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Theorem signstf0 29469
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 12751 . . . . . 6  |-  ( # `  <" K "> )  =  1
21oveq2i 6306 . . . . 5  |-  ( 0..^ ( # `  <" K "> )
)  =  ( 0..^ 1 )
3 fzo01 12002 . . . . 5  |-  ( 0..^ 1 )  =  {
0 }
42, 3eqtri 2475 . . . 4  |-  ( 0..^ ( # `  <" K "> )
)  =  { 0 }
54a1i 11 . . 3  |-  ( K  e.  RR  ->  (
0..^ ( # `  <" K "> )
)  =  { 0 } )
6 simpr 463 . . . . . 6  |-  ( ( K  e.  RR  /\  n  e.  ( 0..^ ( # `  <" K "> )
) )  ->  n  e.  ( 0..^ ( # `  <" K "> ) ) )
76, 4syl6eleq 2541 . . . . 5  |-  ( ( K  e.  RR  /\  n  e.  ( 0..^ ( # `  <" K "> )
) )  ->  n  e.  { 0 } )
8 elsn 3984 . . . . 5  |-  ( n  e.  { 0 }  <-> 
n  =  0 )
97, 8sylib 200 . . . 4  |-  ( ( K  e.  RR  /\  n  e.  ( 0..^ ( # `  <" K "> )
) )  ->  n  =  0 )
10 oveq2 6303 . . . . . . . . 9  |-  ( n  =  0  ->  (
0 ... n )  =  ( 0 ... 0
) )
11 0z 10955 . . . . . . . . . 10  |-  0  e.  ZZ
12 fzsn 11847 . . . . . . . . . 10  |-  ( 0  e.  ZZ  ->  (
0 ... 0 )  =  { 0 } )
1311, 12ax-mp 5 . . . . . . . . 9  |-  ( 0 ... 0 )  =  { 0 }
1410, 13syl6eq 2503 . . . . . . . 8  |-  ( n  =  0  ->  (
0 ... n )  =  { 0 } )
1514mpteq1d 4487 . . . . . . 7  |-  ( n  =  0  ->  (
i  e.  ( 0 ... n )  |->  (sgn
`  ( <" K "> `  i )
) )  =  ( i  e.  { 0 }  |->  (sgn `  ( <" K "> `  i ) ) ) )
1615oveq2d 6311 . . . . . 6  |-  ( n  =  0  ->  ( W  gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( <" K "> `  i
) ) ) )  =  ( W  gsumg  ( i  e.  { 0 } 
|->  (sgn `  ( <" K "> `  i
) ) ) ) )
1716adantl 468 . . . . 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 29458 . . . . . . . 8  |-  W  e. 
Mnd
2120a1i 11 . . . . . . 7  |-  ( K  e.  RR  ->  W  e.  Mnd )
22 0re 9648 . . . . . . . 8  |-  0  e.  RR
2322a1i 11 . . . . . . 7  |-  ( K  e.  RR  ->  0  e.  RR )
24 s1fv 12755 . . . . . . . . . 10  |-  ( K  e.  RR  ->  ( <" K "> `  0 )  =  K )
25 id 22 . . . . . . . . . 10  |-  ( K  e.  RR  ->  K  e.  RR )
2624, 25eqeltrd 2531 . . . . . . . . 9  |-  ( K  e.  RR  ->  ( <" K "> `  0 )  e.  RR )
2726rexrd 9695 . . . . . . . 8  |-  ( K  e.  RR  ->  ( <" K "> `  0 )  e.  RR* )
28 sgncl 29421 . . . . . . . 8  |-  ( (
<" K "> `  0 )  e.  RR*  ->  (sgn `  ( <" K "> `  0
) )  e.  { -u 1 ,  0 ,  1 } )
2927, 28syl 17 . . . . . . 7  |-  ( K  e.  RR  ->  (sgn `  ( <" K "> `  0 )
)  e.  { -u
1 ,  0 ,  1 } )
3018, 19signswbase 29455 . . . . . . . 8  |-  { -u
1 ,  0 ,  1 }  =  (
Base `  W )
31 fveq2 5870 . . . . . . . . 9  |-  ( i  =  0  ->  ( <" K "> `  i )  =  (
<" K "> `  0 ) )
3231fveq2d 5874 . . . . . . . 8  |-  ( i  =  0  ->  (sgn `  ( <" K "> `  i )
)  =  (sgn `  ( <" K "> `  0 ) ) )
3330, 32gsumsn 17599 . . . . . . 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 1269 . . . . . 6  |-  ( K  e.  RR  ->  ( W  gsumg  ( i  e.  {
0 }  |->  (sgn `  ( <" K "> `  i ) ) ) )  =  (sgn
`  ( <" K "> `  0 )
) )
3534adantr 467 . . . . 5  |-  ( ( K  e.  RR  /\  n  =  0 )  ->  ( W  gsumg  ( i  e.  { 0 } 
|->  (sgn `  ( <" K "> `  i
) ) ) )  =  (sgn `  ( <" K "> `  0 ) ) )
3624fveq2d 5874 . . . . . 6  |-  ( K  e.  RR  ->  (sgn `  ( <" K "> `  0 )
)  =  (sgn `  K ) )
3736adantr 467 . . . . 5  |-  ( ( K  e.  RR  /\  n  =  0 )  ->  (sgn `  ( <" K "> `  0 ) )  =  (sgn `  K )
)
3817, 35, 373eqtrd 2491 . . . 4  |-  ( ( K  e.  RR  /\  n  =  0 )  ->  ( W  gsumg  ( i  e.  ( 0 ... n )  |->  (sgn `  ( <" K "> `  i ) ) ) )  =  (sgn
`  K ) )
399, 38syldan 473 . . 3  |-  ( ( K  e.  RR  /\  n  e.  ( 0..^ ( # `  <" K "> )
) )  ->  ( W  gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( <" K "> `  i
) ) ) )  =  (sgn `  K
) )
405, 39mpteq12dva 4483 . 2  |-  ( K  e.  RR  ->  (
n  e.  ( 0..^ ( # `  <" K "> )
)  |->  ( W  gsumg  ( i  e.  ( 0 ... n )  |->  (sgn `  ( <" K "> `  i ) ) ) ) )  =  ( n  e.  {
0 }  |->  (sgn `  K ) ) )
41 s1cl 12748 . . 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 29464 . . 3  |-  ( <" K ">  e. Word  RR  ->  ( T `  <" K "> )  =  (
n  e.  ( 0..^ ( # `  <" K "> )
)  |->  ( W  gsumg  ( i  e.  ( 0 ... n )  |->  (sgn `  ( <" K "> `  i ) ) ) ) ) )
4541, 44syl 17 . 2  |-  ( K  e.  RR  ->  ( T `  <" K "> )  =  ( n  e.  ( 0..^ ( # `  <" K "> )
)  |->  ( W  gsumg  ( i  e.  ( 0 ... n )  |->  (sgn `  ( <" K "> `  i ) ) ) ) ) )
46 sgnclre 29422 . . . 4  |-  ( K  e.  RR  ->  (sgn `  K )  e.  RR )
47 s1val 12744 . . . 4  |-  ( (sgn
`  K )  e.  RR  ->  <" (sgn `  K ) ">  =  { <. 0 ,  (sgn
`  K ) >. } )
4846, 47syl 17 . . 3  |-  ( K  e.  RR  ->  <" (sgn `  K ) ">  =  { <. 0 ,  (sgn
`  K ) >. } )
49 fmptsn 6089 . . . 4  |-  ( ( 0  e.  RR  /\  (sgn `  K )  e.  RR )  ->  { <. 0 ,  (sgn `  K
) >. }  =  ( n  e.  { 0 }  |->  (sgn `  K
) ) )
5022, 46, 49sylancr 670 . . 3  |-  ( K  e.  RR  ->  { <. 0 ,  (sgn `  K
) >. }  =  ( n  e.  { 0 }  |->  (sgn `  K
) ) )
5148, 50eqtrd 2487 . 2  |-  ( K  e.  RR  ->  <" (sgn `  K ) ">  =  ( n  e. 
{ 0 }  |->  (sgn
`  K ) ) )
5240, 45, 513eqtr4d 2497 1  |-  ( K  e.  RR  ->  ( T `  <" K "> )  =  <" (sgn `  K ) "> )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1446    e. wcel 1889    =/= wne 2624   ifcif 3883   {csn 3970   {cpr 3972   {ctp 3974   <.cop 3976    |-> cmpt 4464   ` cfv 5585  (class class class)co 6295    |-> cmpt2 6297   RRcr 9543   0cc0 9544   1c1 9545   RR*cxr 9679    - cmin 9865   -ucneg 9866   ZZcz 10944   ...cfz 11791  ..^cfzo 11922   #chash 12522  Word cword 12663   <"cs1 12666  sgncsgn 13161   sum_csu 13764   ndxcnx 15130   Basecbs 15133   +g cplusg 15202    gsumg cgsu 15351   Mndcmnd 16547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6920  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-oi 8030  df-card 8378  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-fzo 11923  df-seq 12221  df-hash 12523  df-word 12671  df-s1 12674  df-sgn 13162  df-struct 15135  df-ndx 15136  df-slot 15137  df-base 15138  df-plusg 15215  df-0g 15352  df-gsum 15353  df-mgm 16500  df-sgrp 16539  df-mnd 16549  df-mulg 16688  df-cntz 16983
This theorem is referenced by:  signsvtn0  29471  signstfvneq0  29473
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