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Theorem signsvtn0 29459
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 4097 . . . . . . . . . . . 12  |-  ( F  e.  (Word  RR  \  { (/) } )  <->  ( F  e. Word  RR  /\  F  =/=  (/) ) )
21biimpi 198 . . . . . . . . . . 11  |-  ( F  e.  (Word  RR  \  { (/) } )  -> 
( F  e. Word  RR  /\  F  =/=  (/) ) )
32adantr 467 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F  e. Word  RR  /\  F  =/=  (/) ) )
43simpld 461 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  F  e. Word  RR )
54adantr 467 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  F  e. Word  RR )
6 wrdf 12676 . . . . . . . 8  |-  ( F  e. Word  RR  ->  F :
( 0..^ ( # `  F ) ) --> RR )
75, 6syl 17 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  F : ( 0..^ (
# `  F )
) --> RR )
8 lennncl 12688 . . . . . . . . . 10  |-  ( ( F  e. Word  RR  /\  F  =/=  (/) )  ->  ( # `
 F )  e.  NN )
93, 8syl 17 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( # `  F )  e.  NN )
109adantr 467 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  ( # `
 F )  e.  NN )
11 lbfzo0 11955 . . . . . . . 8  |-  ( 0  e.  ( 0..^ (
# `  F )
)  <->  ( # `  F
)  e.  NN )
1210, 11sylibr 216 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  0  e.  ( 0..^ ( # `  F ) ) )
137, 12ffvelrnd 6023 . . . . . 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 29457 . . . . . 6  |-  ( ( F `  0 )  e.  RR  ->  ( T `  <" ( F `  0 ) "> )  =  <" (sgn `  ( F `  0 ) ) "> )
1913, 18syl 17 . . . . 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 463 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  N  =  1 )
2220, 21syl5eqr 2499 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  ( # `
 F )  =  1 )
23 eqs1 12750 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  ( # `  F )  =  1 )  ->  F  =  <" ( F `  0 ) "> )
245, 22, 23syl2anc 667 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  F  =  <" ( F `
 0 ) "> )
2524fveq2d 5869 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  ( T `  F )  =  ( T `  <" ( F ` 
0 ) "> ) )
26 oveq1 6297 . . . . . . . . . 10  |-  ( N  =  1  ->  ( N  -  1 )  =  ( 1  -  1 ) )
27 1m1e0 10678 . . . . . . . . . 10  |-  ( 1  -  1 )  =  0
2826, 27syl6eq 2501 . . . . . . . . 9  |-  ( N  =  1  ->  ( N  -  1 )  =  0 )
2928fveq2d 5869 . . . . . . . 8  |-  ( N  =  1  ->  ( F `  ( N  -  1 ) )  =  ( F ` 
0 ) )
3029fveq2d 5869 . . . . . . 7  |-  ( N  =  1  ->  (sgn `  ( F `  ( N  -  1 ) ) )  =  (sgn
`  ( F ` 
0 ) ) )
3130s1eqd 12740 . . . . . 6  |-  ( N  =  1  ->  <" (sgn `  ( F `  ( N  -  1 ) ) ) ">  =  <" (sgn `  ( F `  0 ) ) "> )
3221, 31syl 17 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  <" (sgn `  ( F `  ( N  -  1 ) ) ) ">  =  <" (sgn `  ( F `  0 ) ) "> )
3319, 25, 323eqtr4d 2495 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  ( T `  F )  =  <" (sgn `  ( F `  ( N  -  1 ) ) ) "> )
3421, 28syl 17 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  ( N  -  1 )  =  0 )
3533, 34fveq12d 5871 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  (
( T `  F
) `  ( N  -  1 ) )  =  ( <" (sgn `  ( F `  ( N  -  1 ) ) ) "> `  0 ) )
364, 6syl 17 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  F : ( 0..^ (
# `  F )
) --> RR )
3720oveq1i 6300 . . . . . . . . 9  |-  ( N  -  1 )  =  ( ( # `  F
)  -  1 )
38 fzo0end 12003 . . . . . . . . . 10  |-  ( (
# `  F )  e.  NN  ->  ( ( # `
 F )  - 
1 )  e.  ( 0..^ ( # `  F
) ) )
393, 8, 383syl 18 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( ( # `  F
)  -  1 )  e.  ( 0..^ (
# `  F )
) )
4037, 39syl5eqel 2533 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( N  -  1 )  e.  ( 0..^ ( # `  F
) ) )
4136, 40ffvelrnd 6023 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F `  ( N  -  1 ) )  e.  RR )
4241rexrd 9690 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F `  ( N  -  1 ) )  e.  RR* )
43 sgncl 29409 . . . . . 6  |-  ( ( F `  ( N  -  1 ) )  e.  RR*  ->  (sgn `  ( F `  ( N  -  1 ) ) )  e.  { -u
1 ,  0 ,  1 } )
4442, 43syl 17 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
(sgn `  ( F `  ( N  -  1 ) ) )  e. 
{ -u 1 ,  0 ,  1 } )
4544adantr 467 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  (sgn `  ( F `  ( N  -  1 ) ) )  e.  { -u 1 ,  0 ,  1 } )
46 s1fv 12748 . . . 4  |-  ( (sgn
`  ( F `  ( N  -  1
) ) )  e. 
{ -u 1 ,  0 ,  1 }  ->  (
<" (sgn `  ( F `  ( N  -  1 ) ) ) "> `  0
)  =  (sgn `  ( F `  ( N  -  1 ) ) ) )
4745, 46syl 17 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  ( <" (sgn `  ( F `  ( N  -  1 ) ) ) "> `  0
)  =  (sgn `  ( F `  ( N  -  1 ) ) ) )
4835, 47eqtrd 2485 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  (
( T `  F
) `  ( N  -  1 ) )  =  (sgn `  ( F `  ( N  -  1 ) ) ) )
49 fzossfz 11938 . . . . . . . . . 10  |-  ( 0..^ ( # `  F
) )  C_  (
0 ... ( # `  F
) )
5049, 39sseldi 3430 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( ( # `  F
)  -  1 )  e.  ( 0 ... ( # `  F
) ) )
51 swrd0val 12777 . . . . . . . . 9  |-  ( ( F  e. Word  RR  /\  ( ( # `  F
)  -  1 )  e.  ( 0 ... ( # `  F
) ) )  -> 
( F substr  <. 0 ,  ( ( # `  F
)  -  1 )
>. )  =  ( F  |`  ( 0..^ ( ( # `  F
)  -  1 ) ) ) )
524, 50, 51syl2anc 667 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F substr  <. 0 ,  ( ( # `  F
)  -  1 )
>. )  =  ( F  |`  ( 0..^ ( ( # `  F
)  -  1 ) ) ) )
5352oveq1d 6305 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( ( F substr  <. 0 ,  ( ( # `  F )  -  1 ) >. ) ++  <" ( lastS  `  F ) "> )  =  ( ( F  |`  ( 0..^ ( ( # `  F
)  -  1 ) ) ) ++  <" ( lastS  `  F ) "> ) )
54 swrdccatwrd 12824 . . . . . . . . 9  |-  ( ( F  e. Word  RR  /\  F  =/=  (/) )  ->  (
( F substr  <. 0 ,  ( ( # `  F
)  -  1 )
>. ) ++  <" ( lastS  `  F ) "> )  =  F )
5554eqcomd 2457 . . . . . . . 8  |-  ( ( F  e. Word  RR  /\  F  =/=  (/) )  ->  F  =  ( ( F substr  <. 0 ,  ( (
# `  F )  -  1 ) >.
) ++  <" ( lastS  `  F
) "> )
)
563, 55syl 17 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  F  =  ( ( F substr  <. 0 ,  ( ( # `  F
)  -  1 )
>. ) ++  <" ( lastS  `  F ) "> ) )
5737oveq2i 6301 . . . . . . . . . 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 5105 . . . . . . . 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 5869 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F `  ( N  -  1 ) )  =  ( F `
 ( ( # `  F )  -  1 ) ) )
62 lsw 12711 . . . . . . . . . . 11  |-  ( F  e.  (Word  RR  \  { (/) } )  -> 
( lastS  `  F )  =  ( F `  (
( # `  F )  -  1 ) ) )
6362adantr 467 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( lastS  `  F )  =  ( F `  (
( # `  F )  -  1 ) ) )
6461, 63eqtr4d 2488 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F `  ( N  -  1 ) )  =  ( lastS  `  F
) )
6564s1eqd 12740 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  <" ( F `  ( N  -  1
) ) ">  =  <" ( lastS  `  F
) "> )
6659, 65oveq12d 6308 . . . . . . 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 2495 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  F  =  ( ( F  |`  ( 0..^ ( N  -  1 ) ) ) ++  <" ( F `  ( N  -  1 ) ) "> ) )
6867fveq2d 5869 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( T `  F
)  =  ( T `
 ( ( F  |`  ( 0..^ ( N  -  1 ) ) ) ++  <" ( F `
 ( N  - 
1 ) ) "> ) ) )
69 ffn 5728 . . . . . . . . . . 11  |-  ( F : ( 0..^ (
# `  F )
) --> RR  ->  F  Fn  ( 0..^ ( # `  F ) ) )
704, 6, 693syl 18 . . . . . . . . . 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 6306 . . . . . . . . . . 11  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( 0..^ N )  =  ( 0..^ (
# `  F )
) )
7372fneq2d 5667 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F  Fn  (
0..^ N )  <->  F  Fn  ( 0..^ ( # `  F
) ) ) )
7470, 73mpbird 236 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  F  Fn  ( 0..^ N ) )
7520, 9syl5eqel 2533 . . . . . . . . . . 11  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  N  e.  NN )
7675nnnn0d 10925 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  N  e.  NN0 )
77 nn0z 10960 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  N  e.  ZZ )
78 fzossrbm1 11947 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
0..^ ( N  - 
1 ) )  C_  ( 0..^ N ) )
7976, 77, 783syl 18 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( 0..^ ( N  -  1 ) ) 
C_  ( 0..^ N ) )
80 fnssres 5689 . . . . . . . . 9  |-  ( ( F  Fn  ( 0..^ N )  /\  (
0..^ ( N  - 
1 ) )  C_  ( 0..^ N ) )  ->  ( F  |`  ( 0..^ ( N  - 
1 ) ) )  Fn  ( 0..^ ( N  -  1 ) ) )
8174, 79, 80syl2anc 667 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F  |`  (
0..^ ( N  - 
1 ) ) )  Fn  ( 0..^ ( N  -  1 ) ) )
82 hashfn 12554 . . . . . . . 8  |-  ( ( F  |`  ( 0..^ ( N  -  1 ) ) )  Fn  ( 0..^ ( N  -  1 ) )  ->  ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  =  ( # `  (
0..^ ( N  - 
1 ) ) ) )
8381, 82syl 17 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  =  (
# `  ( 0..^ ( N  -  1
) ) ) )
84 nnm1nn0 10911 . . . . . . . 8  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
85 hashfzo0 12602 . . . . . . . 8  |-  ( ( N  -  1 )  e.  NN0  ->  ( # `  ( 0..^ ( N  -  1 ) ) )  =  ( N  -  1 ) )
8675, 84, 853syl 18 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( # `  ( 0..^ ( N  -  1 ) ) )  =  ( N  -  1 ) )
8783, 86eqtrd 2485 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  =  ( N  -  1 ) )
8887eqcomd 2457 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( N  -  1 )  =  ( # `  ( F  |`  (
0..^ ( N  - 
1 ) ) ) ) )
8968, 88fveq12d 5871 . . . 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 467 . . 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 2488 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F substr  <. 0 ,  ( ( # `  F
)  -  1 )
>. )  =  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )
92 swrdcl 12775 . . . . . . . . 9  |-  ( F  e. Word  RR  ->  ( F substr  <. 0 ,  ( (
# `  F )  -  1 ) >.
)  e. Word  RR )
934, 92syl 17 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F substr  <. 0 ,  ( ( # `  F
)  -  1 )
>. )  e. Word  RR )
9491, 93eqeltrrd 2530 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F  |`  (
0..^ ( N  - 
1 ) ) )  e. Word  RR )
9594adantr 467 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( F  |`  ( 0..^ ( N  -  1 ) ) )  e. Word  RR )
9687adantr 467 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( # `
 ( F  |`  ( 0..^ ( N  - 
1 ) ) ) )  =  ( N  -  1 ) )
9775adantr 467 . . . . . . . . . 10  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  N  e.  NN )
9897nncnd 10625 . . . . . . . . 9  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  N  e.  CC )
99 1cnd 9659 . . . . . . . . 9  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  1  e.  CC )
100 simpr 463 . . . . . . . . 9  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  N  =/=  1 )
10198, 99, 100subne0d 9995 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( N  -  1 )  =/=  0 )
10296, 101eqnetrd 2691 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( # `
 ( F  |`  ( 0..^ ( N  - 
1 ) ) ) )  =/=  0 )
103 fveq2 5865 . . . . . . . . 9  |-  ( ( F  |`  ( 0..^ ( N  -  1 ) ) )  =  (/)  ->  ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  =  ( # `  (/) ) )
104 hash0 12548 . . . . . . . . 9  |-  ( # `  (/) )  =  0
105103, 104syl6eq 2501 . . . . . . . 8  |-  ( ( F  |`  ( 0..^ ( N  -  1 ) ) )  =  (/)  ->  ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  =  0 )
106105necon3i 2656 . . . . . . 7  |-  ( (
# `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  =/=  0  ->  ( F  |`  (
0..^ ( N  - 
1 ) ) )  =/=  (/) )
107102, 106syl 17 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( F  |`  ( 0..^ ( N  -  1 ) ) )  =/=  (/) )
10895, 107jca 535 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  (
( F  |`  (
0..^ ( N  - 
1 ) ) )  e. Word  RR  /\  ( F  |`  ( 0..^ ( N  -  1 ) ) )  =/=  (/) ) )
109 eldifsn 4097 . . . . 5  |-  ( ( F  |`  ( 0..^ ( N  -  1 ) ) )  e.  (Word  RR  \  { (/)
} )  <->  ( ( F  |`  ( 0..^ ( N  -  1 ) ) )  e. Word  RR  /\  ( F  |`  (
0..^ ( N  - 
1 ) ) )  =/=  (/) ) )
110108, 109sylibr 216 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( F  |`  ( 0..^ ( N  -  1 ) ) )  e.  (Word 
RR  \  { (/) } ) )
11141adantr 467 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( F `  ( N  -  1 ) )  e.  RR )
11214, 15, 16, 17signstfvn 29458 . . . 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 667 . . 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 12688 . . . . . 6  |-  ( ( ( F  |`  (
0..^ ( N  - 
1 ) ) )  e. Word  RR  /\  ( F  |`  ( 0..^ ( N  -  1 ) ) )  =/=  (/) )  -> 
( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  e.  NN )
115 fzo0end 12003 . . . . . 6  |-  ( (
# `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  e.  NN  ->  ( ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  - 
1 )  e.  ( 0..^ ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) ) ) )
116108, 114, 1153syl 18 . . . . 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 29454 . . . . 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 667 . . . 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 467 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  (sgn `  ( F `  ( N  -  1 ) ) )  e.  { -u 1 ,  0 ,  1 } )
120 simpr 463 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F `  ( N  -  1 ) )  =/=  0 )
121 sgn0bi 29418 . . . . . . . 8  |-  ( ( F `  ( N  -  1 ) )  e.  RR*  ->  ( (sgn
`  ( F `  ( N  -  1
) ) )  =  0  <->  ( F `  ( N  -  1
) )  =  0 ) )
122121necon3bid 2668 . . . . . . 7  |-  ( ( F `  ( N  -  1 ) )  e.  RR*  ->  ( (sgn
`  ( F `  ( N  -  1
) ) )  =/=  0  <->  ( F `  ( N  -  1
) )  =/=  0
) )
123122biimpar 488 . . . . . 6  |-  ( ( ( F `  ( N  -  1 ) )  e.  RR*  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
(sgn `  ( F `  ( N  -  1 ) ) )  =/=  0 )
12442, 120, 123syl2anc 667 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
(sgn `  ( F `  ( N  -  1 ) ) )  =/=  0 )
125124adantr 467 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  (sgn `  ( F `  ( N  -  1 ) ) )  =/=  0
)
12614, 15signswlid 29448 . . . 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 1267 . . 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 2489 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  (
( T `  F
) `  ( N  -  1 ) )  =  (sgn `  ( F `  ( N  -  1 ) ) ) )
12948, 128pm2.61dane 2711 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 371    = wceq 1444    e. wcel 1887    =/= wne 2622    \ cdif 3401    C_ wss 3404   (/)c0 3731   ifcif 3881   {csn 3968   {cpr 3970   {ctp 3972   <.cop 3974    |-> cmpt 4461    |` cres 4836    Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290    |-> cmpt2 6292   RRcr 9538   0cc0 9539   1c1 9540   RR*cxr 9674    - cmin 9860   -ucneg 9861   NNcn 10609   NN0cn0 10869   ZZcz 10937   ...cfz 11784  ..^cfzo 11915   #chash 12515  Word cword 12656   lastS clsw 12657   ++ cconcat 12658   <"cs1 12659   substr csubstr 12660  sgncsgn 13149   sum_csu 13752   ndxcnx 15118   Basecbs 15121   +g cplusg 15190    gsumg cgsu 15339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-seq 12214  df-hash 12516  df-word 12664  df-lsw 12665  df-concat 12666  df-s1 12667  df-substr 12668  df-sgn 13150  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-plusg 15203  df-0g 15340  df-gsum 15341  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-mulg 16676  df-cntz 16971
This theorem is referenced by:  signsvfpn  29474  signsvfnn  29475
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