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Theorem signsvtn0 28343
Description: If the last letter is non zero, then this is the zero-skipping sign. (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 ) )
signsvtn0.1  |-  N  =  ( # `  F
)
Assertion
Ref Expression
signsvtn0  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( ( T `  F ) `  ( N  -  1 ) )  =  (sgn `  ( F `  ( N  -  1 ) ) ) )
Distinct variable groups:    a, b,  .+^    f, i, n, F    f, W, i, n    F, a, b, f, i, n    N, a    f, b, i, n, N    T, a,
b
Allowed substitution hints:    .+^ ( f, i,
j, n)    T( f,
i, j, n)    F( j)    N( j)    V( f, i, j, n, a, b)    W( j, a, b)

Proof of Theorem signsvtn0
StepHypRef Expression
1 eldifsn 4158 . . . . . . . . . . . 12  |-  ( F  e.  (Word  RR  \  { (/) } )  <->  ( F  e. Word  RR  /\  F  =/=  (/) ) )
21biimpi 194 . . . . . . . . . . 11  |-  ( F  e.  (Word  RR  \  { (/) } )  -> 
( F  e. Word  RR  /\  F  =/=  (/) ) )
32adantr 465 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F  e. Word  RR  /\  F  =/=  (/) ) )
43simpld 459 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  F  e. Word  RR )
54adantr 465 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  F  e. Word  RR )
6 wrdf 12534 . . . . . . . 8  |-  ( F  e. Word  RR  ->  F :
( 0..^ ( # `  F ) ) --> RR )
75, 6syl 16 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  F : ( 0..^ (
# `  F )
) --> RR )
8 lennncl 12544 . . . . . . . . . 10  |-  ( ( F  e. Word  RR  /\  F  =/=  (/) )  ->  ( # `
 F )  e.  NN )
93, 8syl 16 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( # `  F )  e.  NN )
109adantr 465 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  ( # `
 F )  e.  NN )
11 lbfzo0 11842 . . . . . . . 8  |-  ( 0  e.  ( 0..^ (
# `  F )
)  <->  ( # `  F
)  e.  NN )
1210, 11sylibr 212 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  0  e.  ( 0..^ ( # `  F ) ) )
137, 12ffvelrnd 6033 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  ( F `  0 )  e.  RR )
14 signsv.p . . . . . . 7  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
15 signsv.w . . . . . . 7  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
16 signsv.t . . . . . . 7  |-  T  =  ( f  e. Word  RR  |->  ( n  e.  (
0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) ) )
17 signsv.v . . . . . . 7  |-  V  =  ( f  e. Word  RR  |->  sum_ j  e.  ( 1..^ ( # `  f
) ) if ( ( ( T `  f ) `  j
)  =/=  ( ( T `  f ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
1814, 15, 16, 17signstf0 28341 . . . . . 6  |-  ( ( F `  0 )  e.  RR  ->  ( T `  <" ( F `  0 ) "> )  =  <" (sgn `  ( F `  0 ) ) "> )
1913, 18syl 16 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  ( T `  <" ( F `  0 ) "> )  =  <" (sgn `  ( F `  0 ) ) "> )
20 signsvtn0.1 . . . . . . . 8  |-  N  =  ( # `  F
)
21 simpr 461 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  N  =  1 )
2220, 21syl5eqr 2522 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  ( # `
 F )  =  1 )
23 eqs1 12601 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  ( # `  F )  =  1 )  ->  F  =  <" ( F `  0 ) "> )
245, 22, 23syl2anc 661 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  F  =  <" ( F `
 0 ) "> )
2524fveq2d 5876 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  ( T `  F )  =  ( T `  <" ( F ` 
0 ) "> ) )
26 oveq1 6302 . . . . . . . . . 10  |-  ( N  =  1  ->  ( N  -  1 )  =  ( 1  -  1 ) )
27 1m1e0 10616 . . . . . . . . . 10  |-  ( 1  -  1 )  =  0
2826, 27syl6eq 2524 . . . . . . . . 9  |-  ( N  =  1  ->  ( N  -  1 )  =  0 )
2928fveq2d 5876 . . . . . . . 8  |-  ( N  =  1  ->  ( F `  ( N  -  1 ) )  =  ( F ` 
0 ) )
3029fveq2d 5876 . . . . . . 7  |-  ( N  =  1  ->  (sgn `  ( F `  ( N  -  1 ) ) )  =  (sgn
`  ( F ` 
0 ) ) )
3130s1eqd 12593 . . . . . 6  |-  ( N  =  1  ->  <" (sgn `  ( F `  ( N  -  1 ) ) ) ">  =  <" (sgn `  ( F `  0 ) ) "> )
3221, 31syl 16 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  <" (sgn `  ( F `  ( N  -  1 ) ) ) ">  =  <" (sgn `  ( F `  0 ) ) "> )
3319, 25, 323eqtr4d 2518 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  ( T `  F )  =  <" (sgn `  ( F `  ( N  -  1 ) ) ) "> )
3421, 28syl 16 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  ( N  -  1 )  =  0 )
3533, 34fveq12d 5878 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  (
( T `  F
) `  ( N  -  1 ) )  =  ( <" (sgn `  ( F `  ( N  -  1 ) ) ) "> `  0 ) )
364, 6syl 16 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  F : ( 0..^ (
# `  F )
) --> RR )
3720oveq1i 6305 . . . . . . . . 9  |-  ( N  -  1 )  =  ( ( # `  F
)  -  1 )
38 fzo0end 11884 . . . . . . . . . 10  |-  ( (
# `  F )  e.  NN  ->  ( ( # `
 F )  - 
1 )  e.  ( 0..^ ( # `  F
) ) )
393, 8, 383syl 20 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( ( # `  F
)  -  1 )  e.  ( 0..^ (
# `  F )
) )
4037, 39syl5eqel 2559 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( N  -  1 )  e.  ( 0..^ ( # `  F
) ) )
4136, 40ffvelrnd 6033 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F `  ( N  -  1 ) )  e.  RR )
4241rexrd 9655 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F `  ( N  -  1 ) )  e.  RR* )
43 sgncl 28293 . . . . . 6  |-  ( ( F `  ( N  -  1 ) )  e.  RR*  ->  (sgn `  ( F `  ( N  -  1 ) ) )  e.  { -u
1 ,  0 ,  1 } )
4442, 43syl 16 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
(sgn `  ( F `  ( N  -  1 ) ) )  e. 
{ -u 1 ,  0 ,  1 } )
4544adantr 465 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  (sgn `  ( F `  ( N  -  1 ) ) )  e.  { -u 1 ,  0 ,  1 } )
46 s1fv 12599 . . . 4  |-  ( (sgn
`  ( F `  ( N  -  1
) ) )  e. 
{ -u 1 ,  0 ,  1 }  ->  (
<" (sgn `  ( F `  ( N  -  1 ) ) ) "> `  0
)  =  (sgn `  ( F `  ( N  -  1 ) ) ) )
4745, 46syl 16 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  ( <" (sgn `  ( F `  ( N  -  1 ) ) ) "> `  0
)  =  (sgn `  ( F `  ( N  -  1 ) ) ) )
4835, 47eqtrd 2508 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  (
( T `  F
) `  ( N  -  1 ) )  =  (sgn `  ( F `  ( N  -  1 ) ) ) )
49 fzossfz 11826 . . . . . . . . . 10  |-  ( 0..^ ( # `  F
) )  C_  (
0 ... ( # `  F
) )
5049, 39sseldi 3507 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( ( # `  F
)  -  1 )  e.  ( 0 ... ( # `  F
) ) )
51 swrd0val 12628 . . . . . . . . 9  |-  ( ( F  e. Word  RR  /\  ( ( # `  F
)  -  1 )  e.  ( 0 ... ( # `  F
) ) )  -> 
( F substr  <. 0 ,  ( ( # `  F
)  -  1 )
>. )  =  ( F  |`  ( 0..^ ( ( # `  F
)  -  1 ) ) ) )
524, 50, 51syl2anc 661 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F substr  <. 0 ,  ( ( # `  F
)  -  1 )
>. )  =  ( F  |`  ( 0..^ ( ( # `  F
)  -  1 ) ) ) )
5352oveq1d 6310 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( ( F substr  <. 0 ,  ( ( # `  F )  -  1 ) >. ) concat  <" ( F `  ( ( # `
 F )  - 
1 ) ) "> )  =  ( ( F  |`  (
0..^ ( ( # `  F )  -  1 ) ) ) concat  <" ( F `  (
( # `  F )  -  1 ) ) "> ) )
54 wrdeqcats1 12679 . . . . . . . 8  |-  ( ( F  e. Word  RR  /\  F  =/=  (/) )  ->  F  =  ( ( F substr  <. 0 ,  ( (
# `  F )  -  1 ) >.
) concat  <" ( F `
 ( ( # `  F )  -  1 ) ) "> ) )
553, 54syl 16 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  F  =  ( ( F substr  <. 0 ,  ( ( # `  F
)  -  1 )
>. ) concat  <" ( F `  ( ( # `
 F )  - 
1 ) ) "> ) )
5637oveq2i 6306 . . . . . . . . . 10  |-  ( 0..^ ( N  -  1 ) )  =  ( 0..^ ( ( # `  F )  -  1 ) )
5756a1i 11 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( 0..^ ( N  -  1 ) )  =  ( 0..^ ( ( # `  F
)  -  1 ) ) )
5857reseq2d 5279 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F  |`  (
0..^ ( N  - 
1 ) ) )  =  ( F  |`  ( 0..^ ( ( # `  F )  -  1 ) ) ) )
5937a1i 11 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( N  -  1 )  =  ( (
# `  F )  -  1 ) )
6059fveq2d 5876 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F `  ( N  -  1 ) )  =  ( F `
 ( ( # `  F )  -  1 ) ) )
6160s1eqd 12593 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  <" ( F `  ( N  -  1
) ) ">  =  <" ( F `
 ( ( # `  F )  -  1 ) ) "> )
6258, 61oveq12d 6313 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( ( F  |`  ( 0..^ ( N  - 
1 ) ) ) concat  <" ( F `  ( N  -  1
) ) "> )  =  ( ( F  |`  ( 0..^ ( ( # `  F
)  -  1 ) ) ) concat  <" ( F `  ( ( # `
 F )  - 
1 ) ) "> ) )
6353, 55, 623eqtr4d 2518 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  F  =  ( ( F  |`  ( 0..^ ( N  -  1 ) ) ) concat  <" ( F `  ( N  -  1 ) ) "> ) )
6463fveq2d 5876 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( T `  F
)  =  ( T `
 ( ( F  |`  ( 0..^ ( N  -  1 ) ) ) concat  <" ( F `
 ( N  - 
1 ) ) "> ) ) )
65 ffn 5737 . . . . . . . . . . 11  |-  ( F : ( 0..^ (
# `  F )
) --> RR  ->  F  Fn  ( 0..^ ( # `  F ) ) )
664, 6, 653syl 20 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  F  Fn  ( 0..^ ( # `  F
) ) )
6720a1i 11 . . . . . . . . . . . 12  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  N  =  ( # `  F
) )
6867oveq2d 6311 . . . . . . . . . . 11  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( 0..^ N )  =  ( 0..^ (
# `  F )
) )
6968fneq2d 5678 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F  Fn  (
0..^ N )  <->  F  Fn  ( 0..^ ( # `  F
) ) ) )
7066, 69mpbird 232 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  F  Fn  ( 0..^ N ) )
7120, 9syl5eqel 2559 . . . . . . . . . . 11  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  N  e.  NN )
7271nnnn0d 10864 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  N  e.  NN0 )
73 nn0z 10899 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  N  e.  ZZ )
74 fzossrbm1 11834 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
0..^ ( N  - 
1 ) )  C_  ( 0..^ N ) )
7572, 73, 743syl 20 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( 0..^ ( N  -  1 ) ) 
C_  ( 0..^ N ) )
76 fnssres 5700 . . . . . . . . 9  |-  ( ( F  Fn  ( 0..^ N )  /\  (
0..^ ( N  - 
1 ) )  C_  ( 0..^ N ) )  ->  ( F  |`  ( 0..^ ( N  - 
1 ) ) )  Fn  ( 0..^ ( N  -  1 ) ) )
7770, 75, 76syl2anc 661 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F  |`  (
0..^ ( N  - 
1 ) ) )  Fn  ( 0..^ ( N  -  1 ) ) )
78 hashfn 12423 . . . . . . . 8  |-  ( ( F  |`  ( 0..^ ( N  -  1 ) ) )  Fn  ( 0..^ ( N  -  1 ) )  ->  ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  =  ( # `  (
0..^ ( N  - 
1 ) ) ) )
7977, 78syl 16 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  =  (
# `  ( 0..^ ( N  -  1
) ) ) )
80 nnm1nn0 10849 . . . . . . . 8  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
81 hashfzo0 12468 . . . . . . . 8  |-  ( ( N  -  1 )  e.  NN0  ->  ( # `  ( 0..^ ( N  -  1 ) ) )  =  ( N  -  1 ) )
8271, 80, 813syl 20 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( # `  ( 0..^ ( N  -  1 ) ) )  =  ( N  -  1 ) )
8379, 82eqtrd 2508 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  =  ( N  -  1 ) )
8483eqcomd 2475 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( N  -  1 )  =  ( # `  ( F  |`  (
0..^ ( N  - 
1 ) ) ) ) )
8564, 84fveq12d 5878 . . . 4  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( ( T `  F ) `  ( N  -  1 ) )  =  ( ( T `  ( ( F  |`  ( 0..^ ( N  -  1 ) ) ) concat  <" ( F `  ( N  -  1 ) ) "> )
) `  ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) ) ) )
8685adantr 465 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  (
( T `  F
) `  ( N  -  1 ) )  =  ( ( T `
 ( ( F  |`  ( 0..^ ( N  -  1 ) ) ) concat  <" ( F `
 ( N  - 
1 ) ) "> ) ) `  ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) ) ) )
8752, 58eqtr4d 2511 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F substr  <. 0 ,  ( ( # `  F
)  -  1 )
>. )  =  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )
88 swrdcl 12626 . . . . . . . . . 10  |-  ( F  e. Word  RR  ->  ( F substr  <. 0 ,  ( (
# `  F )  -  1 ) >.
)  e. Word  RR )
894, 88syl 16 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F substr  <. 0 ,  ( ( # `  F
)  -  1 )
>. )  e. Word  RR )
9087, 89eqeltrrd 2556 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F  |`  (
0..^ ( N  - 
1 ) ) )  e. Word  RR )
9190adantr 465 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( F  |`  ( 0..^ ( N  -  1 ) ) )  e. Word  RR )
9283adantr 465 . . . . . . . . 9  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( # `
 ( F  |`  ( 0..^ ( N  - 
1 ) ) ) )  =  ( N  -  1 ) )
9371adantr 465 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  N  e.  NN )
9493nncnd 10564 . . . . . . . . . 10  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  N  e.  CC )
95 1cnd 9624 . . . . . . . . . 10  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  1  e.  CC )
96 simpr 461 . . . . . . . . . 10  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  N  =/=  1 )
9794, 95, 96subne0d 9951 . . . . . . . . 9  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( N  -  1 )  =/=  0 )
9892, 97eqnetrd 2760 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( # `
 ( F  |`  ( 0..^ ( N  - 
1 ) ) ) )  =/=  0 )
99 fveq2 5872 . . . . . . . . . 10  |-  ( ( F  |`  ( 0..^ ( N  -  1 ) ) )  =  (/)  ->  ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  =  ( # `  (/) ) )
100 hash0 12417 . . . . . . . . . 10  |-  ( # `  (/) )  =  0
10199, 100syl6eq 2524 . . . . . . . . 9  |-  ( ( F  |`  ( 0..^ ( N  -  1 ) ) )  =  (/)  ->  ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  =  0 )
102101necon3i 2707 . . . . . . . 8  |-  ( (
# `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  =/=  0  ->  ( F  |`  (
0..^ ( N  - 
1 ) ) )  =/=  (/) )
10398, 102syl 16 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( F  |`  ( 0..^ ( N  -  1 ) ) )  =/=  (/) )
10491, 103jca 532 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  (
( F  |`  (
0..^ ( N  - 
1 ) ) )  e. Word  RR  /\  ( F  |`  ( 0..^ ( N  -  1 ) ) )  =/=  (/) ) )
105 eldifsn 4158 . . . . . 6  |-  ( ( F  |`  ( 0..^ ( N  -  1 ) ) )  e.  (Word  RR  \  { (/)
} )  <->  ( ( F  |`  ( 0..^ ( N  -  1 ) ) )  e. Word  RR  /\  ( F  |`  (
0..^ ( N  - 
1 ) ) )  =/=  (/) ) )
106104, 105sylibr 212 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( F  |`  ( 0..^ ( N  -  1 ) ) )  e.  (Word 
RR  \  { (/) } ) )
10741adantr 465 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( F `  ( N  -  1 ) )  e.  RR )
10814, 15, 16, 17signstfvn 28342 . . . . 5  |-  ( ( ( F  |`  (
0..^ ( N  - 
1 ) ) )  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  e.  RR )  -> 
( ( T `  ( ( F  |`  ( 0..^ ( N  - 
1 ) ) ) concat  <" ( F `  ( N  -  1
) ) "> ) ) `  ( # `
 ( F  |`  ( 0..^ ( N  - 
1 ) ) ) ) )  =  ( ( ( T `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) ) `
 ( ( # `  ( F  |`  (
0..^ ( N  - 
1 ) ) ) )  -  1 ) )  .+^  (sgn `  ( F `  ( N  -  1 ) ) ) ) )
109106, 107, 108syl2anc 661 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  (
( T `  (
( F  |`  (
0..^ ( N  - 
1 ) ) ) concat  <" ( F `  ( N  -  1
) ) "> ) ) `  ( # `
 ( F  |`  ( 0..^ ( N  - 
1 ) ) ) ) )  =  ( ( ( T `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) ) `
 ( ( # `  ( F  |`  (
0..^ ( N  - 
1 ) ) ) )  -  1 ) )  .+^  (sgn `  ( F `  ( N  -  1 ) ) ) ) )
110 lennncl 12544 . . . . . . 7  |-  ( ( ( F  |`  (
0..^ ( N  - 
1 ) ) )  e. Word  RR  /\  ( F  |`  ( 0..^ ( N  -  1 ) ) )  =/=  (/) )  -> 
( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  e.  NN )
111 fzo0end 11884 . . . . . . 7  |-  ( (
# `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  e.  NN  ->  ( ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  - 
1 )  e.  ( 0..^ ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) ) ) )
112104, 110, 1113syl 20 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  (
( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  -  1 )  e.  ( 0..^ ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) ) ) )
11314, 15, 16, 17signstcl 28338 . . . . . 6  |-  ( ( ( F  |`  (
0..^ ( N  - 
1 ) ) )  e. Word  RR  /\  (
( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  -  1 )  e.  ( 0..^ ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) ) ) )  ->  ( ( T `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) ) `  (
( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  -  1 ) )  e.  { -u 1 ,  0 ,  1 } )
11491, 112, 113syl2anc 661 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  (
( T `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) ) `  ( ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  - 
1 ) )  e. 
{ -u 1 ,  0 ,  1 } )
11544adantr 465 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  (sgn `  ( F `  ( N  -  1 ) ) )  e.  { -u 1 ,  0 ,  1 } )
116 simpr 461 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F `  ( N  -  1 ) )  =/=  0 )
117 sgn0bi 28302 . . . . . . . . 9  |-  ( ( F `  ( N  -  1 ) )  e.  RR*  ->  ( (sgn
`  ( F `  ( N  -  1
) ) )  =  0  <->  ( F `  ( N  -  1
) )  =  0 ) )
118117necon3bid 2725 . . . . . . . 8  |-  ( ( F `  ( N  -  1 ) )  e.  RR*  ->  ( (sgn
`  ( F `  ( N  -  1
) ) )  =/=  0  <->  ( F `  ( N  -  1
) )  =/=  0
) )
119118biimpar 485 . . . . . . 7  |-  ( ( ( F `  ( N  -  1 ) )  e.  RR*  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
(sgn `  ( F `  ( N  -  1 ) ) )  =/=  0 )
12042, 116, 119syl2anc 661 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
(sgn `  ( F `  ( N  -  1 ) ) )  =/=  0 )
121120adantr 465 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  (sgn `  ( F `  ( N  -  1 ) ) )  =/=  0
)
12214, 15signswlid 28332 . . . . 5  |-  ( ( ( ( ( T `
 ( F  |`  ( 0..^ ( N  - 
1 ) ) ) ) `  ( (
# `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  -  1 ) )  e.  { -u 1 ,  0 ,  1 }  /\  (sgn `  ( F `  ( N  -  1 ) ) )  e.  { -u 1 ,  0 ,  1 } )  /\  (sgn `  ( F `  ( N  -  1
) ) )  =/=  0 )  ->  (
( ( T `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) ) `
 ( ( # `  ( F  |`  (
0..^ ( N  - 
1 ) ) ) )  -  1 ) )  .+^  (sgn `  ( F `  ( N  -  1 ) ) ) )  =  (sgn
`  ( F `  ( N  -  1
) ) ) )
123114, 115, 121, 122syl21anc 1227 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  (
( ( T `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) ) `
 ( ( # `  ( F  |`  (
0..^ ( N  - 
1 ) ) ) )  -  1 ) )  .+^  (sgn `  ( F `  ( N  -  1 ) ) ) )  =  (sgn
`  ( F `  ( N  -  1
) ) ) )
124109, 123eqtrd 2508 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  (
( T `  (
( F  |`  (
0..^ ( N  - 
1 ) ) ) concat  <" ( F `  ( N  -  1
) ) "> ) ) `  ( # `
 ( F  |`  ( 0..^ ( N  - 
1 ) ) ) ) )  =  (sgn
`  ( F `  ( N  -  1
) ) ) )
12586, 124eqtrd 2508 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  (
( T `  F
) `  ( N  -  1 ) )  =  (sgn `  ( F `  ( N  -  1 ) ) ) )
12648, 125pm2.61dane 2785 1  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( ( T `  F ) `  ( N  -  1 ) )  =  (sgn `  ( F `  ( N  -  1 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3478    C_ wss 3481   (/)c0 3790   ifcif 3945   {csn 4033   {cpr 4035   {ctp 4037   <.cop 4039    |-> cmpt 4511    |` cres 5007    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   RRcr 9503   0cc0 9504   1c1 9505   RR*cxr 9639    - cmin 9817   -ucneg 9818   NNcn 10548   NN0cn0 10807   ZZcz 10876   ...cfz 11684  ..^cfzo 11804   #chash 12385  Word cword 12515   concat cconcat 12517   <"cs1 12518   substr csubstr 12519  sgncsgn 12899   sum_csu 13488   ndxcnx 14504   Basecbs 14507   +g cplusg 14572    gsumg cgsu 14713
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-seq 12088  df-hash 12386  df-word 12523  df-concat 12525  df-s1 12526  df-substr 12527  df-sgn 12900  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-plusg 14585  df-0g 14714  df-gsum 14715  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mulg 15932  df-cntz 16227
This theorem is referenced by:  signsvfpn  28358  signsvfnn  28359
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