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Theorem signstfvn 28182
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 concat  <" 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 28167 . . . 4  |-  { -u
1 ,  0 ,  1 }  =  (
Base `  W )
41, 2signswmnd 28170 . . . . 5  |-  W  e. 
Mnd
54a1i 11 . . . 4  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  W  e.  Mnd )
6 eldifi 3626 . . . . . . . . 9  |-  ( F  e.  (Word  RR  \  { (/) } )  ->  F  e. Word  RR )
7 lencl 12527 . . . . . . . . 9  |-  ( F  e. Word  RR  ->  ( # `  F )  e.  NN0 )
86, 7syl 16 . . . . . . . 8  |-  ( F  e.  (Word  RR  \  { (/) } )  -> 
( # `  F )  e.  NN0 )
9 eldifsn 4152 . . . . . . . . 9  |-  ( F  e.  (Word  RR  \  { (/) } )  <->  ( F  e. Word  RR  /\  F  =/=  (/) ) )
10 hasheq0 12400 . . . . . . . . . . 11  |-  ( F  e. Word  RR  ->  ( (
# `  F )  =  0  <->  F  =  (/) ) )
1110necon3bid 2725 . . . . . . . . . 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 10808 . . . . . . . 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 10836 . . . . . 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 11115 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
2018, 19syl6eleq 2565 . . . 4  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( # `  F
)  -  1 )  e.  ( ZZ>= `  0
) )
21 s1cl 12576 . . . . . . . . . 10  |-  ( K  e.  RR  ->  <" K ">  e. Word  RR )
22 ccatcl 12557 . . . . . . . . . 10  |-  ( ( F  e. Word  RR  /\  <" K ">  e. Word  RR )  ->  ( F concat  <" K "> )  e. Word  RR )
236, 21, 22syl2an 477 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( F concat  <" K "> )  e. Word  RR )
2423adantr 465 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  ( F concat  <" K "> )  e. Word  RR )
25 wrdf 12518 . . . . . . . 8  |-  ( ( F concat  <" K "> )  e. Word  RR  ->  ( F concat  <" K "> ) : ( 0..^ ( # `  ( F concat  <" K "> ) ) ) --> RR )
2624, 25syl 16 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  ( F concat  <" K "> ) : ( 0..^ (
# `  ( F concat  <" K "> ) ) ) --> RR )
278adantr 465 . . . . . . . . . . . 12  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( # `  F
)  e.  NN0 )
2827nn0zd 10963 . . . . . . . . . . 11  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( # `  F
)  e.  ZZ )
29 fzoval 11797 . . . . . . . . . . 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 11813 . . . . . . . . . 10  |-  ( 0..^ ( # `  F
) )  C_  (
0 ... ( # `  F
) )
3230, 31syl6eqssr 3555 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( 0 ... (
( # `  F )  -  1 ) ) 
C_  ( 0 ... ( # `  F
) ) )
33 ccatlen 12558 . . . . . . . . . . . . 13  |-  ( ( F  e. Word  RR  /\  <" K ">  e. Word  RR )  ->  ( # `
 ( F concat  <" K "> ) )  =  ( ( # `  F
)  +  ( # `  <" K "> ) ) )
346, 21, 33syl2an 477 . . . . . . . . . . . 12  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( # `  ( F concat  <" K "> ) )  =  ( ( # `  F
)  +  ( # `  <" K "> ) ) )
35 s1len 12579 . . . . . . . . . . . . 13  |-  ( # `  <" K "> )  =  1
3635oveq2i 6294 . . . . . . . . . . . 12  |-  ( (
# `  F )  +  ( # `  <" K "> )
)  =  ( (
# `  F )  +  1 )
3734, 36syl6eq 2524 . . . . . . . . . . 11  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( # `  ( F concat  <" K "> ) )  =  ( ( # `  F
)  +  1 ) )
3837oveq2d 6299 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( 0..^ ( # `  ( F concat  <" K "> ) ) )  =  ( 0..^ ( ( # `  F
)  +  1 ) ) )
3928peano2zd 10968 . . . . . . . . . . 11  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( # `  F
)  +  1 )  e.  ZZ )
40 fzoval 11797 . . . . . . . . . . 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 10853 . . . . . . . . . . . 12  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( # `  F
)  e.  CC )
43 ax-1cn 9549 . . . . . . . . . . . . 13  |-  1  e.  CC
4443a1i 11 . . . . . . . . . . . 12  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  1  e.  CC )
4542, 44pncand 9930 . . . . . . . . . . 11  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( ( # `  F )  +  1 )  -  1 )  =  ( # `  F
) )
4645oveq2d 6299 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( 0 ... (
( ( # `  F
)  +  1 )  -  1 ) )  =  ( 0 ... ( # `  F
) ) )
4738, 41, 463eqtrd 2512 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( 0..^ ( # `  ( F concat  <" K "> ) ) )  =  ( 0 ... ( # `  F
) ) )
4832, 47sseqtr4d 3541 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( 0 ... (
( # `  F )  -  1 ) ) 
C_  ( 0..^ (
# `  ( F concat  <" K "> ) ) ) )
4948sselda 3504 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  i  e.  ( 0..^ ( # `  ( F concat  <" K "> ) ) ) )
5026, 49ffvelrnd 6021 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  ( ( F concat  <" K "> ) `  i )  e.  RR )
5150rexrd 9642 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  ( ( F concat  <" K "> ) `  i )  e.  RR* )
52 sgncl 28133 . . . . 5  |-  ( ( ( F concat  <" K "> ) `  i
)  e.  RR*  ->  (sgn
`  ( ( F concat  <" K "> ) `  i )
)  e.  { -u
1 ,  0 ,  1 } )
5351, 52syl 16 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  (sgn `  (
( F concat  <" K "> ) `  i
) )  e.  { -u 1 ,  0 ,  1 } )
541, 2signswplusg 28168 . . . 4  |-  .+^  =  ( +g  `  W )
55 simpr 461 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  K  e.  RR )
5655rexrd 9642 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  K  e.  RR* )
57 sgncl 28133 . . . . 5  |-  ( K  e.  RR*  ->  (sgn `  K )  e.  { -u 1 ,  0 ,  1 } )
5856, 57syl 16 . . . 4  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  (sgn `  K )  e.  { -u 1 ,  0 ,  1 } )
59 simpr 461 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  =  ( ( ( # `  F
)  -  1 )  +  1 ) )  ->  i  =  ( ( ( # `  F
)  -  1 )  +  1 ) )
6042, 44npcand 9933 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( ( # `  F )  -  1 )  +  1 )  =  ( # `  F
) )
6160adantr 465 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  =  ( ( ( # `  F
)  -  1 )  +  1 ) )  ->  ( ( (
# `  F )  -  1 )  +  1 )  =  (
# `  F )
)
6259, 61eqtrd 2508 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  =  ( ( ( # `  F
)  -  1 )  +  1 ) )  ->  i  =  (
# `  F )
)
6362fveq2d 5869 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  =  ( ( ( # `  F
)  -  1 )  +  1 ) )  ->  ( ( F concat  <" K "> ) `  i )  =  ( ( F concat  <" K "> ) `  ( # `  F
) ) )
646adantr 465 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  F  e. Word  RR )
6555, 21syl 16 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  <" K ">  e. Word  RR )
66 c0ex 9589 . . . . . . . . . . . . 13  |-  0  e.  _V
6766snid 4055 . . . . . . . . . . . 12  |-  0  e.  { 0 }
68 fzo01 11864 . . . . . . . . . . . 12  |-  ( 0..^ 1 )  =  {
0 }
6967, 68eleqtrri 2554 . . . . . . . . . . 11  |-  0  e.  ( 0..^ 1 )
7035oveq2i 6294 . . . . . . . . . . 11  |-  ( 0..^ ( # `  <" K "> )
)  =  ( 0..^ 1 )
7169, 70eleqtrri 2554 . . . . . . . . . 10  |-  0  e.  ( 0..^ ( # `  <" K "> ) )
7271a1i 11 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  0  e.  ( 0..^ ( # `  <" K "> )
) )
73 ccatval3 12561 . . . . . . . . 9  |-  ( ( F  e. Word  RR  /\  <" K ">  e. Word  RR  /\  0  e.  ( 0..^ ( # `  <" K "> ) ) )  -> 
( ( F concat  <" K "> ) `  (
0  +  ( # `  F ) ) )  =  ( <" K "> `  0 )
)
7464, 65, 72, 73syl3anc 1228 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( F concat  <" K "> ) `  (
0  +  ( # `  F ) ) )  =  ( <" K "> `  0 )
)
7542addid2d 9779 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( 0  +  (
# `  F )
)  =  ( # `  F ) )
7675fveq2d 5869 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( F concat  <" K "> ) `  (
0  +  ( # `  F ) ) )  =  ( ( F concat  <" K "> ) `  ( # `  F
) ) )
77 s1fv 12581 . . . . . . . . 9  |-  ( K  e.  RR  ->  ( <" K "> `  0 )  =  K )
7855, 77syl 16 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( <" K "> `  0 )  =  K )
7974, 76, 783eqtr3d 2516 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( F concat  <" K "> ) `  ( # `
 F ) )  =  K )
8079adantr 465 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  =  ( ( ( # `  F
)  -  1 )  +  1 ) )  ->  ( ( F concat  <" K "> ) `  ( # `  F
) )  =  K )
8163, 80eqtrd 2508 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  =  ( ( ( # `  F
)  -  1 )  +  1 ) )  ->  ( ( F concat  <" K "> ) `  i )  =  K )
8281fveq2d 5869 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  =  ( ( ( # `  F
)  -  1 )  +  1 ) )  ->  (sgn `  (
( F concat  <" K "> ) `  i
) )  =  (sgn
`  K ) )
833, 5, 20, 53, 54, 58, 82gsumnunsn 28149 . . 3  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( W  gsumg  ( i  e.  ( 0 ... ( ( ( # `  F
)  -  1 )  +  1 ) ) 
|->  (sgn `  ( ( F concat  <" K "> ) `  i ) ) ) )  =  ( ( W  gsumg  ( i  e.  ( 0 ... ( ( # `  F
)  -  1 ) )  |->  (sgn `  (
( F concat  <" K "> ) `  i
) ) ) ) 
.+^  (sgn `  K )
) )
8460oveq2d 6299 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( 0 ... (
( ( # `  F
)  -  1 )  +  1 ) )  =  ( 0 ... ( # `  F
) ) )
8584mpteq1d 4528 . . . 4  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( i  e.  ( 0 ... ( ( ( # `  F
)  -  1 )  +  1 ) ) 
|->  (sgn `  ( ( F concat  <" K "> ) `  i ) ) )  =  ( i  e.  ( 0 ... ( # `  F
) )  |->  (sgn `  ( ( F concat  <" K "> ) `  i
) ) ) )
8685oveq2d 6299 . . 3  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( W  gsumg  ( i  e.  ( 0 ... ( ( ( # `  F
)  -  1 )  +  1 ) ) 
|->  (sgn `  ( ( F concat  <" K "> ) `  i ) ) ) )  =  ( W  gsumg  ( i  e.  ( 0 ... ( # `  F ) )  |->  (sgn
`  ( ( F concat  <" K "> ) `  i )
) ) ) )
8764adantr 465 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  F  e. Word  RR )
8865adantr 465 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  <" K ">  e. Word  RR )
8930eleq2d 2537 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( i  e.  ( 0..^ ( # `  F
) )  <->  i  e.  ( 0 ... (
( # `  F )  -  1 ) ) ) )
9089biimpar 485 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  i  e.  ( 0..^ ( # `  F
) ) )
91 ccatval1 12559 . . . . . . . 8  |-  ( ( F  e. Word  RR  /\  <" K ">  e. Word  RR  /\  i  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F concat  <" K "> ) `  i )  =  ( F `  i ) )
9287, 88, 90, 91syl3anc 1228 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  ( ( F concat  <" K "> ) `  i )  =  ( F `  i ) )
9392fveq2d 5869 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  (sgn `  (
( F concat  <" K "> ) `  i
) )  =  (sgn
`  ( F `  i ) ) )
9493mpteq2dva 4533 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) 
|->  (sgn `  ( ( F concat  <" K "> ) `  i ) ) )  =  ( i  e.  ( 0 ... ( ( # `  F )  -  1 ) )  |->  (sgn `  ( F `  i ) ) ) )
9594oveq2d 6299 . . . 4  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( W  gsumg  ( i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) 
|->  (sgn `  ( ( F concat  <" K "> ) `  i ) ) ) )  =  ( W  gsumg  ( i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) 
|->  (sgn `  ( F `  i ) ) ) ) )
9695oveq1d 6298 . . 3  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( W  gsumg  ( i  e.  ( 0 ... ( ( # `  F
)  -  1 ) )  |->  (sgn `  (
( F concat  <" K "> ) `  i
) ) ) ) 
.+^  (sgn `  K )
)  =  ( ( W  gsumg  ( i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) 
|->  (sgn `  ( F `  i ) ) ) )  .+^  (sgn `  K
) ) )
9783, 86, 963eqtr3d 2516 . 2  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( W  gsumg  ( i  e.  ( 0 ... ( # `  F ) )  |->  (sgn
`  ( ( F concat  <" K "> ) `  i )
) ) )  =  ( ( W  gsumg  ( i  e.  ( 0 ... ( ( # `  F
)  -  1 ) )  |->  (sgn `  ( F `  i )
) ) )  .+^  (sgn `  K ) ) )
98 eqidd 2468 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( # `  F
)  =  ( # `  F ) )
9998olcd 393 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( # `  F
)  e.  ( 0..^ ( # `  F
) )  \/  ( # `
 F )  =  ( # `  F
) ) )
10027, 19syl6eleq 2565 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( # `  F
)  e.  ( ZZ>= ` 
0 ) )
101 fzosplitsni 11887 . . . . . 6  |-  ( (
# `  F )  e.  ( ZZ>= `  0 )  ->  ( ( # `  F
)  e.  ( 0..^ ( ( # `  F
)  +  1 ) )  <->  ( ( # `  F )  e.  ( 0..^ ( # `  F
) )  \/  ( # `
 F )  =  ( # `  F
) ) ) )
102100, 101syl 16 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( # `  F
)  e.  ( 0..^ ( ( # `  F
)  +  1 ) )  <->  ( ( # `  F )  e.  ( 0..^ ( # `  F
) )  \/  ( # `
 F )  =  ( # `  F
) ) ) )
10399, 102mpbird 232 . . . 4  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( # `  F
)  e.  ( 0..^ ( ( # `  F
)  +  1 ) ) )
104103, 38eleqtrrd 2558 . . 3  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( # `  F
)  e.  ( 0..^ ( # `  ( F concat  <" K "> ) ) ) )
105 signsv.t . . . 4  |-  T  =  ( f  e. Word  RR  |->  ( n  e.  (
0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) ) )
106 signsv.v . . . 4  |-  V  =  ( f  e. Word  RR  |->  sum_ j  e.  ( 1..^ ( # `  f
) ) if ( ( ( T `  f ) `  j
)  =/=  ( ( T `  f ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
1071, 2, 105, 106signstfval 28177 . . 3  |-  ( ( ( F concat  <" K "> )  e. Word  RR  /\  ( # `  F
)  e.  ( 0..^ ( # `  ( F concat  <" K "> ) ) ) )  ->  ( ( T `
 ( F concat  <" K "> ) ) `  ( # `  F ) )  =  ( W 
gsumg  ( i  e.  ( 0 ... ( # `  F ) )  |->  (sgn
`  ( ( F concat  <" K "> ) `  i )
) ) ) )
10823, 104, 107syl2anc 661 . 2  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( T `  ( F concat  <" K "> ) ) `  ( # `  F ) )  =  ( W 
gsumg  ( i  e.  ( 0 ... ( # `  F ) )  |->  (sgn
`  ( ( F concat  <" K "> ) `  i )
) ) ) )
109 fzo0end 11871 . . . . . 6  |-  ( (
# `  F )  e.  NN  ->  ( ( # `
 F )  - 
1 )  e.  ( 0..^ ( # `  F
) ) )
11015, 109syl 16 . . . . 5  |-  ( F  e.  (Word  RR  \  { (/) } )  -> 
( ( # `  F
)  -  1 )  e.  ( 0..^ (
# `  F )
) )
1111, 2, 105, 106signstfval 28177 . . . . 5  |-  ( ( F  e. Word  RR  /\  ( ( # `  F
)  -  1 )  e.  ( 0..^ (
# `  F )
) )  ->  (
( T `  F
) `  ( ( # `
 F )  - 
1 ) )  =  ( W  gsumg  ( i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) 
|->  (sgn `  ( F `  i ) ) ) ) )
1126, 110, 111syl2anc 661 . . . 4  |-  ( F  e.  (Word  RR  \  { (/) } )  -> 
( ( T `  F ) `  (
( # `  F )  -  1 ) )  =  ( W  gsumg  ( i  e.  ( 0 ... ( ( # `  F
)  -  1 ) )  |->  (sgn `  ( F `  i )
) ) ) )
113112adantr 465 . . 3  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( T `  F ) `  (
( # `  F )  -  1 ) )  =  ( W  gsumg  ( i  e.  ( 0 ... ( ( # `  F
)  -  1 ) )  |->  (sgn `  ( F `  i )
) ) ) )
114113oveq1d 6298 . 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
) ) )
11597, 108, 1143eqtr4d 2518 1  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( T `  ( F concat  <" 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 1379    e. wcel 1767    =/= wne 2662    \ cdif 3473   (/)c0 3785   ifcif 3939   {csn 4027   {cpr 4029   {ctp 4031   <.cop 4033    |-> cmpt 4505   -->wf 5583   ` cfv 5587  (class class class)co 6283    |-> cmpt2 6285   CCcc 9489   RRcr 9490   0cc0 9491   1c1 9492    + caddc 9494   RR*cxr 9626    - cmin 9804   -ucneg 9805   NNcn 10535   NN0cn0 10794   ZZcz 10863   ZZ>=cuz 11081   ...cfz 11671  ..^cfzo 11791   #chash 12372  Word cword 12499   concat cconcat 12501   <"cs1 12502  sgncsgn 12881   sum_csu 13470   ndxcnx 14486   Basecbs 14489   +g cplusg 14554    gsumg cgsu 14695   Mndcmnd 15725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8319  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-nn 10536  df-2 10593  df-n0 10795  df-z 10864  df-uz 11082  df-fz 11672  df-fzo 11792  df-seq 12075  df-hash 12373  df-word 12507  df-concat 12509  df-s1 12510  df-sgn 12882  df-struct 14491  df-ndx 14492  df-slot 14493  df-base 14494  df-plusg 14567  df-0g 14696  df-gsum 14697  df-mnd 15731
This theorem is referenced by:  signsvtn0  28183  signstfvneq0  28185  signstfveq0  28190  signsvfn  28195
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