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Theorem signsvtn0 28791
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 4141 . . . . . . . . . . . 12  |-  ( F  e.  (Word  RR  \  { (/) } )  <->  ( F  e. Word  RR  /\  F  =/=  (/) ) )
21biimpi 194 . . . . . . . . . . 11  |-  ( F  e.  (Word  RR  \  { (/) } )  -> 
( F  e. Word  RR  /\  F  =/=  (/) ) )
32adantr 463 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F  e. Word  RR  /\  F  =/=  (/) ) )
43simpld 457 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  F  e. Word  RR )
54adantr 463 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  F  e. Word  RR )
6 wrdf 12538 . . . . . . . 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 12550 . . . . . . . . . 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 463 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  ( # `
 F )  e.  NN )
11 lbfzo0 11839 . . . . . . . 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 6008 . . . . . 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 28789 . . . . . 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 459 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  N  =  1 )
2220, 21syl5eqr 2509 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  ( # `
 F )  =  1 )
23 eqs1 12610 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  ( # `  F )  =  1 )  ->  F  =  <" ( F `  0 ) "> )
245, 22, 23syl2anc 659 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  F  =  <" ( F `
 0 ) "> )
2524fveq2d 5852 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  ( T `  F )  =  ( T `  <" ( F ` 
0 ) "> ) )
26 oveq1 6277 . . . . . . . . . 10  |-  ( N  =  1  ->  ( N  -  1 )  =  ( 1  -  1 ) )
27 1m1e0 10600 . . . . . . . . . 10  |-  ( 1  -  1 )  =  0
2826, 27syl6eq 2511 . . . . . . . . 9  |-  ( N  =  1  ->  ( N  -  1 )  =  0 )
2928fveq2d 5852 . . . . . . . 8  |-  ( N  =  1  ->  ( F `  ( N  -  1 ) )  =  ( F ` 
0 ) )
3029fveq2d 5852 . . . . . . 7  |-  ( N  =  1  ->  (sgn `  ( F `  ( N  -  1 ) ) )  =  (sgn
`  ( F ` 
0 ) ) )
3130s1eqd 12602 . . . . . 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 2505 . . . 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 5854 . . 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 6280 . . . . . . . . 9  |-  ( N  -  1 )  =  ( ( # `  F
)  -  1 )
38 fzo0end 11885 . . . . . . . . . 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 2546 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( N  -  1 )  e.  ( 0..^ ( # `  F
) ) )
4136, 40ffvelrnd 6008 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F `  ( N  -  1 ) )  e.  RR )
4241rexrd 9632 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F `  ( N  -  1 ) )  e.  RR* )
43 sgncl 28741 . . . . . 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 463 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  (sgn `  ( F `  ( N  -  1 ) ) )  e.  { -u 1 ,  0 ,  1 } )
46 s1fv 12608 . . . 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 2495 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  (
( T `  F
) `  ( N  -  1 ) )  =  (sgn `  ( F `  ( N  -  1 ) ) ) )
49 fzossfz 11822 . . . . . . . . . 10  |-  ( 0..^ ( # `  F
) )  C_  (
0 ... ( # `  F
) )
5049, 39sseldi 3487 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( ( # `  F
)  -  1 )  e.  ( 0 ... ( # `  F
) ) )
51 swrd0val 12637 . . . . . . . . 9  |-  ( ( F  e. Word  RR  /\  ( ( # `  F
)  -  1 )  e.  ( 0 ... ( # `  F
) ) )  -> 
( F substr  <. 0 ,  ( ( # `  F
)  -  1 )
>. )  =  ( F  |`  ( 0..^ ( ( # `  F
)  -  1 ) ) ) )
524, 50, 51syl2anc 659 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F substr  <. 0 ,  ( ( # `  F
)  -  1 )
>. )  =  ( F  |`  ( 0..^ ( ( # `  F
)  -  1 ) ) ) )
5352oveq1d 6285 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( ( F substr  <. 0 ,  ( ( # `  F )  -  1 ) >. ) ++  <" ( lastS  `  F ) "> )  =  ( ( F  |`  ( 0..^ ( ( # `  F
)  -  1 ) ) ) ++  <" ( lastS  `  F ) "> ) )
54 swrdccatwrd 12684 . . . . . . . . 9  |-  ( ( F  e. Word  RR  /\  F  =/=  (/) )  ->  (
( F substr  <. 0 ,  ( ( # `  F
)  -  1 )
>. ) ++  <" ( lastS  `  F ) "> )  =  F )
5554eqcomd 2462 . . . . . . . 8  |-  ( ( F  e. Word  RR  /\  F  =/=  (/) )  ->  F  =  ( ( F substr  <. 0 ,  ( (
# `  F )  -  1 ) >.
) ++  <" ( lastS  `  F
) "> )
)
563, 55syl 16 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  F  =  ( ( F substr  <. 0 ,  ( ( # `  F
)  -  1 )
>. ) ++  <" ( lastS  `  F ) "> ) )
5737oveq2i 6281 . . . . . . . . . 10  |-  ( 0..^ ( N  -  1 ) )  =  ( 0..^ ( ( # `  F )  -  1 ) )
5857a1i 11 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( 0..^ ( N  -  1 ) )  =  ( 0..^ ( ( # `  F
)  -  1 ) ) )
5958reseq2d 5262 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F  |`  (
0..^ ( N  - 
1 ) ) )  =  ( F  |`  ( 0..^ ( ( # `  F )  -  1 ) ) ) )
6037a1i 11 . . . . . . . . . . 11  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( N  -  1 )  =  ( (
# `  F )  -  1 ) )
6160fveq2d 5852 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F `  ( N  -  1 ) )  =  ( F `
 ( ( # `  F )  -  1 ) ) )
62 lsw 12573 . . . . . . . . . . 11  |-  ( F  e.  (Word  RR  \  { (/) } )  -> 
( lastS  `  F )  =  ( F `  (
( # `  F )  -  1 ) ) )
6362adantr 463 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( lastS  `  F )  =  ( F `  (
( # `  F )  -  1 ) ) )
6461, 63eqtr4d 2498 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F `  ( N  -  1 ) )  =  ( lastS  `  F
) )
6564s1eqd 12602 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  <" ( F `  ( N  -  1
) ) ">  =  <" ( lastS  `  F
) "> )
6659, 65oveq12d 6288 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( ( F  |`  ( 0..^ ( N  - 
1 ) ) ) ++ 
<" ( F `  ( N  -  1
) ) "> )  =  ( ( F  |`  ( 0..^ ( ( # `  F
)  -  1 ) ) ) ++  <" ( lastS  `  F ) "> ) )
6753, 56, 663eqtr4d 2505 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  F  =  ( ( F  |`  ( 0..^ ( N  -  1 ) ) ) ++  <" ( F `  ( N  -  1 ) ) "> ) )
6867fveq2d 5852 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( T `  F
)  =  ( T `
 ( ( F  |`  ( 0..^ ( N  -  1 ) ) ) ++  <" ( F `
 ( N  - 
1 ) ) "> ) ) )
69 ffn 5713 . . . . . . . . . . 11  |-  ( F : ( 0..^ (
# `  F )
) --> RR  ->  F  Fn  ( 0..^ ( # `  F ) ) )
704, 6, 693syl 20 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  F  Fn  ( 0..^ ( # `  F
) ) )
7120a1i 11 . . . . . . . . . . . 12  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  N  =  ( # `  F
) )
7271oveq2d 6286 . . . . . . . . . . 11  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( 0..^ N )  =  ( 0..^ (
# `  F )
) )
7372fneq2d 5654 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F  Fn  (
0..^ N )  <->  F  Fn  ( 0..^ ( # `  F
) ) ) )
7470, 73mpbird 232 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  F  Fn  ( 0..^ N ) )
7520, 9syl5eqel 2546 . . . . . . . . . . 11  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  N  e.  NN )
7675nnnn0d 10848 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  N  e.  NN0 )
77 nn0z 10883 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  N  e.  ZZ )
78 fzossrbm1 11831 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
0..^ ( N  - 
1 ) )  C_  ( 0..^ N ) )
7976, 77, 783syl 20 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( 0..^ ( N  -  1 ) ) 
C_  ( 0..^ N ) )
80 fnssres 5676 . . . . . . . . 9  |-  ( ( F  Fn  ( 0..^ N )  /\  (
0..^ ( N  - 
1 ) )  C_  ( 0..^ N ) )  ->  ( F  |`  ( 0..^ ( N  - 
1 ) ) )  Fn  ( 0..^ ( N  -  1 ) ) )
8174, 79, 80syl2anc 659 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F  |`  (
0..^ ( N  - 
1 ) ) )  Fn  ( 0..^ ( N  -  1 ) ) )
82 hashfn 12426 . . . . . . . 8  |-  ( ( F  |`  ( 0..^ ( N  -  1 ) ) )  Fn  ( 0..^ ( N  -  1 ) )  ->  ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  =  ( # `  (
0..^ ( N  - 
1 ) ) ) )
8381, 82syl 16 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  =  (
# `  ( 0..^ ( N  -  1
) ) ) )
84 nnm1nn0 10833 . . . . . . . 8  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
85 hashfzo0 12472 . . . . . . . 8  |-  ( ( N  -  1 )  e.  NN0  ->  ( # `  ( 0..^ ( N  -  1 ) ) )  =  ( N  -  1 ) )
8675, 84, 853syl 20 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( # `  ( 0..^ ( N  -  1 ) ) )  =  ( N  -  1 ) )
8783, 86eqtrd 2495 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  =  ( N  -  1 ) )
8887eqcomd 2462 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( N  -  1 )  =  ( # `  ( F  |`  (
0..^ ( N  - 
1 ) ) ) ) )
8968, 88fveq12d 5854 . . . 4  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( ( T `  F ) `  ( N  -  1 ) )  =  ( ( T `  ( ( F  |`  ( 0..^ ( N  -  1 ) ) ) ++  <" ( F `  ( N  -  1 ) ) "> )
) `  ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) ) ) )
9089adantr 463 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  (
( T `  F
) `  ( N  -  1 ) )  =  ( ( T `
 ( ( F  |`  ( 0..^ ( N  -  1 ) ) ) ++  <" ( F `
 ( N  - 
1 ) ) "> ) ) `  ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) ) ) )
9152, 59eqtr4d 2498 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F substr  <. 0 ,  ( ( # `  F
)  -  1 )
>. )  =  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )
92 swrdcl 12635 . . . . . . . . 9  |-  ( F  e. Word  RR  ->  ( F substr  <. 0 ,  ( (
# `  F )  -  1 ) >.
)  e. Word  RR )
934, 92syl 16 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F substr  <. 0 ,  ( ( # `  F
)  -  1 )
>. )  e. Word  RR )
9491, 93eqeltrrd 2543 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F  |`  (
0..^ ( N  - 
1 ) ) )  e. Word  RR )
9594adantr 463 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( F  |`  ( 0..^ ( N  -  1 ) ) )  e. Word  RR )
9687adantr 463 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( # `
 ( F  |`  ( 0..^ ( N  - 
1 ) ) ) )  =  ( N  -  1 ) )
9775adantr 463 . . . . . . . . . 10  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  N  e.  NN )
9897nncnd 10547 . . . . . . . . 9  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  N  e.  CC )
99 1cnd 9601 . . . . . . . . 9  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  1  e.  CC )
100 simpr 459 . . . . . . . . 9  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  N  =/=  1 )
10198, 99, 100subne0d 9931 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( N  -  1 )  =/=  0 )
10296, 101eqnetrd 2747 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( # `
 ( F  |`  ( 0..^ ( N  - 
1 ) ) ) )  =/=  0 )
103 fveq2 5848 . . . . . . . . 9  |-  ( ( F  |`  ( 0..^ ( N  -  1 ) ) )  =  (/)  ->  ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  =  ( # `  (/) ) )
104 hash0 12420 . . . . . . . . 9  |-  ( # `  (/) )  =  0
105103, 104syl6eq 2511 . . . . . . . 8  |-  ( ( F  |`  ( 0..^ ( N  -  1 ) ) )  =  (/)  ->  ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  =  0 )
106105necon3i 2694 . . . . . . 7  |-  ( (
# `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  =/=  0  ->  ( F  |`  (
0..^ ( N  - 
1 ) ) )  =/=  (/) )
107102, 106syl 16 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( F  |`  ( 0..^ ( N  -  1 ) ) )  =/=  (/) )
10895, 107jca 530 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  (
( F  |`  (
0..^ ( N  - 
1 ) ) )  e. Word  RR  /\  ( F  |`  ( 0..^ ( N  -  1 ) ) )  =/=  (/) ) )
109 eldifsn 4141 . . . . 5  |-  ( ( F  |`  ( 0..^ ( N  -  1 ) ) )  e.  (Word  RR  \  { (/)
} )  <->  ( ( F  |`  ( 0..^ ( N  -  1 ) ) )  e. Word  RR  /\  ( F  |`  (
0..^ ( N  - 
1 ) ) )  =/=  (/) ) )
110108, 109sylibr 212 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( F  |`  ( 0..^ ( N  -  1 ) ) )  e.  (Word 
RR  \  { (/) } ) )
11141adantr 463 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( F `  ( N  -  1 ) )  e.  RR )
11214, 15, 16, 17signstfvn 28790 . . . 4  |-  ( ( ( F  |`  (
0..^ ( N  - 
1 ) ) )  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  e.  RR )  -> 
( ( T `  ( ( F  |`  ( 0..^ ( N  - 
1 ) ) ) ++ 
<" ( F `  ( N  -  1
) ) "> ) ) `  ( # `
 ( F  |`  ( 0..^ ( N  - 
1 ) ) ) ) )  =  ( ( ( T `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) ) `
 ( ( # `  ( F  |`  (
0..^ ( N  - 
1 ) ) ) )  -  1 ) )  .+^  (sgn `  ( F `  ( N  -  1 ) ) ) ) )
113110, 111, 112syl2anc 659 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  (
( T `  (
( F  |`  (
0..^ ( N  - 
1 ) ) ) ++ 
<" ( F `  ( N  -  1
) ) "> ) ) `  ( # `
 ( F  |`  ( 0..^ ( N  - 
1 ) ) ) ) )  =  ( ( ( T `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) ) `
 ( ( # `  ( F  |`  (
0..^ ( N  - 
1 ) ) ) )  -  1 ) )  .+^  (sgn `  ( F `  ( N  -  1 ) ) ) ) )
114 lennncl 12550 . . . . . 6  |-  ( ( ( F  |`  (
0..^ ( N  - 
1 ) ) )  e. Word  RR  /\  ( F  |`  ( 0..^ ( N  -  1 ) ) )  =/=  (/) )  -> 
( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  e.  NN )
115 fzo0end 11885 . . . . . 6  |-  ( (
# `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  e.  NN  ->  ( ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  - 
1 )  e.  ( 0..^ ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) ) ) )
116108, 114, 1153syl 20 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  (
( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  -  1 )  e.  ( 0..^ ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) ) ) )
11714, 15, 16, 17signstcl 28786 . . . . 5  |-  ( ( ( 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 } )
11895, 116, 117syl2anc 659 . . . 4  |-  ( ( ( 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 } )
11944adantr 463 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  (sgn `  ( F `  ( N  -  1 ) ) )  e.  { -u 1 ,  0 ,  1 } )
120 simpr 459 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F `  ( N  -  1 ) )  =/=  0 )
121 sgn0bi 28750 . . . . . . . 8  |-  ( ( F `  ( N  -  1 ) )  e.  RR*  ->  ( (sgn
`  ( F `  ( N  -  1
) ) )  =  0  <->  ( F `  ( N  -  1
) )  =  0 ) )
122121necon3bid 2712 . . . . . . 7  |-  ( ( F `  ( N  -  1 ) )  e.  RR*  ->  ( (sgn
`  ( F `  ( N  -  1
) ) )  =/=  0  <->  ( F `  ( N  -  1
) )  =/=  0
) )
123122biimpar 483 . . . . . 6  |-  ( ( ( F `  ( N  -  1 ) )  e.  RR*  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
(sgn `  ( F `  ( N  -  1 ) ) )  =/=  0 )
12442, 120, 123syl2anc 659 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
(sgn `  ( F `  ( N  -  1 ) ) )  =/=  0 )
125124adantr 463 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  (sgn `  ( F `  ( N  -  1 ) ) )  =/=  0
)
12614, 15signswlid 28780 . . . 4  |-  ( ( ( ( ( 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
) ) ) )
127118, 119, 125, 126syl21anc 1225 . . 3  |-  ( ( ( 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
) ) ) )
12890, 113, 1273eqtrd 2499 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  (
( T `  F
) `  ( N  -  1 ) )  =  (sgn `  ( F `  ( N  -  1 ) ) ) )
12948, 128pm2.61dane 2772 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 367    = wceq 1398    e. wcel 1823    =/= wne 2649    \ cdif 3458    C_ wss 3461   (/)c0 3783   ifcif 3929   {csn 4016   {cpr 4018   {ctp 4020   <.cop 4022    |-> cmpt 4497    |` cres 4990    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   RRcr 9480   0cc0 9481   1c1 9482   RR*cxr 9616    - cmin 9796   -ucneg 9797   NNcn 10531   NN0cn0 10791   ZZcz 10860   ...cfz 11675  ..^cfzo 11799   #chash 12387  Word cword 12518   lastS clsw 12519   ++ cconcat 12520   <"cs1 12521   substr csubstr 12522  sgncsgn 13001   sum_csu 13590   ndxcnx 14713   Basecbs 14716   +g cplusg 14784    gsumg cgsu 14930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12090  df-hash 12388  df-word 12526  df-lsw 12527  df-concat 12528  df-s1 12529  df-substr 12530  df-sgn 13002  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-plusg 14797  df-0g 14931  df-gsum 14932  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-mulg 16259  df-cntz 16554
This theorem is referenced by:  signsvfpn  28806  signsvfnn  28807
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