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Theorem signsvtn0 27135
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 4111 . . . . . . . . . . . 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 12361 . . . . . . . 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 12371 . . . . . . . . . 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 11706 . . . . . . . 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 5956 . . . . . 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 27133 . . . . . 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 . . . . . . . . 9  |-  N  =  ( # `  F
)
2120a1i 11 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  N  =  ( # `  F
) )
22 simpr 461 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  N  =  1 )
2321, 22eqtr3d 2497 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  ( # `
 F )  =  1 )
24 eqs1 12421 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  ( # `  F )  =  1 )  ->  F  =  <" ( F `  0 ) "> )
255, 23, 24syl2anc 661 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  F  =  <" ( F `
 0 ) "> )
2625fveq2d 5806 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  ( T `  F )  =  ( T `  <" ( F ` 
0 ) "> ) )
27 oveq1 6210 . . . . . . . . . 10  |-  ( N  =  1  ->  ( N  -  1 )  =  ( 1  -  1 ) )
28 1m1e0 10504 . . . . . . . . . 10  |-  ( 1  -  1 )  =  0
2927, 28syl6eq 2511 . . . . . . . . 9  |-  ( N  =  1  ->  ( N  -  1 )  =  0 )
3029fveq2d 5806 . . . . . . . 8  |-  ( N  =  1  ->  ( F `  ( N  -  1 ) )  =  ( F ` 
0 ) )
3130fveq2d 5806 . . . . . . 7  |-  ( N  =  1  ->  (sgn `  ( F `  ( N  -  1 ) ) )  =  (sgn
`  ( F ` 
0 ) ) )
3231s1eqd 12413 . . . . . 6  |-  ( N  =  1  ->  <" (sgn `  ( F `  ( N  -  1 ) ) ) ">  =  <" (sgn `  ( F `  0 ) ) "> )
3322, 32syl 16 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  <" (sgn `  ( F `  ( N  -  1 ) ) ) ">  =  <" (sgn `  ( F `  0 ) ) "> )
3419, 26, 333eqtr4d 2505 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  ( T `  F )  =  <" (sgn `  ( F `  ( N  -  1 ) ) ) "> )
3522, 29syl 16 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  ( N  -  1 )  =  0 )
3634, 35fveq12d 5808 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  (
( T `  F
) `  ( N  -  1 ) )  =  ( <" (sgn `  ( F `  ( N  -  1 ) ) ) "> `  0 ) )
374, 6syl 16 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  F : ( 0..^ (
# `  F )
) --> RR )
3820oveq1i 6213 . . . . . . . . 9  |-  ( N  -  1 )  =  ( ( # `  F
)  -  1 )
39 fzo0end 11739 . . . . . . . . . 10  |-  ( (
# `  F )  e.  NN  ->  ( ( # `
 F )  - 
1 )  e.  ( 0..^ ( # `  F
) ) )
403, 8, 393syl 20 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( ( # `  F
)  -  1 )  e.  ( 0..^ (
# `  F )
) )
4138, 40syl5eqel 2546 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( N  -  1 )  e.  ( 0..^ ( # `  F
) ) )
4237, 41ffvelrnd 5956 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F `  ( N  -  1 ) )  e.  RR )
4342rexrd 9547 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F `  ( N  -  1 ) )  e.  RR* )
44 sgncl 27085 . . . . . 6  |-  ( ( F `  ( N  -  1 ) )  e.  RR*  ->  (sgn `  ( F `  ( N  -  1 ) ) )  e.  { -u
1 ,  0 ,  1 } )
4543, 44syl 16 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
(sgn `  ( F `  ( N  -  1 ) ) )  e. 
{ -u 1 ,  0 ,  1 } )
4645adantr 465 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  (sgn `  ( F `  ( N  -  1 ) ) )  e.  { -u 1 ,  0 ,  1 } )
47 s1fv 12419 . . . 4  |-  ( (sgn
`  ( F `  ( N  -  1
) ) )  e. 
{ -u 1 ,  0 ,  1 }  ->  (
<" (sgn `  ( F `  ( N  -  1 ) ) ) "> `  0
)  =  (sgn `  ( F `  ( N  -  1 ) ) ) )
4846, 47syl 16 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  ( <" (sgn `  ( F `  ( N  -  1 ) ) ) "> `  0
)  =  (sgn `  ( F `  ( N  -  1 ) ) ) )
4936, 48eqtrd 2495 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  = 
1 )  ->  (
( T `  F
) `  ( N  -  1 ) )  =  (sgn `  ( F `  ( N  -  1 ) ) ) )
50 fzossfz 11690 . . . . . . . . . 10  |-  ( 0..^ ( # `  F
) )  C_  (
0 ... ( # `  F
) )
5150, 40sseldi 3465 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( ( # `  F
)  -  1 )  e.  ( 0 ... ( # `  F
) ) )
52 swrd0val 12438 . . . . . . . . 9  |-  ( ( F  e. Word  RR  /\  ( ( # `  F
)  -  1 )  e.  ( 0 ... ( # `  F
) ) )  -> 
( F substr  <. 0 ,  ( ( # `  F
)  -  1 )
>. )  =  ( F  |`  ( 0..^ ( ( # `  F
)  -  1 ) ) ) )
534, 51, 52syl2anc 661 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F substr  <. 0 ,  ( ( # `  F
)  -  1 )
>. )  =  ( F  |`  ( 0..^ ( ( # `  F
)  -  1 ) ) ) )
5453oveq1d 6218 . . . . . . 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 ) ) "> ) )
55 wrdeqcats1 12489 . . . . . . . 8  |-  ( ( F  e. Word  RR  /\  F  =/=  (/) )  ->  F  =  ( ( F substr  <. 0 ,  ( (
# `  F )  -  1 ) >.
) concat  <" ( F `
 ( ( # `  F )  -  1 ) ) "> ) )
563, 55syl 16 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  F  =  ( ( F substr  <. 0 ,  ( ( # `  F
)  -  1 )
>. ) concat  <" ( F `  ( ( # `
 F )  - 
1 ) ) "> ) )
5738oveq2i 6214 . . . . . . . . . 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 5221 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F  |`  (
0..^ ( N  - 
1 ) ) )  =  ( F  |`  ( 0..^ ( ( # `  F )  -  1 ) ) ) )
6038a1i 11 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( N  -  1 )  =  ( (
# `  F )  -  1 ) )
6160fveq2d 5806 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F `  ( N  -  1 ) )  =  ( F `
 ( ( # `  F )  -  1 ) ) )
6261s1eqd 12413 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  <" ( F `  ( N  -  1
) ) ">  =  <" ( F `
 ( ( # `  F )  -  1 ) ) "> )
6359, 62oveq12d 6221 . . . . . . 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 ) ) "> ) )
6454, 56, 633eqtr4d 2505 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  F  =  ( ( F  |`  ( 0..^ ( N  -  1 ) ) ) concat  <" ( F `  ( N  -  1 ) ) "> ) )
6564fveq2d 5806 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( T `  F
)  =  ( T `
 ( ( F  |`  ( 0..^ ( N  -  1 ) ) ) concat  <" ( F `
 ( N  - 
1 ) ) "> ) ) )
66 ffn 5670 . . . . . . . . . . 11  |-  ( F : ( 0..^ (
# `  F )
) --> RR  ->  F  Fn  ( 0..^ ( # `  F ) ) )
6737, 66syl 16 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  F  Fn  ( 0..^ ( # `  F
) ) )
6820a1i 11 . . . . . . . . . . . 12  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  N  =  ( # `  F
) )
6968oveq2d 6219 . . . . . . . . . . 11  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( 0..^ N )  =  ( 0..^ (
# `  F )
) )
7069fneq2d 5613 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F  Fn  (
0..^ N )  <->  F  Fn  ( 0..^ ( # `  F
) ) ) )
7167, 70mpbird 232 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  F  Fn  ( 0..^ N ) )
7220, 9syl5eqel 2546 . . . . . . . . . . 11  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  N  e.  NN )
7372nnnn0d 10750 . . . . . . . . . 10  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  ->  N  e.  NN0 )
74 fzossrbm1 11698 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  ( 0..^ ( N  -  1 ) )  C_  (
0..^ N ) )
7573, 74syl 16 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( 0..^ ( N  -  1 ) ) 
C_  ( 0..^ N ) )
76 fnssres 5635 . . . . . . . . 9  |-  ( ( F  Fn  ( 0..^ N )  /\  (
0..^ ( N  - 
1 ) )  C_  ( 0..^ N ) )  ->  ( F  |`  ( 0..^ ( N  - 
1 ) ) )  Fn  ( 0..^ ( N  -  1 ) ) )
7771, 75, 76syl2anc 661 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F  |`  (
0..^ ( N  - 
1 ) ) )  Fn  ( 0..^ ( N  -  1 ) ) )
78 hashfn 12259 . . . . . . . 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 10735 . . . . . . . . 9  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
8172, 80syl 16 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( N  -  1 )  e.  NN0 )
82 hashfzo0 12312 . . . . . . . 8  |-  ( ( N  -  1 )  e.  NN0  ->  ( # `  ( 0..^ ( N  -  1 ) ) )  =  ( N  -  1 ) )
8381, 82syl 16 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( # `  ( 0..^ ( N  -  1 ) ) )  =  ( N  -  1 ) )
8479, 83eqtrd 2495 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  =  ( N  -  1 ) )
8584eqcomd 2462 . . . . 5  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( N  -  1 )  =  ( # `  ( F  |`  (
0..^ ( N  - 
1 ) ) ) ) )
8665, 85fveq12d 5808 . . . 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 ) ) ) ) ) )
8786adantr 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 ) ) ) ) ) )
8853, 59eqtr4d 2498 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F substr  <. 0 ,  ( ( # `  F
)  -  1 )
>. )  =  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )
89 swrdcl 12436 . . . . . . . . . 10  |-  ( F  e. Word  RR  ->  ( F substr  <. 0 ,  ( (
# `  F )  -  1 ) >.
)  e. Word  RR )
904, 89syl 16 . . . . . . . . 9  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F substr  <. 0 ,  ( ( # `  F
)  -  1 )
>. )  e. Word  RR )
9188, 90eqeltrrd 2543 . . . . . . . 8  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F  |`  (
0..^ ( N  - 
1 ) ) )  e. Word  RR )
9291adantr 465 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( F  |`  ( 0..^ ( N  -  1 ) ) )  e. Word  RR )
9384adantr 465 . . . . . . . . 9  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( # `
 ( F  |`  ( 0..^ ( N  - 
1 ) ) ) )  =  ( N  -  1 ) )
9472adantr 465 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  N  e.  NN )
9594nncnd 10452 . . . . . . . . . 10  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  N  e.  CC )
96 ax-1cn 9454 . . . . . . . . . . 11  |-  1  e.  CC
9796a1i 11 . . . . . . . . . 10  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  1  e.  CC )
98 simpr 461 . . . . . . . . . 10  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  N  =/=  1 )
9995, 97, 98subne0d 9842 . . . . . . . . 9  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( N  -  1 )  =/=  0 )
10093, 99eqnetrd 2745 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( # `
 ( F  |`  ( 0..^ ( N  - 
1 ) ) ) )  =/=  0 )
101 fveq2 5802 . . . . . . . . . 10  |-  ( ( F  |`  ( 0..^ ( N  -  1 ) ) )  =  (/)  ->  ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  =  ( # `  (/) ) )
102 hash0 12255 . . . . . . . . . 10  |-  ( # `  (/) )  =  0
103101, 102syl6eq 2511 . . . . . . . . 9  |-  ( ( F  |`  ( 0..^ ( N  -  1 ) ) )  =  (/)  ->  ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  =  0 )
104103necon3i 2692 . . . . . . . 8  |-  ( (
# `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  =/=  0  ->  ( F  |`  (
0..^ ( N  - 
1 ) ) )  =/=  (/) )
105100, 104syl 16 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( F  |`  ( 0..^ ( N  -  1 ) ) )  =/=  (/) )
10692, 105jca 532 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  (
( F  |`  (
0..^ ( N  - 
1 ) ) )  e. Word  RR  /\  ( F  |`  ( 0..^ ( N  -  1 ) ) )  =/=  (/) ) )
107 eldifsn 4111 . . . . . 6  |-  ( ( F  |`  ( 0..^ ( N  -  1 ) ) )  e.  (Word  RR  \  { (/)
} )  <->  ( ( F  |`  ( 0..^ ( N  -  1 ) ) )  e. Word  RR  /\  ( F  |`  (
0..^ ( N  - 
1 ) ) )  =/=  (/) ) )
108106, 107sylibr 212 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( F  |`  ( 0..^ ( N  -  1 ) ) )  e.  (Word 
RR  \  { (/) } ) )
10942adantr 465 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( F `  ( N  -  1 ) )  e.  RR )
11014, 15, 16, 17signstfvn 27134 . . . . 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 ) ) ) ) )
111108, 109, 110syl2anc 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 ) ) ) ) )
112106simpld 459 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  ( F  |`  ( 0..^ ( N  -  1 ) ) )  e. Word  RR )
113 lennncl 12371 . . . . . . 7  |-  ( ( ( F  |`  (
0..^ ( N  - 
1 ) ) )  e. Word  RR  /\  ( F  |`  ( 0..^ ( N  -  1 ) ) )  =/=  (/) )  -> 
( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  e.  NN )
114 fzo0end 11739 . . . . . . 7  |-  ( (
# `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  e.  NN  ->  ( ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  - 
1 )  e.  ( 0..^ ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) ) ) )
115106, 113, 1143syl 20 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  (
( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) )  -  1 )  e.  ( 0..^ ( # `  ( F  |`  ( 0..^ ( N  -  1 ) ) ) ) ) )
11614, 15, 16, 17signstcl 27130 . . . . . 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 } )
117112, 115, 116syl2anc 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 } )
11845adantr 465 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  (sgn `  ( F `  ( N  -  1 ) ) )  e.  { -u 1 ,  0 ,  1 } )
119 simpr 461 . . . . . . 7  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( F `  ( N  -  1 ) )  =/=  0 )
120 sgn0bi 27094 . . . . . . . . 9  |-  ( ( F `  ( N  -  1 ) )  e.  RR*  ->  ( (sgn
`  ( F `  ( N  -  1
) ) )  =  0  <->  ( F `  ( N  -  1
) )  =  0 ) )
121120necon3bid 2710 . . . . . . . 8  |-  ( ( F `  ( N  -  1 ) )  e.  RR*  ->  ( (sgn
`  ( F `  ( N  -  1
) ) )  =/=  0  <->  ( F `  ( N  -  1
) )  =/=  0
) )
122121biimpar 485 . . . . . . 7  |-  ( ( ( F `  ( N  -  1 ) )  e.  RR*  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
(sgn `  ( F `  ( N  -  1 ) ) )  =/=  0 )
12343, 119, 122syl2anc 661 . . . . . 6  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
(sgn `  ( F `  ( N  -  1 ) ) )  =/=  0 )
124123adantr 465 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  (sgn `  ( F `  ( N  -  1 ) ) )  =/=  0
)
12514, 15signswlid 27124 . . . . 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
) ) ) )
126117, 118, 124, 125syl21anc 1218 . . . 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
) ) ) )
127111, 126eqtrd 2495 . . 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
) ) ) )
12887, 127eqtrd 2495 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F `  ( N  -  1
) )  =/=  0
)  /\  N  =/=  1 )  ->  (
( T `  F
) `  ( N  -  1 ) )  =  (sgn `  ( F `  ( N  -  1 ) ) ) )
129 exmidne 2658 . . 3  |-  ( N  =  1  \/  N  =/=  1 )
130129a1i 11 . 2  |-  ( ( F  e.  (Word  RR  \  { (/) } )  /\  ( F `  ( N  -  1 ) )  =/=  0 )  -> 
( N  =  1  \/  N  =/=  1
) )
13149, 128, 130mpjaodan 784 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    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648    \ cdif 3436    C_ wss 3439   (/)c0 3748   ifcif 3902   {csn 3988   {cpr 3990   {ctp 3992   <.cop 3994    |-> cmpt 4461    |` cres 4953    Fn wfn 5524   -->wf 5525   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   CCcc 9394   RRcr 9395   0cc0 9396   1c1 9397   RR*cxr 9531    - cmin 9709   -ucneg 9710   NNcn 10436   NN0cn0 10693   ...cfz 11557  ..^cfzo 11668   #chash 12223  Word cword 12342   concat cconcat 12344   <"cs1 12345   substr csubstr 12346  sgncsgn 12696   sum_csu 13284   ndxcnx 14292   Basecbs 14295   +g cplusg 14360    gsumg cgsu 14501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-oi 7838  df-card 8223  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-fzo 11669  df-seq 11927  df-hash 12224  df-word 12350  df-concat 12352  df-s1 12353  df-substr 12354  df-sgn 12697  df-struct 14297  df-ndx 14298  df-slot 14299  df-base 14300  df-plusg 14373  df-0g 14502  df-gsum 14503  df-mnd 15537  df-mulg 15670  df-cntz 15957
This theorem is referenced by:  signsvfpn  27150  signsvfnn  27151
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