =  (
n  e.  ( 0..^ ( # `  <" K "> )
)  |->  ( W  gsumg  ( i  e.  ( 0 ... n )  |->  (sgn `  ( <" K "> `  i ) ) ) ) ) )
4541, 44syl 16 . 2  |-  ( K  e.  RR  ->  ( T `  <" K "> )  =  ( n  e.  ( 0..^ ( # `  <" K "> )
)  |->  ( W  gsumg  ( i  e.  ( 0 ... n )  |->  (sgn `  ( <" K "> `  i ) ) ) ) ) )
46 sgnclre 28104 . . . 4  |-  ( K  e.  RR  ->  (sgn `  K )  e.  RR )
47 s1val 12561 . . . 4  |-  ( (sgn
`  K )  e.  RR  ->  <" (sgn `  K ) ">  =  { <. 0 ,  (sgn
`  K ) >. } )
4846, 47syl 16 . . 3  |-  ( K  e.  RR  ->  <" (sgn `  K ) ">  =  { <. 0 ,  (sgn
`  K ) >. } )
49 fmptsn 6072 . . . 4  |-  ( ( 0  e.  RR  /\  (sgn `  K )  e.  RR )  ->  { <. 0 ,  (sgn `  K
) >. }  =  ( n  e.  { 0 }  |->  (sgn `  K
) ) )
5022, 46, 49sylancr 663 . . 3  |-  ( K  e.  RR  ->  { <. 0 ,  (sgn `  K
) >. }  =  ( n  e.  { 0 }  |->  (sgn `  K
) ) )
5148, 50eqtrd 2501 . 2  |-  ( K  e.  RR  ->  <" (sgn `  K ) ">  =  ( n  e. 
{ 0 }  |->  (sgn
`  K ) ) )
5240, 45, 513eqtr4d 2511 1  |-  ( K  e.  RR  ->  ( T `  <" K "> )  =  <" (sgn `  K ) "> )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655   ifcif 3932   {csn 4020   {cpr 4022   {ctp 4024   <.cop 4026    |-> cmpt 4498   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   RRcr 9480   0cc0 9481   1c1 9482   RR*cxr 9616    - cmin 9794   -ucneg 9795   ZZcz 10853   ...cfz 11661  ..^cfzo 11781   #chash 12360  Word cword 12487   <"cs1 12490  sgncsgn 12869   sum_csu 13457   ndxcnx 14476   Basecbs 14479   +g cplusg 14544    gsumg cgsu 14685   Mndcmnd 15715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-seq 12064  df-hash 12361  df-word 12495  df-s1 12498  df-sgn 12870  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-plusg 14557  df-0g 14686  df-gsum 14687  df-mnd 15721  df-mulg 15854  df-cntz 16143
This theorem is referenced by:  signsvtn0  28153  signstfvneq0  28155
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