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Theorem signstfvn 28723
Description: Zero-skipping sign in a word compared to a shorter 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
signstfvn  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( T `  ( F ++  <" K "> ) ) `  ( # `  F ) )  =  ( ( ( T `  F
) `  ( ( # `
 F )  - 
1 ) )  .+^  (sgn `  K ) ) )
Distinct variable groups:    a, b,  .+^    f, i, n, F    f, K, i, n    f, W, i, n
Allowed substitution hints:    .+^ ( f, i,
j, n)    T( f,
i, j, n, a, b)    F( j, a, b)    K( j, a, b)    V( f, i, j, n, a, b)    W( j, a, b)

Proof of Theorem signstfvn
StepHypRef Expression
1 signsv.p . . . . 5  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
2 signsv.w . . . . 5  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
31, 2signswbase 28708 . . . 4  |-  { -u
1 ,  0 ,  1 }  =  (
Base `  W )
41, 2signswmnd 28711 . . . . 5  |-  W  e. 
Mnd
54a1i 11 . . . 4  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  W  e.  Mnd )
6 eldifi 3622 . . . . . . . . 9  |-  ( F  e.  (Word  RR  \  { (/) } )  ->  F  e. Word  RR )
7 lencl 12569 . . . . . . . . 9  |-  ( F  e. Word  RR  ->  ( # `  F )  e.  NN0 )
86, 7syl 16 . . . . . . . 8  |-  ( F  e.  (Word  RR  \  { (/) } )  -> 
( # `  F )  e.  NN0 )
9 eldifsn 4157 . . . . . . . . 9  |-  ( F  e.  (Word  RR  \  { (/) } )  <->  ( F  e. Word  RR  /\  F  =/=  (/) ) )
10 hasheq0 12436 . . . . . . . . . . 11  |-  ( F  e. Word  RR  ->  ( (
# `  F )  =  0  <->  F  =  (/) ) )
1110necon3bid 2715 . . . . . . . . . 10  |-  ( F  e. Word  RR  ->  ( (
# `  F )  =/=  0  <->  F  =/=  (/) ) )
1211biimpar 485 . . . . . . . . 9  |-  ( ( F  e. Word  RR  /\  F  =/=  (/) )  ->  ( # `
 F )  =/=  0 )
139, 12sylbi 195 . . . . . . . 8  |-  ( F  e.  (Word  RR  \  { (/) } )  -> 
( # `  F )  =/=  0 )
14 elnnne0 10830 . . . . . . . 8  |-  ( (
# `  F )  e.  NN  <->  ( ( # `  F )  e.  NN0  /\  ( # `  F
)  =/=  0 ) )
158, 13, 14sylanbrc 664 . . . . . . 7  |-  ( F  e.  (Word  RR  \  { (/) } )  -> 
( # `  F )  e.  NN )
1615adantr 465 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( # `  F
)  e.  NN )
17 nnm1nn0 10858 . . . . . 6  |-  ( (
# `  F )  e.  NN  ->  ( ( # `
 F )  - 
1 )  e.  NN0 )
1816, 17syl 16 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( # `  F
)  -  1 )  e.  NN0 )
19 nn0uz 11140 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
2018, 19syl6eleq 2555 . . . 4  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( # `  F
)  -  1 )  e.  ( ZZ>= `  0
) )
21 s1cl 12623 . . . . . . . . . 10  |-  ( K  e.  RR  ->  <" K ">  e. Word  RR )
22 ccatcl 12602 . . . . . . . . . 10  |-  ( ( F  e. Word  RR  /\  <" K ">  e. Word  RR )  ->  ( F ++  <" K "> )  e. Word  RR )
236, 21, 22syl2an 477 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( F ++  <" K "> )  e. Word  RR )
2423adantr 465 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  ( F ++  <" K "> )  e. Word  RR )
25 wrdf 12558 . . . . . . . 8  |-  ( ( F ++  <" K "> )  e. Word  RR  ->  ( F ++  <" K "> ) : ( 0..^ ( # `  ( F ++  <" K "> ) ) ) --> RR )
2624, 25syl 16 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  ( F ++  <" K "> ) : ( 0..^ (
# `  ( F ++  <" K "> ) ) ) --> RR )
278adantr 465 . . . . . . . . . . . 12  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( # `  F
)  e.  NN0 )
2827nn0zd 10988 . . . . . . . . . . 11  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( # `  F
)  e.  ZZ )
29 fzoval 11827 . . . . . . . . . . 11  |-  ( (
# `  F )  e.  ZZ  ->  ( 0..^ ( # `  F
) )  =  ( 0 ... ( (
# `  F )  -  1 ) ) )
3028, 29syl 16 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( 0..^ ( # `  F ) )  =  ( 0 ... (
( # `  F )  -  1 ) ) )
31 fzossfz 11844 . . . . . . . . . 10  |-  ( 0..^ ( # `  F
) )  C_  (
0 ... ( # `  F
) )
3230, 31syl6eqssr 3550 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( 0 ... (
( # `  F )  -  1 ) ) 
C_  ( 0 ... ( # `  F
) ) )
33 ccatlen 12603 . . . . . . . . . . . . 13  |-  ( ( F  e. Word  RR  /\  <" K ">  e. Word  RR )  ->  ( # `
 ( F ++  <" K "> )
)  =  ( (
# `  F )  +  ( # `  <" K "> )
) )
346, 21, 33syl2an 477 . . . . . . . . . . . 12  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( # `  ( F ++  <" K "> ) )  =  ( ( # `  F
)  +  ( # `  <" K "> ) ) )
35 s1len 12626 . . . . . . . . . . . . 13  |-  ( # `  <" K "> )  =  1
3635oveq2i 6307 . . . . . . . . . . . 12  |-  ( (
# `  F )  +  ( # `  <" K "> )
)  =  ( (
# `  F )  +  1 )
3734, 36syl6eq 2514 . . . . . . . . . . 11  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( # `  ( F ++  <" K "> ) )  =  ( ( # `  F
)  +  1 ) )
3837oveq2d 6312 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( 0..^ ( # `  ( F ++  <" K "> ) ) )  =  ( 0..^ ( ( # `  F
)  +  1 ) ) )
3928peano2zd 10993 . . . . . . . . . . 11  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( # `  F
)  +  1 )  e.  ZZ )
40 fzoval 11827 . . . . . . . . . . 11  |-  ( ( ( # `  F
)  +  1 )  e.  ZZ  ->  (
0..^ ( ( # `  F )  +  1 ) )  =  ( 0 ... ( ( ( # `  F
)  +  1 )  -  1 ) ) )
4139, 40syl 16 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( 0..^ ( (
# `  F )  +  1 ) )  =  ( 0 ... ( ( ( # `  F )  +  1 )  -  1 ) ) )
4227nn0cnd 10875 . . . . . . . . . . . 12  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( # `  F
)  e.  CC )
43 1cnd 9629 . . . . . . . . . . . 12  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  1  e.  CC )
4442, 43pncand 9951 . . . . . . . . . . 11  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( ( # `  F )  +  1 )  -  1 )  =  ( # `  F
) )
4544oveq2d 6312 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( 0 ... (
( ( # `  F
)  +  1 )  -  1 ) )  =  ( 0 ... ( # `  F
) ) )
4638, 41, 453eqtrd 2502 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( 0..^ ( # `  ( F ++  <" K "> ) ) )  =  ( 0 ... ( # `  F
) ) )
4732, 46sseqtr4d 3536 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( 0 ... (
( # `  F )  -  1 ) ) 
C_  ( 0..^ (
# `  ( F ++  <" K "> ) ) ) )
4847sselda 3499 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  i  e.  ( 0..^ ( # `  ( F ++  <" K "> ) ) ) )
4926, 48ffvelrnd 6033 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  ( ( F ++  <" K "> ) `  i )  e.  RR )
5049rexrd 9660 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  ( ( F ++  <" K "> ) `  i )  e.  RR* )
51 sgncl 28674 . . . . 5  |-  ( ( ( F ++  <" K "> ) `  i
)  e.  RR*  ->  (sgn
`  ( ( F ++ 
<" K "> ) `  i )
)  e.  { -u
1 ,  0 ,  1 } )
5250, 51syl 16 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  (sgn `  (
( F ++  <" K "> ) `  i
) )  e.  { -u 1 ,  0 ,  1 } )
531, 2signswplusg 28709 . . . 4  |-  .+^  =  ( +g  `  W )
54 simpr 461 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  K  e.  RR )
5554rexrd 9660 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  K  e.  RR* )
56 sgncl 28674 . . . . 5  |-  ( K  e.  RR*  ->  (sgn `  K )  e.  { -u 1 ,  0 ,  1 } )
5755, 56syl 16 . . . 4  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  (sgn `  K )  e.  { -u 1 ,  0 ,  1 } )
58 simpr 461 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  =  ( ( ( # `  F
)  -  1 )  +  1 ) )  ->  i  =  ( ( ( # `  F
)  -  1 )  +  1 ) )
5942, 43npcand 9954 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( ( # `  F )  -  1 )  +  1 )  =  ( # `  F
) )
6059adantr 465 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  =  ( ( ( # `  F
)  -  1 )  +  1 ) )  ->  ( ( (
# `  F )  -  1 )  +  1 )  =  (
# `  F )
)
6158, 60eqtrd 2498 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  =  ( ( ( # `  F
)  -  1 )  +  1 ) )  ->  i  =  (
# `  F )
)
6261fveq2d 5876 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  =  ( ( ( # `  F
)  -  1 )  +  1 ) )  ->  ( ( F ++ 
<" K "> ) `  i )  =  ( ( F ++ 
<" K "> ) `  ( # `  F
) ) )
636adantr 465 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  F  e. Word  RR )
6454, 21syl 16 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  <" K ">  e. Word  RR )
65 c0ex 9607 . . . . . . . . . . . . 13  |-  0  e.  _V
6665snid 4060 . . . . . . . . . . . 12  |-  0  e.  { 0 }
67 fzo01 11900 . . . . . . . . . . . 12  |-  ( 0..^ 1 )  =  {
0 }
6866, 67eleqtrri 2544 . . . . . . . . . . 11  |-  0  e.  ( 0..^ 1 )
6935oveq2i 6307 . . . . . . . . . . 11  |-  ( 0..^ ( # `  <" K "> )
)  =  ( 0..^ 1 )
7068, 69eleqtrri 2544 . . . . . . . . . 10  |-  0  e.  ( 0..^ ( # `  <" K "> ) )
7170a1i 11 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  0  e.  ( 0..^ ( # `  <" K "> )
) )
72 ccatval3 12606 . . . . . . . . 9  |-  ( ( F  e. Word  RR  /\  <" K ">  e. Word  RR  /\  0  e.  ( 0..^ ( # `  <" K "> ) ) )  -> 
( ( F ++  <" K "> ) `  ( 0  +  (
# `  F )
) )  =  (
<" K "> `  0 ) )
7363, 64, 71, 72syl3anc 1228 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( F ++  <" K "> ) `  ( 0  +  (
# `  F )
) )  =  (
<" K "> `  0 ) )
7442addid2d 9798 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( 0  +  (
# `  F )
)  =  ( # `  F ) )
7574fveq2d 5876 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( F ++  <" K "> ) `  ( 0  +  (
# `  F )
) )  =  ( ( F ++  <" K "> ) `  ( # `
 F ) ) )
76 s1fv 12628 . . . . . . . . 9  |-  ( K  e.  RR  ->  ( <" K "> `  0 )  =  K )
7754, 76syl 16 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( <" K "> `  0 )  =  K )
7873, 75, 773eqtr3d 2506 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( F ++  <" K "> ) `  ( # `  F
) )  =  K )
7978adantr 465 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  =  ( ( ( # `  F
)  -  1 )  +  1 ) )  ->  ( ( F ++ 
<" K "> ) `  ( # `  F
) )  =  K )
8062, 79eqtrd 2498 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  =  ( ( ( # `  F
)  -  1 )  +  1 ) )  ->  ( ( F ++ 
<" K "> ) `  i )  =  K )
8180fveq2d 5876 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  =  ( ( ( # `  F
)  -  1 )  +  1 ) )  ->  (sgn `  (
( F ++  <" K "> ) `  i
) )  =  (sgn
`  K ) )
823, 5, 20, 52, 53, 57, 81gsumnunsn 28690 . . 3  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( W  gsumg  ( i  e.  ( 0 ... ( ( ( # `  F
)  -  1 )  +  1 ) ) 
|->  (sgn `  ( ( F ++  <" K "> ) `  i ) ) ) )  =  ( ( W  gsumg  ( i  e.  ( 0 ... ( ( # `  F
)  -  1 ) )  |->  (sgn `  (
( F ++  <" K "> ) `  i
) ) ) ) 
.+^  (sgn `  K )
) )
8359oveq2d 6312 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( 0 ... (
( ( # `  F
)  -  1 )  +  1 ) )  =  ( 0 ... ( # `  F
) ) )
8483mpteq1d 4538 . . . 4  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( i  e.  ( 0 ... ( ( ( # `  F
)  -  1 )  +  1 ) ) 
|->  (sgn `  ( ( F ++  <" K "> ) `  i ) ) )  =  ( i  e.  ( 0 ... ( # `  F
) )  |->  (sgn `  ( ( F ++  <" K "> ) `  i ) ) ) )
8584oveq2d 6312 . . 3  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( W  gsumg  ( i  e.  ( 0 ... ( ( ( # `  F
)  -  1 )  +  1 ) ) 
|->  (sgn `  ( ( F ++  <" K "> ) `  i ) ) ) )  =  ( W  gsumg  ( i  e.  ( 0 ... ( # `  F ) )  |->  (sgn
`  ( ( F ++ 
<" K "> ) `  i )
) ) ) )
8663adantr 465 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  F  e. Word  RR )
8764adantr 465 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  <" K ">  e. Word  RR )
8830eleq2d 2527 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( i  e.  ( 0..^ ( # `  F
) )  <->  i  e.  ( 0 ... (
( # `  F )  -  1 ) ) ) )
8988biimpar 485 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  i  e.  ( 0..^ ( # `  F
) ) )
90 ccatval1 12604 . . . . . . . 8  |-  ( ( F  e. Word  RR  /\  <" K ">  e. Word  RR  /\  i  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F ++ 
<" K "> ) `  i )  =  ( F `  i ) )
9186, 87, 89, 90syl3anc 1228 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  ( ( F ++  <" K "> ) `  i )  =  ( F `  i ) )
9291fveq2d 5876 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  (sgn `  (
( F ++  <" K "> ) `  i
) )  =  (sgn
`  ( F `  i ) ) )
9392mpteq2dva 4543 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) 
|->  (sgn `  ( ( F ++  <" K "> ) `  i ) ) )  =  ( i  e.  ( 0 ... ( ( # `  F )  -  1 ) )  |->  (sgn `  ( F `  i ) ) ) )
9493oveq2d 6312 . . . 4  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( W  gsumg  ( i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) 
|->  (sgn `  ( ( F ++  <" K "> ) `  i ) ) ) )  =  ( W  gsumg  ( i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) 
|->  (sgn `  ( F `  i ) ) ) ) )
9594oveq1d 6311 . . 3  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( W  gsumg  ( i  e.  ( 0 ... ( ( # `  F
)  -  1 ) )  |->  (sgn `  (
( F ++  <" K "> ) `  i
) ) ) ) 
.+^  (sgn `  K )
)  =  ( ( W  gsumg  ( i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) 
|->  (sgn `  ( F `  i ) ) ) )  .+^  (sgn `  K
) ) )
9682, 85, 953eqtr3d 2506 . 2  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( W  gsumg  ( i  e.  ( 0 ... ( # `  F ) )  |->  (sgn
`  ( ( F ++ 
<" K "> ) `  i )
) ) )  =  ( ( W  gsumg  ( i  e.  ( 0 ... ( ( # `  F
)  -  1 ) )  |->  (sgn `  ( F `  i )
) ) )  .+^  (sgn `  K ) ) )
97 eqidd 2458 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( # `  F
)  =  ( # `  F ) )
9897olcd 393 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( # `  F
)  e.  ( 0..^ ( # `  F
) )  \/  ( # `
 F )  =  ( # `  F
) ) )
9927, 19syl6eleq 2555 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( # `  F
)  e.  ( ZZ>= ` 
0 ) )
100 fzosplitsni 11923 . . . . . 6  |-  ( (
# `  F )  e.  ( ZZ>= `  0 )  ->  ( ( # `  F
)  e.  ( 0..^ ( ( # `  F
)  +  1 ) )  <->  ( ( # `  F )  e.  ( 0..^ ( # `  F
) )  \/  ( # `
 F )  =  ( # `  F
) ) ) )
10199, 100syl 16 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( # `  F
)  e.  ( 0..^ ( ( # `  F
)  +  1 ) )  <->  ( ( # `  F )  e.  ( 0..^ ( # `  F
) )  \/  ( # `
 F )  =  ( # `  F
) ) ) )
10298, 101mpbird 232 . . . 4  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( # `  F
)  e.  ( 0..^ ( ( # `  F
)  +  1 ) ) )
103102, 38eleqtrrd 2548 . . 3  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( # `  F
)  e.  ( 0..^ ( # `  ( F ++  <" K "> ) ) ) )
104 signsv.t . . . 4  |-  T  =  ( f  e. Word  RR  |->  ( n  e.  (
0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) ) )
105 signsv.v . . . 4  |-  V  =  ( f  e. Word  RR  |->  sum_ j  e.  ( 1..^ ( # `  f
) ) if ( ( ( T `  f ) `  j
)  =/=  ( ( T `  f ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
1061, 2, 104, 105signstfval 28718 . . 3  |-  ( ( ( F ++  <" K "> )  e. Word  RR  /\  ( # `  F
)  e.  ( 0..^ ( # `  ( F ++  <" K "> ) ) ) )  ->  ( ( T `
 ( F ++  <" K "> )
) `  ( # `  F
) )  =  ( W  gsumg  ( i  e.  ( 0 ... ( # `  F ) )  |->  (sgn
`  ( ( F ++ 
<" K "> ) `  i )
) ) ) )
10723, 103, 106syl2anc 661 . 2  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( T `  ( F ++  <" K "> ) ) `  ( # `  F ) )  =  ( W 
gsumg  ( i  e.  ( 0 ... ( # `  F ) )  |->  (sgn
`  ( ( F ++ 
<" K "> ) `  i )
) ) ) )
108 fzo0end 11907 . . . . . 6  |-  ( (
# `  F )  e.  NN  ->  ( ( # `
 F )  - 
1 )  e.  ( 0..^ ( # `  F
) ) )
10915, 108syl 16 . . . . 5  |-  ( F  e.  (Word  RR  \  { (/) } )  -> 
( ( # `  F
)  -  1 )  e.  ( 0..^ (
# `  F )
) )
1101, 2, 104, 105signstfval 28718 . . . . 5  |-  ( ( F  e. Word  RR  /\  ( ( # `  F
)  -  1 )  e.  ( 0..^ (
# `  F )
) )  ->  (
( T `  F
) `  ( ( # `
 F )  - 
1 ) )  =  ( W  gsumg  ( i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) 
|->  (sgn `  ( F `  i ) ) ) ) )
1116, 109, 110syl2anc 661 . . . 4  |-  ( F  e.  (Word  RR  \  { (/) } )  -> 
( ( T `  F ) `  (
( # `  F )  -  1 ) )  =  ( W  gsumg  ( i  e.  ( 0 ... ( ( # `  F
)  -  1 ) )  |->  (sgn `  ( F `  i )
) ) ) )
112111adantr 465 . . 3  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( T `  F ) `  (
( # `  F )  -  1 ) )  =  ( W  gsumg  ( i  e.  ( 0 ... ( ( # `  F
)  -  1 ) )  |->  (sgn `  ( F `  i )
) ) ) )
113112oveq1d 6311 . 2  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( ( T `
 F ) `  ( ( # `  F
)  -  1 ) )  .+^  (sgn `  K
) )  =  ( ( W  gsumg  ( i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) 
|->  (sgn `  ( F `  i ) ) ) )  .+^  (sgn `  K
) ) )
11496, 107, 1133eqtr4d 2508 1  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( T `  ( F ++  <" K "> ) ) `  ( # `  F ) )  =  ( ( ( T `  F
) `  ( ( # `
 F )  - 
1 ) )  .+^  (sgn `  K ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652    \ cdif 3468   (/)c0 3793   ifcif 3944   {csn 4032   {cpr 4034   {ctp 4036   <.cop 4038    |-> cmpt 4515   -->wf 5590   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512   RR*cxr 9644    - cmin 9824   -ucneg 9825   NNcn 10556   NN0cn0 10816   ZZcz 10885   ZZ>=cuz 11106   ...cfz 11697  ..^cfzo 11821   #chash 12408  Word cword 12538   ++ cconcat 12540   <"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-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-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-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  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-card 8337  df-cda 8565  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-concat 12548  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
This theorem is referenced by:  signsvtn0  28724  signstfvneq0  28726  signstfveq0  28731  signsvfn  28736
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