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Theorem signstf0 27114
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 12415 . . . . . 6  |-  ( # `  <" K "> )  =  1
21oveq2i 6212 . . . . 5  |-  ( 0..^ ( # `  <" K "> )
)  =  ( 0..^ 1 )
3 fzo01 11730 . . . . 5  |-  ( 0..^ 1 )  =  {
0 }
42, 3eqtri 2483 . . . 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 2552 . . . . 5  |-  ( ( K  e.  RR  /\  n  e.  ( 0..^ ( # `  <" K "> )
) )  ->  n  e.  { 0 } )
8 elsn 4000 . . . . 5  |-  ( n  e.  { 0 }  <-> 
n  =  0 )
97, 8sylib 196 . . . 4  |-  ( ( K  e.  RR  /\  n  e.  ( 0..^ ( # `  <" K "> )
) )  ->  n  =  0 )
10 oveq2 6209 . . . . . . . . 9  |-  ( n  =  0  ->  (
0 ... n )  =  ( 0 ... 0
) )
11 0z 10769 . . . . . . . . . 10  |-  0  e.  ZZ
12 fzsn 11618 . . . . . . . . . 10  |-  ( 0  e.  ZZ  ->  (
0 ... 0 )  =  { 0 } )
1311, 12ax-mp 5 . . . . . . . . 9  |-  ( 0 ... 0 )  =  { 0 }
1410, 13syl6eq 2511 . . . . . . . 8  |-  ( n  =  0  ->  (
0 ... n )  =  { 0 } )
1514mpteq1d 4482 . . . . . . 7  |-  ( n  =  0  ->  (
i  e.  ( 0 ... n )  |->  (sgn
`  ( <" K "> `  i )
) )  =  ( i  e.  { 0 }  |->  (sgn `  ( <" K "> `  i ) ) ) )
1615oveq2d 6217 . . . . . 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 27103 . . . . . . . 8  |-  W  e. 
Mnd
2120a1i 11 . . . . . . 7  |-  ( K  e.  RR  ->  W  e.  Mnd )
22 0re 9498 . . . . . . . 8  |-  0  e.  RR
2322a1i 11 . . . . . . 7  |-  ( K  e.  RR  ->  0  e.  RR )
24 s1fv 12417 . . . . . . . . . 10  |-  ( K  e.  RR  ->  ( <" K "> `  0 )  =  K )
25 id 22 . . . . . . . . . 10  |-  ( K  e.  RR  ->  K  e.  RR )
2624, 25eqeltrd 2542 . . . . . . . . 9  |-  ( K  e.  RR  ->  ( <" K "> `  0 )  e.  RR )
2726rexrd 9545 . . . . . . . 8  |-  ( K  e.  RR  ->  ( <" K "> `  0 )  e.  RR* )
28 sgncl 27066 . . . . . . . 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 27100 . . . . . . . 8  |-  { -u
1 ,  0 ,  1 }  =  (
Base `  W )
31 fveq2 5800 . . . . . . . . 9  |-  ( i  =  0  ->  ( <" K "> `  i )  =  (
<" K "> `  0 ) )
3231fveq2d 5804 . . . . . . . 8  |-  ( i  =  0  ->  (sgn `  ( <" K "> `  i )
)  =  (sgn `  ( <" K "> `  0 ) ) )
3330, 32gsumsn 16572 . . . . . . 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 1219 . . . . . 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 5804 . . . . . 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 2499 . . . 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 4478 . 2  |-  ( K  e.  RR  ->  (
n  e.  ( 0..^ ( # `  <" K "> )
)  |->  ( W  gsumg  ( i  e.  ( 0 ... n )  |->  (sgn `  ( <" K "> `  i ) ) ) ) )  =  ( n  e.  {
0 }  |->  (sgn `  K ) ) )
41 s1cl 12412 . . 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 27109 . . 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 27067 . . . 4  |-  ( K  e.  RR  ->  (sgn `  K )  e.  RR )
47 s1val 12409 . . . 4  |-  ( (sgn
`  K )  e.  RR  ->  <" (sgn `  K ) ">  =  { <. 0 ,  (sgn
`  K ) >. } )
4846, 47syl 16 . . 3  |-  ( K  e.  RR  ->  <" (sgn `  K ) ">  =  { <. 0 ,  (sgn
`  K ) >. } )
49 fmptsn 6009 . . . 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 2495 . 2  |-  ( K  e.  RR  ->  <" (sgn `  K ) ">  =  ( n  e. 
{ 0 }  |->  (sgn
`  K ) ) )
5240, 45, 513eqtr4d 2505 1  |-  ( K  e.  RR  ->  ( T `  <" K "> )  =  <" (sgn `  K ) "> )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   ifcif 3900   {csn 3986   {cpr 3988   {ctp 3990   <.cop 3992    |-> cmpt 4459   ` cfv 5527  (class class class)co 6201    |-> cmpt2 6203   RRcr 9393   0cc0 9394   1c1 9395   RR*cxr 9529    - cmin 9707   -ucneg 9708   ZZcz 10758   ...cfz 11555  ..^cfzo 11666   #chash 12221  Word cword 12340   <"cs1 12343  sgncsgn 12694   sum_csu 13282   ndxcnx 14290   Basecbs 14293   +g cplusg 14358    gsumg cgsu 14499   Mndcmnd 15529
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 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-supp 6802  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-oi 7836  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667  df-seq 11925  df-hash 12222  df-word 12348  df-s1 12351  df-sgn 12695  df-struct 14295  df-ndx 14296  df-slot 14297  df-base 14298  df-plusg 14371  df-0g 14500  df-gsum 14501  df-mnd 15535  df-mulg 15668  df-cntz 15955
This theorem is referenced by:  signsvtn0  27116  signstfvneq0  27118
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