Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  signstfveq0 Structured version   Unicode version

Theorem signstfveq0 26981
Description: In case the last letter is zero, the zero skipping sign is the same as the previous letter. (Contributed by Thierry Arnoux, 11-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 ) )
signstfveq0.1  |-  N  =  ( # `  F
)
Assertion
Ref Expression
signstfveq0  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( T `  F ) `  ( N  -  1 ) )  =  ( ( T `  F
) `  ( N  -  2 ) ) )
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 signstfveq0
StepHypRef Expression
1 simpll 753 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  F  e.  (Word  RR  \  { (/) } ) )
21eldifad 3343 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  F  e. Word  RR )
3 swrdcl 12318 . . . . . 6  |-  ( F  e. Word  RR  ->  ( F substr  <. 0 ,  ( N  -  1 ) >.
)  e. Word  RR )
42, 3syl 16 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( F substr  <.
0 ,  ( N  -  1 ) >.
)  e. Word  RR )
5 1nn0 10598 . . . . . . . . . . . . 13  |-  1  e.  NN0
65a1i 11 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  1  e.  NN0 )
76nn0red 10640 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  1  e.  RR )
8 2re 10394 . . . . . . . . . . . . . 14  |-  2  e.  RR
98a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  2  e.  RR )
10 signstfveq0.1 . . . . . . . . . . . . . . 15  |-  N  =  ( # `  F
)
11 lencl 12252 . . . . . . . . . . . . . . . 16  |-  ( F  e. Word  RR  ->  ( # `  F )  e.  NN0 )
122, 11syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( # `  F
)  e.  NN0 )
1310, 12syl5eqel 2527 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  N  e.  NN0 )
1413nn0red 10640 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  N  e.  RR )
15 1le2 10538 . . . . . . . . . . . . . 14  |-  1  <_  2
1615a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  1  <_  2 )
17 signsv.p . . . . . . . . . . . . . . . 16  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
18 signsv.w . . . . . . . . . . . . . . . 16  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
19 signsv.t . . . . . . . . . . . . . . . 16  |-  T  =  ( f  e. Word  RR  |->  ( n  e.  (
0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) ) )
20 signsv.v . . . . . . . . . . . . . . . 16  |-  V  =  ( f  e. Word  RR  |->  sum_ j  e.  ( 1..^ ( # `  f
) ) if ( ( ( T `  f ) `  j
)  =/=  ( ( T `  f ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
2117, 18, 19, 20, 10signstfveq0a 26980 . . . . . . . . . . . . . . 15  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  N  e.  ( ZZ>= `  2 )
)
22 eluz2 10870 . . . . . . . . . . . . . . 15  |-  ( N  e.  ( ZZ>= `  2
)  <->  ( 2  e.  ZZ  /\  N  e.  ZZ  /\  2  <_  N ) )
2321, 22sylib 196 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( 2  e.  ZZ  /\  N  e.  ZZ  /\  2  <_  N ) )
2423simp3d 1002 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  2  <_  N )
257, 9, 14, 16, 24letrd 9531 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  1  <_  N )
266, 25jca 532 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( 1  e.  NN0  /\  1  <_  N ) )
27 fznn0 11527 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( 1  e.  ( 0 ... N )  <->  ( 1  e.  NN0  /\  1  <_  N ) ) )
2813, 27syl 16 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( 1  e.  ( 0 ... N )  <->  ( 1  e.  NN0  /\  1  <_  N ) ) )
2926, 28mpbird 232 . . . . . . . . . 10  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  1  e.  ( 0 ... N
) )
30 fznn0sub2 11491 . . . . . . . . . 10  |-  ( 1  e.  ( 0 ... N )  ->  ( N  -  1 )  e.  ( 0 ... N ) )
3129, 30syl 16 . . . . . . . . 9  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( N  -  1 )  e.  ( 0 ... N
) )
3210oveq2i 6105 . . . . . . . . 9  |-  ( 0 ... N )  =  ( 0 ... ( # `
 F ) )
3331, 32syl6eleq 2533 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( N  -  1 )  e.  ( 0 ... ( # `
 F ) ) )
34 swrd0len 12321 . . . . . . . 8  |-  ( ( F  e. Word  RR  /\  ( N  -  1
)  e.  ( 0 ... ( # `  F
) ) )  -> 
( # `  ( F substr  <. 0 ,  ( N  -  1 ) >.
) )  =  ( N  -  1 ) )
352, 33, 34syl2anc 661 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( # `  ( F substr  <. 0 ,  ( N  -  1 )
>. ) )  =  ( N  -  1 ) )
36 uz2m1nn 10932 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( N  -  1 )  e.  NN )
3721, 36syl 16 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( N  -  1 )  e.  NN )
3835, 37eqeltrd 2517 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( # `  ( F substr  <. 0 ,  ( N  -  1 )
>. ) )  e.  NN )
39 nnne0 10357 . . . . . . 7  |-  ( (
# `  ( F substr  <.
0 ,  ( N  -  1 ) >.
) )  e.  NN  ->  ( # `  ( F substr  <. 0 ,  ( N  -  1 )
>. ) )  =/=  0
)
40 fveq2 5694 . . . . . . . . 9  |-  ( ( F substr  <. 0 ,  ( N  -  1 )
>. )  =  (/)  ->  ( # `
 ( F substr  <. 0 ,  ( N  - 
1 ) >. )
)  =  ( # `  (/) ) )
41 hash0 12138 . . . . . . . . 9  |-  ( # `  (/) )  =  0
4240, 41syl6eq 2491 . . . . . . . 8  |-  ( ( F substr  <. 0 ,  ( N  -  1 )
>. )  =  (/)  ->  ( # `
 ( F substr  <. 0 ,  ( N  - 
1 ) >. )
)  =  0 )
4342necon3i 2653 . . . . . . 7  |-  ( (
# `  ( F substr  <.
0 ,  ( N  -  1 ) >.
) )  =/=  0  ->  ( F substr  <. 0 ,  ( N  - 
1 ) >. )  =/=  (/) )
4439, 43syl 16 . . . . . 6  |-  ( (
# `  ( F substr  <.
0 ,  ( N  -  1 ) >.
) )  e.  NN  ->  ( F substr  <. 0 ,  ( N  - 
1 ) >. )  =/=  (/) )
4538, 44syl 16 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( F substr  <.
0 ,  ( N  -  1 ) >.
)  =/=  (/) )
464, 45jca 532 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( F substr  <. 0 ,  ( N  -  1 )
>. )  e. Word  RR  /\  ( F substr  <. 0 ,  ( N  -  1 ) >. )  =/=  (/) ) )
47 eldifsn 4003 . . . 4  |-  ( ( F substr  <. 0 ,  ( N  -  1 )
>. )  e.  (Word  RR  \  { (/) } )  <-> 
( ( F substr  <. 0 ,  ( N  - 
1 ) >. )  e. Word  RR  /\  ( F substr  <. 0 ,  ( N  -  1 ) >.
)  =/=  (/) ) )
4846, 47sylibr 212 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( F substr  <.
0 ,  ( N  -  1 ) >.
)  e.  (Word  RR  \  { (/) } ) )
49 simpr 461 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( F `  ( N  -  1 ) )  =  0 )
50 0re 9389 . . . 4  |-  0  e.  RR
5149, 50syl6eqel 2531 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( F `  ( N  -  1 ) )  e.  RR )
5217, 18, 19, 20signstfvn 26973 . . 3  |-  ( ( ( F substr  <. 0 ,  ( N  - 
1 ) >. )  e.  (Word  RR  \  { (/)
} )  /\  ( F `  ( N  -  1 ) )  e.  RR )  -> 
( ( T `  ( ( F substr  <. 0 ,  ( N  - 
1 ) >. ) concat  <" ( F `  ( N  -  1
) ) "> ) ) `  ( # `
 ( F substr  <. 0 ,  ( N  - 
1 ) >. )
) )  =  ( ( ( T `  ( F substr  <. 0 ,  ( N  -  1 ) >. ) ) `  ( ( # `  ( F substr  <. 0 ,  ( N  -  1 )
>. ) )  -  1 ) )  .+^  (sgn `  ( F `  ( N  -  1 ) ) ) ) )
5348, 51, 52syl2anc 661 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( T `  ( ( F substr  <. 0 ,  ( N  -  1 )
>. ) concat  <" ( F `  ( N  -  1 ) ) "> ) ) `
 ( # `  ( F substr  <. 0 ,  ( N  -  1 )
>. ) ) )  =  ( ( ( T `
 ( F substr  <. 0 ,  ( N  - 
1 ) >. )
) `  ( ( # `
 ( F substr  <. 0 ,  ( N  - 
1 ) >. )
)  -  1 ) )  .+^  (sgn `  ( F `  ( N  -  1 ) ) ) ) )
54 eldifsn 4003 . . . . . . . . 9  |-  ( F  e.  (Word  RR  \  { (/) } )  <->  ( F  e. Word  RR  /\  F  =/=  (/) ) )
551, 54sylib 196 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( F  e. Word  RR  /\  F  =/=  (/) ) )
5655simprd 463 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  F  =/=  (/) )
57 wrdeqcats1 12371 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  F  =/=  (/) )  ->  F  =  ( ( F substr  <. 0 ,  ( (
# `  F )  -  1 ) >.
) concat  <" ( F `
 ( ( # `  F )  -  1 ) ) "> ) )
582, 56, 57syl2anc 661 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  F  =  ( ( F substr  <. 0 ,  ( ( # `  F )  -  1 ) >. ) concat  <" ( F `  ( ( # `
 F )  - 
1 ) ) "> ) )
5910oveq1i 6104 . . . . . . . . 9  |-  ( N  -  1 )  =  ( ( # `  F
)  -  1 )
6059opeq2i 4066 . . . . . . . 8  |-  <. 0 ,  ( N  - 
1 ) >.  =  <. 0 ,  ( ( # `
 F )  - 
1 ) >.
6160oveq2i 6105 . . . . . . 7  |-  ( F substr  <. 0 ,  ( N  -  1 ) >.
)  =  ( F substr  <. 0 ,  ( (
# `  F )  -  1 ) >.
)
6259fveq2i 5697 . . . . . . . 8  |-  ( F `
 ( N  - 
1 ) )  =  ( F `  (
( # `  F )  -  1 ) )
63 s1eq 12294 . . . . . . . 8  |-  ( ( F `  ( N  -  1 ) )  =  ( F `  ( ( # `  F
)  -  1 ) )  ->  <" ( F `  ( N  -  1 ) ) ">  =  <" ( F `  (
( # `  F )  -  1 ) ) "> )
6462, 63ax-mp 5 . . . . . . 7  |-  <" ( F `  ( N  -  1 ) ) ">  =  <" ( F `  (
( # `  F )  -  1 ) ) ">
6561, 64oveq12i 6106 . . . . . 6  |-  ( ( F substr  <. 0 ,  ( N  -  1 )
>. ) concat  <" ( F `  ( N  -  1 ) ) "> )  =  ( ( F substr  <. 0 ,  ( ( # `  F )  -  1 ) >. ) concat  <" ( F `  ( ( # `
 F )  - 
1 ) ) "> )
6658, 65syl6eqr 2493 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  F  =  ( ( F substr  <. 0 ,  ( N  - 
1 ) >. ) concat  <" ( F `  ( N  -  1
) ) "> ) )
6766fveq2d 5698 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( T `  F )  =  ( T `  ( ( F substr  <. 0 ,  ( N  -  1 )
>. ) concat  <" ( F `  ( N  -  1 ) ) "> ) ) )
6867eqcomd 2448 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( T `  ( ( F substr  <. 0 ,  ( N  - 
1 ) >. ) concat  <" ( F `  ( N  -  1
) ) "> ) )  =  ( T `  F ) )
6968, 35fveq12d 5700 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( T `  ( ( F substr  <. 0 ,  ( N  -  1 )
>. ) concat  <" ( F `  ( N  -  1 ) ) "> ) ) `
 ( # `  ( F substr  <. 0 ,  ( N  -  1 )
>. ) ) )  =  ( ( T `  F ) `  ( N  -  1 ) ) )
7013nn0cnd 10641 . . . . . . . . . 10  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  N  e.  CC )
71 ax-1cn 9343 . . . . . . . . . . 11  |-  1  e.  CC
7271a1i 11 . . . . . . . . . 10  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  1  e.  CC )
7370, 72, 72subsub4d 9753 . . . . . . . . 9  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( N  -  1 )  -  1 )  =  ( N  -  (
1  +  1 ) ) )
74 1p1e2 10438 . . . . . . . . . 10  |-  ( 1  +  1 )  =  2
7574oveq2i 6105 . . . . . . . . 9  |-  ( N  -  ( 1  +  1 ) )  =  ( N  -  2 )
7673, 75syl6eq 2491 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( N  -  1 )  -  1 )  =  ( N  -  2 ) )
77 fzo0end 11622 . . . . . . . . 9  |-  ( ( N  -  1 )  e.  NN  ->  (
( N  -  1 )  -  1 )  e.  ( 0..^ ( N  -  1 ) ) )
7837, 77syl 16 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( N  -  1 )  -  1 )  e.  ( 0..^ ( N  -  1 ) ) )
7976, 78eqeltrrd 2518 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( N  -  2 )  e.  ( 0..^ ( N  -  1 ) ) )
8035oveq2d 6110 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( 0..^ ( # `  ( F substr  <. 0 ,  ( N  -  1 )
>. ) ) )  =  ( 0..^ ( N  -  1 ) ) )
8179, 80eleqtrrd 2520 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( N  -  2 )  e.  ( 0..^ ( # `  ( F substr  <. 0 ,  ( N  - 
1 ) >. )
) ) )
8217, 18, 19, 20signstfvp 26975 . . . . . 6  |-  ( ( ( F substr  <. 0 ,  ( N  - 
1 ) >. )  e. Word  RR  /\  ( F `
 ( N  - 
1 ) )  e.  RR  /\  ( N  -  2 )  e.  ( 0..^ ( # `  ( F substr  <. 0 ,  ( N  - 
1 ) >. )
) ) )  -> 
( ( T `  ( ( F substr  <. 0 ,  ( N  - 
1 ) >. ) concat  <" ( F `  ( N  -  1
) ) "> ) ) `  ( N  -  2 ) )  =  ( ( T `  ( F substr  <. 0 ,  ( N  -  1 ) >.
) ) `  ( N  -  2 ) ) )
834, 51, 81, 82syl3anc 1218 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( T `  ( ( F substr  <. 0 ,  ( N  -  1 )
>. ) concat  <" ( F `  ( N  -  1 ) ) "> ) ) `
 ( N  - 
2 ) )  =  ( ( T `  ( F substr  <. 0 ,  ( N  -  1 ) >. ) ) `  ( N  -  2
) ) )
8467fveq1d 5696 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( T `  F ) `  ( N  -  2 ) )  =  ( ( T `  (
( F substr  <. 0 ,  ( N  -  1 ) >. ) concat  <" ( F `  ( N  -  1 ) ) "> ) ) `
 ( N  - 
2 ) ) )
8535oveq1d 6109 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( # `
 ( F substr  <. 0 ,  ( N  - 
1 ) >. )
)  -  1 )  =  ( ( N  -  1 )  - 
1 ) )
8685, 73eqtrd 2475 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( # `
 ( F substr  <. 0 ,  ( N  - 
1 ) >. )
)  -  1 )  =  ( N  -  ( 1  +  1 ) ) )
8786, 75syl6eq 2491 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( # `
 ( F substr  <. 0 ,  ( N  - 
1 ) >. )
)  -  1 )  =  ( N  - 
2 ) )
8887fveq2d 5698 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( T `  ( F substr  <.
0 ,  ( N  -  1 ) >.
) ) `  (
( # `  ( F substr  <. 0 ,  ( N  -  1 ) >.
) )  -  1 ) )  =  ( ( T `  ( F substr  <. 0 ,  ( N  -  1 )
>. ) ) `  ( N  -  2 ) ) )
8983, 84, 883eqtr4rd 2486 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( T `  ( F substr  <.
0 ,  ( N  -  1 ) >.
) ) `  (
( # `  ( F substr  <. 0 ,  ( N  -  1 ) >.
) )  -  1 ) )  =  ( ( T `  F
) `  ( N  -  2 ) ) )
90 fveq2 5694 . . . . . 6  |-  ( ( F `  ( N  -  1 ) )  =  0  ->  (sgn `  ( F `  ( N  -  1 ) ) )  =  (sgn
`  0 ) )
91 sgn0 12581 . . . . . 6  |-  (sgn ` 
0 )  =  0
9290, 91syl6eq 2491 . . . . 5  |-  ( ( F `  ( N  -  1 ) )  =  0  ->  (sgn `  ( F `  ( N  -  1 ) ) )  =  0 )
9392adantl 466 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  (sgn `  ( F `  ( N  -  1 ) ) )  =  0 )
9489, 93oveq12d 6112 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( (
( T `  ( F substr  <. 0 ,  ( N  -  1 )
>. ) ) `  (
( # `  ( F substr  <. 0 ,  ( N  -  1 ) >.
) )  -  1 ) )  .+^  (sgn `  ( F `  ( N  -  1 ) ) ) )  =  ( ( ( T `  F ) `  ( N  -  2 ) )  .+^  0 ) )
95 uznn0sub 10895 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( N  -  2 )  e. 
NN0 )
9621, 95syl 16 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( N  -  2 )  e. 
NN0 )
97 eluz2b2 10930 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  2
)  <->  ( N  e.  NN  /\  1  < 
N ) )
9897simplbi 460 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  NN )
9921, 98syl 16 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  N  e.  NN )
100 2rp 10999 . . . . . . . . . 10  |-  2  e.  RR+
101100a1i 11 . . . . . . . . 9  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  2  e.  RR+ )
10214, 101ltsubrpd 11058 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( N  -  2 )  < 
N )
10396, 99, 1023jca 1168 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( N  -  2 )  e.  NN0  /\  N  e.  NN  /\  ( N  -  2 )  < 
N ) )
104 elfzo0 11590 . . . . . . 7  |-  ( ( N  -  2 )  e.  ( 0..^ N )  <->  ( ( N  -  2 )  e. 
NN0  /\  N  e.  NN  /\  ( N  - 
2 )  <  N
) )
105103, 104sylibr 212 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( N  -  2 )  e.  ( 0..^ N ) )
10610oveq2i 6105 . . . . . 6  |-  ( 0..^ N )  =  ( 0..^ ( # `  F
) )
107105, 106syl6eleq 2533 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( N  -  2 )  e.  ( 0..^ ( # `  F ) ) )
10817, 18, 19, 20signstcl 26969 . . . . 5  |-  ( ( F  e. Word  RR  /\  ( N  -  2
)  e.  ( 0..^ ( # `  F
) ) )  -> 
( ( T `  F ) `  ( N  -  2 ) )  e.  { -u
1 ,  0 ,  1 } )
1092, 107, 108syl2anc 661 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( T `  F ) `  ( N  -  2 ) )  e.  { -u 1 ,  0 ,  1 } )
11017, 18signswrid 26962 . . . 4  |-  ( ( ( T `  F
) `  ( N  -  2 ) )  e.  { -u 1 ,  0 ,  1 }  ->  ( (
( T `  F
) `  ( N  -  2 ) ) 
.+^  0 )  =  ( ( T `  F ) `  ( N  -  2 ) ) )
111109, 110syl 16 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( (
( T `  F
) `  ( N  -  2 ) ) 
.+^  0 )  =  ( ( T `  F ) `  ( N  -  2 ) ) )
11294, 111eqtrd 2475 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( (
( T `  ( F substr  <. 0 ,  ( N  -  1 )
>. ) ) `  (
( # `  ( F substr  <. 0 ,  ( N  -  1 ) >.
) )  -  1 ) )  .+^  (sgn `  ( F `  ( N  -  1 ) ) ) )  =  ( ( T `  F
) `  ( N  -  2 ) ) )
11353, 69, 1123eqtr3d 2483 1  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( T `  F ) `  ( N  -  1 ) )  =  ( ( T `  F
) `  ( N  -  2 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2609    \ cdif 3328   (/)c0 3640   ifcif 3794   {csn 3880   {cpr 3882   {ctp 3884   <.cop 3886   class class class wbr 4295    e. cmpt 4353   ` cfv 5421  (class class class)co 6094    e. cmpt2 6096   CCcc 9283   RRcr 9284   0cc0 9285   1c1 9286    + caddc 9288    < clt 9421    <_ cle 9422    - cmin 9598   -ucneg 9599   NNcn 10325   2c2 10374   NN0cn0 10582   ZZcz 10649   ZZ>=cuz 10864   RR+crp 10994   ...cfz 11440  ..^cfzo 11551   #chash 12106  Word cword 12224   concat cconcat 12226   <"cs1 12227   substr csubstr 12228  sgncsgn 12578   sum_csu 13166   ndxcnx 14174   Basecbs 14177   +g cplusg 14241    gsumg cgsu 14382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-1o 6923  df-oadd 6927  df-er 7104  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-card 8112  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-nn 10326  df-2 10383  df-n0 10583  df-z 10650  df-uz 10865  df-rp 10995  df-fz 11441  df-fzo 11552  df-seq 11810  df-hash 12107  df-word 12232  df-concat 12234  df-s1 12235  df-substr 12236  df-sgn 12579  df-struct 14179  df-ndx 14180  df-slot 14181  df-base 14182  df-plusg 14254  df-0g 14383  df-gsum 14384  df-mnd 15418
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator