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Theorem signstfveq0 29040
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 752 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  F  e.  (Word  RR  \  { (/) } ) )
21eldifad 3426 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  F  e. Word  RR )
3 swrdcl 12700 . . . . 5  |-  ( F  e. Word  RR  ->  ( F substr  <. 0 ,  ( N  -  1 ) >.
)  e. Word  RR )
42, 3syl 17 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( F substr  <.
0 ,  ( N  -  1 ) >.
)  e. Word  RR )
5 1nn0 10852 . . . . . . . . . . 11  |-  1  e.  NN0
65a1i 11 . . . . . . . . . 10  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  1  e.  NN0 )
76nn0red 10894 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  1  e.  RR )
8 2re 10646 . . . . . . . . . . . 12  |-  2  e.  RR
98a1i 11 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  2  e.  RR )
10 signstfveq0.1 . . . . . . . . . . . . 13  |-  N  =  ( # `  F
)
11 lencl 12614 . . . . . . . . . . . . . 14  |-  ( F  e. Word  RR  ->  ( # `  F )  e.  NN0 )
122, 11syl 17 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( # `  F
)  e.  NN0 )
1310, 12syl5eqel 2494 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  N  e.  NN0 )
1413nn0red 10894 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  N  e.  RR )
15 1le2 10790 . . . . . . . . . . . 12  |-  1  <_  2
1615a1i 11 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  1  <_  2 )
17 signsv.p . . . . . . . . . . . . . 14  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
18 signsv.w . . . . . . . . . . . . . 14  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
19 signsv.t . . . . . . . . . . . . . 14  |-  T  =  ( f  e. Word  RR  |->  ( n  e.  (
0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) ) )
20 signsv.v . . . . . . . . . . . . . 14  |-  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 29039 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  N  e.  ( ZZ>= `  2 )
)
22 eluz2 11133 . . . . . . . . . . . . 13  |-  ( N  e.  ( ZZ>= `  2
)  <->  ( 2  e.  ZZ  /\  N  e.  ZZ  /\  2  <_  N ) )
2321, 22sylib 196 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( 2  e.  ZZ  /\  N  e.  ZZ  /\  2  <_  N ) )
2423simp3d 1011 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  2  <_  N )
257, 9, 14, 16, 24letrd 9773 . . . . . . . . . 10  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  1  <_  N )
26 fznn0 11825 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  ( 1  e.  ( 0 ... N )  <->  ( 1  e.  NN0  /\  1  <_  N ) ) )
2713, 26syl 17 . . . . . . . . . 10  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( 1  e.  ( 0 ... N )  <->  ( 1  e.  NN0  /\  1  <_  N ) ) )
286, 25, 27mpbir2and 923 . . . . . . . . 9  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  1  e.  ( 0 ... N
) )
29 fznn0sub2 11836 . . . . . . . . 9  |-  ( 1  e.  ( 0 ... N )  ->  ( N  -  1 )  e.  ( 0 ... N ) )
3028, 29syl 17 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( N  -  1 )  e.  ( 0 ... N
) )
3110oveq2i 6289 . . . . . . . 8  |-  ( 0 ... N )  =  ( 0 ... ( # `
 F ) )
3230, 31syl6eleq 2500 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( N  -  1 )  e.  ( 0 ... ( # `
 F ) ) )
33 swrd0len 12703 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  ( N  -  1
)  e.  ( 0 ... ( # `  F
) ) )  -> 
( # `  ( F substr  <. 0 ,  ( N  -  1 ) >.
) )  =  ( N  -  1 ) )
342, 32, 33syl2anc 659 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( # `  ( F substr  <. 0 ,  ( N  -  1 )
>. ) )  =  ( N  -  1 ) )
35 uz2m1nn 11201 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( N  -  1 )  e.  NN )
3621, 35syl 17 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( N  -  1 )  e.  NN )
3734, 36eqeltrd 2490 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( # `  ( F substr  <. 0 ,  ( N  -  1 )
>. ) )  e.  NN )
38 nnne0 10609 . . . . . 6  |-  ( (
# `  ( F substr  <.
0 ,  ( N  -  1 ) >.
) )  e.  NN  ->  ( # `  ( F substr  <. 0 ,  ( N  -  1 )
>. ) )  =/=  0
)
39 fveq2 5849 . . . . . . . 8  |-  ( ( F substr  <. 0 ,  ( N  -  1 )
>. )  =  (/)  ->  ( # `
 ( F substr  <. 0 ,  ( N  - 
1 ) >. )
)  =  ( # `  (/) ) )
40 hash0 12485 . . . . . . . 8  |-  ( # `  (/) )  =  0
4139, 40syl6eq 2459 . . . . . . 7  |-  ( ( F substr  <. 0 ,  ( N  -  1 )
>. )  =  (/)  ->  ( # `
 ( F substr  <. 0 ,  ( N  - 
1 ) >. )
)  =  0 )
4241necon3i 2643 . . . . . 6  |-  ( (
# `  ( F substr  <.
0 ,  ( N  -  1 ) >.
) )  =/=  0  ->  ( F substr  <. 0 ,  ( N  - 
1 ) >. )  =/=  (/) )
4338, 42syl 17 . . . . 5  |-  ( (
# `  ( F substr  <.
0 ,  ( N  -  1 ) >.
) )  e.  NN  ->  ( F substr  <. 0 ,  ( N  - 
1 ) >. )  =/=  (/) )
4437, 43syl 17 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( F substr  <.
0 ,  ( N  -  1 ) >.
)  =/=  (/) )
45 eldifsn 4097 . . . 4  |-  ( ( F substr  <. 0 ,  ( N  -  1 )
>. )  e.  (Word  RR  \  { (/) } )  <-> 
( ( F substr  <. 0 ,  ( N  - 
1 ) >. )  e. Word  RR  /\  ( F substr  <. 0 ,  ( N  -  1 ) >.
)  =/=  (/) ) )
464, 44, 45sylanbrc 662 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( F substr  <.
0 ,  ( N  -  1 ) >.
)  e.  (Word  RR  \  { (/) } ) )
47 simpr 459 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( F `  ( N  -  1 ) )  =  0 )
48 0re 9626 . . . 4  |-  0  e.  RR
4947, 48syl6eqel 2498 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( F `  ( N  -  1 ) )  e.  RR )
5017, 18, 19, 20signstfvn 29032 . . 3  |-  ( ( ( F substr  <. 0 ,  ( N  - 
1 ) >. )  e.  (Word  RR  \  { (/)
} )  /\  ( F `  ( N  -  1 ) )  e.  RR )  -> 
( ( T `  ( ( F substr  <. 0 ,  ( N  - 
1 ) >. ) ++  <" ( F `  ( N  -  1
) ) "> ) ) `  ( # `
 ( F substr  <. 0 ,  ( N  - 
1 ) >. )
) )  =  ( ( ( T `  ( F substr  <. 0 ,  ( N  -  1 ) >. ) ) `  ( ( # `  ( F substr  <. 0 ,  ( N  -  1 )
>. ) )  -  1 ) )  .+^  (sgn `  ( F `  ( N  -  1 ) ) ) ) )
5146, 49, 50syl2anc 659 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( T `  ( ( F substr  <. 0 ,  ( N  -  1 )
>. ) ++  <" ( F `  ( N  -  1 ) ) "> ) ) `
 ( # `  ( F substr  <. 0 ,  ( N  -  1 )
>. ) ) )  =  ( ( ( T `
 ( F substr  <. 0 ,  ( N  - 
1 ) >. )
) `  ( ( # `
 ( F substr  <. 0 ,  ( N  - 
1 ) >. )
)  -  1 ) )  .+^  (sgn `  ( F `  ( N  -  1 ) ) ) ) )
5210oveq1i 6288 . . . . . . . . 9  |-  ( N  -  1 )  =  ( ( # `  F
)  -  1 )
5352opeq2i 4163 . . . . . . . 8  |-  <. 0 ,  ( N  - 
1 ) >.  =  <. 0 ,  ( ( # `
 F )  - 
1 ) >.
5453oveq2i 6289 . . . . . . 7  |-  ( F substr  <. 0 ,  ( N  -  1 ) >.
)  =  ( F substr  <. 0 ,  ( (
# `  F )  -  1 ) >.
)
5554a1i 11 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( F substr  <.
0 ,  ( N  -  1 ) >.
)  =  ( F substr  <. 0 ,  ( (
# `  F )  -  1 ) >.
) )
56 lsw 12638 . . . . . . . . . 10  |-  ( F  e.  (Word  RR  \  { (/) } )  -> 
( lastS  `  F )  =  ( F `  (
( # `  F )  -  1 ) ) )
5756ad2antrr 724 . . . . . . . . 9  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( lastS  `  F
)  =  ( F `
 ( ( # `  F )  -  1 ) ) )
5810eqcomi 2415 . . . . . . . . . . 11  |-  ( # `  F )  =  N
5958oveq1i 6288 . . . . . . . . . 10  |-  ( (
# `  F )  -  1 )  =  ( N  -  1 )
6059fveq2i 5852 . . . . . . . . 9  |-  ( F `
 ( ( # `  F )  -  1 ) )  =  ( F `  ( N  -  1 ) )
6157, 60syl6eq 2459 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( lastS  `  F
)  =  ( F `
 ( N  - 
1 ) ) )
6261s1eqd 12667 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  <" ( lastS  `  F ) ">  =  <" ( F `
 ( N  - 
1 ) ) "> )
6362eqcomd 2410 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  <" ( F `  ( N  -  1 ) ) ">  =  <" ( lastS  `  F ) "> )
6455, 63oveq12d 6296 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( F substr  <. 0 ,  ( N  -  1 )
>. ) ++  <" ( F `  ( N  -  1 ) ) "> )  =  ( ( F substr  <. 0 ,  ( ( # `  F )  -  1 ) >. ) ++  <" ( lastS  `  F ) "> ) )
65 eldifsn 4097 . . . . . . 7  |-  ( F  e.  (Word  RR  \  { (/) } )  <->  ( F  e. Word  RR  /\  F  =/=  (/) ) )
661, 65sylib 196 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( F  e. Word  RR  /\  F  =/=  (/) ) )
67 swrdccatwrd 12749 . . . . . 6  |-  ( ( F  e. Word  RR  /\  F  =/=  (/) )  ->  (
( F substr  <. 0 ,  ( ( # `  F
)  -  1 )
>. ) ++  <" ( lastS  `  F ) "> )  =  F )
6866, 67syl 17 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( F substr  <. 0 ,  ( ( # `  F
)  -  1 )
>. ) ++  <" ( lastS  `  F ) "> )  =  F )
6964, 68eqtrd 2443 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( F substr  <. 0 ,  ( N  -  1 )
>. ) ++  <" ( F `  ( N  -  1 ) ) "> )  =  F )
7069fveq2d 5853 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( T `  ( ( F substr  <. 0 ,  ( N  - 
1 ) >. ) ++  <" ( F `  ( N  -  1
) ) "> ) )  =  ( T `  F ) )
7170, 34fveq12d 5855 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( T `  ( ( F substr  <. 0 ,  ( N  -  1 )
>. ) ++  <" ( F `  ( N  -  1 ) ) "> ) ) `
 ( # `  ( F substr  <. 0 ,  ( N  -  1 )
>. ) ) )  =  ( ( T `  F ) `  ( N  -  1 ) ) )
7213nn0cnd 10895 . . . . . . . . . 10  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  N  e.  CC )
73 1cnd 9642 . . . . . . . . . 10  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  1  e.  CC )
7472, 73, 73subsub4d 9998 . . . . . . . . 9  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( N  -  1 )  -  1 )  =  ( N  -  (
1  +  1 ) ) )
75 1p1e2 10690 . . . . . . . . . 10  |-  ( 1  +  1 )  =  2
7675oveq2i 6289 . . . . . . . . 9  |-  ( N  -  ( 1  +  1 ) )  =  ( N  -  2 )
7774, 76syl6eq 2459 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( N  -  1 )  -  1 )  =  ( N  -  2 ) )
78 fzo0end 11941 . . . . . . . . 9  |-  ( ( N  -  1 )  e.  NN  ->  (
( N  -  1 )  -  1 )  e.  ( 0..^ ( N  -  1 ) ) )
7936, 78syl 17 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( N  -  1 )  -  1 )  e.  ( 0..^ ( N  -  1 ) ) )
8077, 79eqeltrrd 2491 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( N  -  2 )  e.  ( 0..^ ( N  -  1 ) ) )
8134oveq2d 6294 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( 0..^ ( # `  ( F substr  <. 0 ,  ( N  -  1 )
>. ) ) )  =  ( 0..^ ( N  -  1 ) ) )
8280, 81eleqtrrd 2493 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( N  -  2 )  e.  ( 0..^ ( # `  ( F substr  <. 0 ,  ( N  - 
1 ) >. )
) ) )
8317, 18, 19, 20signstfvp 29034 . . . . . 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 ) >. ) ++  <" ( F `  ( N  -  1
) ) "> ) ) `  ( N  -  2 ) )  =  ( ( T `  ( F substr  <. 0 ,  ( N  -  1 ) >.
) ) `  ( N  -  2 ) ) )
844, 49, 82, 83syl3anc 1230 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( T `  ( ( F substr  <. 0 ,  ( N  -  1 )
>. ) ++  <" ( F `  ( N  -  1 ) ) "> ) ) `
 ( N  - 
2 ) )  =  ( ( T `  ( F substr  <. 0 ,  ( N  -  1 ) >. ) ) `  ( N  -  2
) ) )
8569eqcomd 2410 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  F  =  ( ( F substr  <. 0 ,  ( N  - 
1 ) >. ) ++  <" ( F `  ( N  -  1
) ) "> ) )
8685fveq2d 5853 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( T `  F )  =  ( T `  ( ( F substr  <. 0 ,  ( N  -  1 )
>. ) ++  <" ( F `  ( N  -  1 ) ) "> ) ) )
8786fveq1d 5851 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( T `  F ) `  ( N  -  2 ) )  =  ( ( T `  (
( F substr  <. 0 ,  ( N  -  1 ) >. ) ++  <" ( F `  ( N  -  1 ) ) "> ) ) `
 ( N  - 
2 ) ) )
8834oveq1d 6293 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( # `
 ( F substr  <. 0 ,  ( N  - 
1 ) >. )
)  -  1 )  =  ( ( N  -  1 )  - 
1 ) )
8988, 74eqtrd 2443 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( # `
 ( F substr  <. 0 ,  ( N  - 
1 ) >. )
)  -  1 )  =  ( N  -  ( 1  +  1 ) ) )
9089, 76syl6eq 2459 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( # `
 ( F substr  <. 0 ,  ( N  - 
1 ) >. )
)  -  1 )  =  ( N  - 
2 ) )
9190fveq2d 5853 . . . . 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 ) ) )
9284, 87, 913eqtr4rd 2454 . . . 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 ) ) )
93 fveq2 5849 . . . . . 6  |-  ( ( F `  ( N  -  1 ) )  =  0  ->  (sgn `  ( F `  ( N  -  1 ) ) )  =  (sgn
`  0 ) )
94 sgn0 13071 . . . . . 6  |-  (sgn ` 
0 )  =  0
9593, 94syl6eq 2459 . . . . 5  |-  ( ( F `  ( N  -  1 ) )  =  0  ->  (sgn `  ( F `  ( N  -  1 ) ) )  =  0 )
9695adantl 464 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  (sgn `  ( F `  ( N  -  1 ) ) )  =  0 )
9792, 96oveq12d 6296 . . 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 ) )
98 uznn0sub 11158 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( N  -  2 )  e. 
NN0 )
9921, 98syl 17 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( N  -  2 )  e. 
NN0 )
100 eluz2nn 11165 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  NN )
10121, 100syl 17 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  N  e.  NN )
102 2rp 11270 . . . . . . . . 9  |-  2  e.  RR+
103102a1i 11 . . . . . . . 8  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  2  e.  RR+ )
10414, 103ltsubrpd 11332 . . . . . . 7  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( N  -  2 )  < 
N )
105 elfzo0 11895 . . . . . . 7  |-  ( ( N  -  2 )  e.  ( 0..^ N )  <->  ( ( N  -  2 )  e. 
NN0  /\  N  e.  NN  /\  ( N  - 
2 )  <  N
) )
10699, 101, 104, 105syl3anbrc 1181 . . . . . 6  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( N  -  2 )  e.  ( 0..^ N ) )
10710oveq2i 6289 . . . . . 6  |-  ( 0..^ N )  =  ( 0..^ ( # `  F
) )
108106, 107syl6eleq 2500 . . . . 5  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( N  -  2 )  e.  ( 0..^ ( # `  F ) ) )
10917, 18, 19, 20signstcl 29028 . . . . 5  |-  ( ( F  e. Word  RR  /\  ( N  -  2
)  e.  ( 0..^ ( # `  F
) ) )  -> 
( ( T `  F ) `  ( N  -  2 ) )  e.  { -u
1 ,  0 ,  1 } )
1102, 108, 109syl2anc 659 . . . 4  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( ( T `  F ) `  ( N  -  2 ) )  e.  { -u 1 ,  0 ,  1 } )
11117, 18signswrid 29021 . . . 4  |-  ( ( ( T `  F
) `  ( N  -  2 ) )  e.  { -u 1 ,  0 ,  1 }  ->  ( (
( T `  F
) `  ( N  -  2 ) ) 
.+^  0 )  =  ( ( T `  F ) `  ( N  -  2 ) ) )
112110, 111syl 17 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  ( F `  ( N  -  1 ) )  =  0 )  ->  ( (
( T `  F
) `  ( N  -  2 ) ) 
.+^  0 )  =  ( ( T `  F ) `  ( N  -  2 ) ) )
11397, 112eqtrd 2443 . 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 ) ) )
11451, 71, 1133eqtr3d 2451 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598    \ cdif 3411   (/)c0 3738   ifcif 3885   {csn 3972   {cpr 3974   {ctp 3976   <.cop 3978   class class class wbr 4395    |-> cmpt 4453   ` cfv 5569  (class class class)co 6278    |-> cmpt2 6280   RRcr 9521   0cc0 9522   1c1 9523    + caddc 9525    < clt 9658    <_ cle 9659    - cmin 9841   -ucneg 9842   NNcn 10576   2c2 10626   NN0cn0 10836   ZZcz 10905   ZZ>=cuz 11127   RR+crp 11265   ...cfz 11726  ..^cfzo 11854   #chash 12452  Word cword 12583   lastS clsw 12584   ++ cconcat 12585   <"cs1 12586   substr csubstr 12587  sgncsgn 13068   sum_csu 13657   ndxcnx 14838   Basecbs 14841   +g cplusg 14909    gsumg cgsu 15055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-fz 11727  df-fzo 11855  df-seq 12152  df-hash 12453  df-word 12591  df-lsw 12592  df-concat 12593  df-s1 12594  df-substr 12595  df-sgn 13069  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-plusg 14922  df-0g 15056  df-gsum 15057  df-mgm 16196  df-sgrp 16235  df-mnd 16245
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator