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Theorem signstf0 28722
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 12626 . . . . . 6  |-  ( # `  <" K "> )  =  1
21oveq2i 6307 . . . . 5  |-  ( 0..^ ( # `  <" K "> )
)  =  ( 0..^ 1 )
3 fzo01 11900 . . . . 5  |-  ( 0..^ 1 )  =  {
0 }
42, 3eqtri 2486 . . . 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 2555 . . . . 5  |-  ( ( K  e.  RR  /\  n  e.  ( 0..^ ( # `  <" K "> )
) )  ->  n  e.  { 0 } )
8 elsn 4046 . . . . 5  |-  ( n  e.  { 0 }  <-> 
n  =  0 )
97, 8sylib 196 . . . 4  |-  ( ( K  e.  RR  /\  n  e.  ( 0..^ ( # `  <" K "> )
) )  ->  n  =  0 )
10 oveq2 6304 . . . . . . . . 9  |-  ( n  =  0  ->  (
0 ... n )  =  ( 0 ... 0
) )
11 0z 10896 . . . . . . . . . 10  |-  0  e.  ZZ
12 fzsn 11751 . . . . . . . . . 10  |-  ( 0  e.  ZZ  ->  (
0 ... 0 )  =  { 0 } )
1311, 12ax-mp 5 . . . . . . . . 9  |-  ( 0 ... 0 )  =  { 0 }
1410, 13syl6eq 2514 . . . . . . . 8  |-  ( n  =  0  ->  (
0 ... n )  =  { 0 } )
1514mpteq1d 4538 . . . . . . 7  |-  ( n  =  0  ->  (
i  e.  ( 0 ... n )  |->  (sgn
`  ( <" K "> `  i )
) )  =  ( i  e.  { 0 }  |->  (sgn `  ( <" K "> `  i ) ) ) )
1615oveq2d 6312 . . . . . 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 28711 . . . . . . . 8  |-  W  e. 
Mnd
2120a1i 11 . . . . . . 7  |-  ( K  e.  RR  ->  W  e.  Mnd )
22 0re 9613 . . . . . . . 8  |-  0  e.  RR
2322a1i 11 . . . . . . 7  |-  ( K  e.  RR  ->  0  e.  RR )
24 s1fv 12628 . . . . . . . . . 10  |-  ( K  e.  RR  ->  ( <" K "> `  0 )  =  K )
25 id 22 . . . . . . . . . 10  |-  ( K  e.  RR  ->  K  e.  RR )
2624, 25eqeltrd 2545 . . . . . . . . 9  |-  ( K  e.  RR  ->  ( <" K "> `  0 )  e.  RR )
2726rexrd 9660 . . . . . . . 8  |-  ( K  e.  RR  ->  ( <" K "> `  0 )  e.  RR* )
28 sgncl 28674 . . . . . . . 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 28708 . . . . . . . 8  |-  { -u
1 ,  0 ,  1 }  =  (
Base `  W )
31 fveq2 5872 . . . . . . . . 9  |-  ( i  =  0  ->  ( <" K "> `  i )  =  (
<" K "> `  0 ) )
3231fveq2d 5876 . . . . . . . 8  |-  ( i  =  0  ->  (sgn `  ( <" K "> `  i )
)  =  (sgn `  ( <" K "> `  0 ) ) )
3330, 32gsumsn 17108 . . . . . . 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 1228 . . . . . 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 5876 . . . . . 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 2502 . . . 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 4534 . 2  |-  ( K  e.  RR  ->  (
n  e.  ( 0..^ ( # `  <" K "> )
)  |->  ( W  gsumg  ( i  e.  ( 0 ... n )  |->  (sgn `  ( <" K "> `  i ) ) ) ) )  =  ( n  e.  {
0 }  |->  (sgn `  K ) ) )
41 s1cl 12623 . . 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 28717 . . 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 28675 . . . 4  |-  ( K  e.  RR  ->  (sgn `  K )  e.  RR )
47 s1val 12619 . . . 4  |-  ( (sgn
`  K )  e.  RR  ->  <" (sgn `  K ) ">  =  { <. 0 ,  (sgn
`  K ) >. } )
4846, 47syl 16 . . 3  |-  ( K  e.  RR  ->  <" (sgn `  K ) ">  =  { <. 0 ,  (sgn
`  K ) >. } )
49 fmptsn 6092 . . . 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 2498 . 2  |-  ( K  e.  RR  ->  <" (sgn `  K ) ">  =  ( n  e. 
{ 0 }  |->  (sgn
`  K ) ) )
5240, 45, 513eqtr4d 2508 1  |-  ( K  e.  RR  ->  ( T `  <" K "> )  =  <" (sgn `  K ) "> )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   ifcif 3944   {csn 4032   {cpr 4034   {ctp 4036   <.cop 4038    |-> cmpt 4515   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   RRcr 9508   0cc0 9509   1c1 9510   RR*cxr 9644    - cmin 9824   -ucneg 9825   ZZcz 10885   ...cfz 11697  ..^cfzo 11821   #chash 12408  Word cword 12538   <"cs1 12541  sgncsgn 12931   sum_csu 13520   ndxcnx 14641   Basecbs 14644   +g cplusg 14712    gsumg cgsu 14858   Mndcmnd 16046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-seq 12111  df-hash 12409  df-word 12546  df-s1 12549  df-sgn 12932  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-plusg 14725  df-0g 14859  df-gsum 14860  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-mulg 16187  df-cntz 16482
This theorem is referenced by:  signsvtn0  28724  signstfvneq0  28726
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