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Theorem signstfvn 26970
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 26955 . . . 4  |-  { -u
1 ,  0 ,  1 }  =  (
Base `  W )
41, 2signswmnd 26958 . . . . 5  |-  W  e. 
Mnd
54a1i 11 . . . 4  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  W  e.  Mnd )
6 eldifi 3478 . . . . . . . . 9  |-  ( F  e.  (Word  RR  \  { (/) } )  ->  F  e. Word  RR )
7 lencl 12249 . . . . . . . . 9  |-  ( F  e. Word  RR  ->  ( # `  F )  e.  NN0 )
86, 7syl 16 . . . . . . . 8  |-  ( F  e.  (Word  RR  \  { (/) } )  -> 
( # `  F )  e.  NN0 )
9 eldifsn 4000 . . . . . . . . 9  |-  ( F  e.  (Word  RR  \  { (/) } )  <->  ( F  e. Word  RR  /\  F  =/=  (/) ) )
10 hasheq0 12131 . . . . . . . . . . 11  |-  ( F  e. Word  RR  ->  ( (
# `  F )  =  0  <->  F  =  (/) ) )
1110necon3bid 2643 . . . . . . . . . 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 10593 . . . . . . . 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 10621 . . . . . 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 10895 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
2018, 19syl6eleq 2533 . . . 4  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( # `  F
)  -  1 )  e.  ( ZZ>= `  0
) )
21 s1cl 12293 . . . . . . . . . 10  |-  ( K  e.  RR  ->  <" K ">  e. Word  RR )
22 ccatcl 12274 . . . . . . . . . 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 12240 . . . . . . . 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 10745 . . . . . . . . . . 11  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( # `  F
)  e.  ZZ )
29 fzoval 11554 . . . . . . . . . . 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 11570 . . . . . . . . . 10  |-  ( 0..^ ( # `  F
) )  C_  (
0 ... ( # `  F
) )
3230, 31syl6eqssr 3407 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( 0 ... (
( # `  F )  -  1 ) ) 
C_  ( 0 ... ( # `  F
) ) )
33 ccatlen 12275 . . . . . . . . . . . . 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 12296 . . . . . . . . . . . . 13  |-  ( # `  <" K "> )  =  1
3635oveq2i 6102 . . . . . . . . . . . 12  |-  ( (
# `  F )  +  ( # `  <" K "> )
)  =  ( (
# `  F )  +  1 )
3734, 36syl6eq 2491 . . . . . . . . . . 11  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( # `  ( F concat  <" K "> ) )  =  ( ( # `  F
)  +  1 ) )
3837oveq2d 6107 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( 0..^ ( # `  ( F concat  <" K "> ) ) )  =  ( 0..^ ( ( # `  F
)  +  1 ) ) )
3928peano2zd 10750 . . . . . . . . . . 11  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( # `  F
)  +  1 )  e.  ZZ )
40 fzoval 11554 . . . . . . . . . . 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 10638 . . . . . . . . . . . 12  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( # `  F
)  e.  CC )
43 ax-1cn 9340 . . . . . . . . . . . . 13  |-  1  e.  CC
4443a1i 11 . . . . . . . . . . . 12  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  1  e.  CC )
4542, 44pncand 9720 . . . . . . . . . . 11  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( ( # `  F )  +  1 )  -  1 )  =  ( # `  F
) )
4645oveq2d 6107 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( 0 ... (
( ( # `  F
)  +  1 )  -  1 ) )  =  ( 0 ... ( # `  F
) ) )
4738, 41, 463eqtrd 2479 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( 0..^ ( # `  ( F concat  <" K "> ) ) )  =  ( 0 ... ( # `  F
) ) )
4832, 47sseqtr4d 3393 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( 0 ... (
( # `  F )  -  1 ) ) 
C_  ( 0..^ (
# `  ( F concat  <" K "> ) ) ) )
4948sselda 3356 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  i  e.  ( 0..^ ( # `  ( F concat  <" K "> ) ) ) )
5026, 49ffvelrnd 5844 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  ( ( F concat  <" K "> ) `  i )  e.  RR )
5150rexrd 9433 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  ( ( F concat  <" K "> ) `  i )  e.  RR* )
52 sgncl 26921 . . . . 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 26956 . . . 4  |-  .+^  =  ( +g  `  W )
55 simpr 461 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  K  e.  RR )
5655rexrd 9433 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  K  e.  RR* )
57 sgncl 26921 . . . . 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 9723 . . . . . . . . 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 2475 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  =  ( ( ( # `  F
)  -  1 )  +  1 ) )  ->  i  =  (
# `  F )
)
6362fveq2d 5695 . . . . . 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 9380 . . . . . . . . . . . . 13  |-  0  e.  _V
6766snid 3905 . . . . . . . . . . . 12  |-  0  e.  { 0 }
68 fzo01 11612 . . . . . . . . . . . 12  |-  ( 0..^ 1 )  =  {
0 }
6967, 68eleqtrri 2516 . . . . . . . . . . 11  |-  0  e.  ( 0..^ 1 )
7035oveq2i 6102 . . . . . . . . . . 11  |-  ( 0..^ ( # `  <" K "> )
)  =  ( 0..^ 1 )
7169, 70eleqtrri 2516 . . . . . . . . . 10  |-  0  e.  ( 0..^ ( # `  <" K "> ) )
7271a1i 11 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  0  e.  ( 0..^ ( # `  <" K "> )
) )
73 ccatval3 12278 . . . . . . . . 9  |-  ( ( F  e. Word  RR  /\  <" K ">  e. Word  RR  /\  0  e.  ( 0..^ ( # `  <" K "> ) ) )  -> 
( ( F concat  <" K "> ) `  (
0  +  ( # `  F ) ) )  =  ( <" K "> `  0 )
)
7464, 65, 72, 73syl3anc 1218 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( F concat  <" K "> ) `  (
0  +  ( # `  F ) ) )  =  ( <" K "> `  0 )
)
7542addid2d 9570 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( 0  +  (
# `  F )
)  =  ( # `  F ) )
7675fveq2d 5695 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( ( F concat  <" K "> ) `  (
0  +  ( # `  F ) ) )  =  ( ( F concat  <" K "> ) `  ( # `  F
) ) )
77 s1fv 12298 . . . . . . . . 9  |-  ( K  e.  RR  ->  ( <" K "> `  0 )  =  K )
7855, 77syl 16 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( <" K "> `  0 )  =  K )
7974, 76, 783eqtr3d 2483 . . . . . . 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 2475 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  =  ( ( ( # `  F
)  -  1 )  +  1 ) )  ->  ( ( F concat  <" K "> ) `  i )  =  K )
8281fveq2d 5695 . . . 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 26937 . . 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 6107 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( 0 ... (
( ( # `  F
)  -  1 )  +  1 ) )  =  ( 0 ... ( # `  F
) ) )
8584mpteq1d 4373 . . . 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 6107 . . 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 2510 . . . . . . . . 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 12276 . . . . . . . 8  |-  ( ( F  e. Word  RR  /\  <" K ">  e. Word  RR  /\  i  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F concat  <" K "> ) `  i )  =  ( F `  i ) )
9287, 88, 90, 91syl3anc 1218 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  ( ( F concat  <" K "> ) `  i )  =  ( F `  i ) )
9392fveq2d 5695 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  K  e.  RR )  /\  i  e.  ( 0 ... ( (
# `  F )  -  1 ) ) )  ->  (sgn `  (
( F concat  <" K "> ) `  i
) )  =  (sgn
`  ( F `  i ) ) )
9493mpteq2dva 4378 . . . . 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 6107 . . . 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 6106 . . 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 2483 . 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 2444 . . . . . 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 2533 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  K  e.  RR )  ->  ( # `  F
)  e.  ( ZZ>= ` 
0 ) )
101 fzosplitsni 11634 . . . . . 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 2520 . . 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 26965 . . 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 11619 . . . . . 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 26965 . . . . 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 6106 . 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 2485 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 1369    e. wcel 1756    =/= wne 2606    \ cdif 3325   (/)c0 3637   ifcif 3791   {csn 3877   {cpr 3879   {ctp 3881   <.cop 3883    e. cmpt 4350   -->wf 5414   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285   RR*cxr 9417    - cmin 9595   -ucneg 9596   NNcn 10322   NN0cn0 10579   ZZcz 10646   ZZ>=cuz 10861   ...cfz 11437  ..^cfzo 11548   #chash 12103  Word cword 12221   concat cconcat 12223   <"cs1 12224  sgncsgn 12575   sum_csu 13163   ndxcnx 14171   Basecbs 14174   +g cplusg 14238    gsumg cgsu 14379   Mndcmnd 15409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-seq 11807  df-hash 12104  df-word 12229  df-concat 12231  df-s1 12232  df-sgn 12576  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-plusg 14251  df-0g 14380  df-gsum 14381  df-mnd 15415
This theorem is referenced by:  signsvtn0  26971  signstfvneq0  26973  signstfveq0  26978  signsvfn  26983
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