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Theorem signstfvneq0 29463
Description: In case the first letter is not zero, the zero skipping sign is never zero. (Contributed by Thierry Arnoux, 10-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 ) )
Assertion
Ref Expression
signstfvneq0  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( ( T `  F ) `  N
)  =/=  0 )
Distinct variable groups:    a, b,  .+^    f, i, n, F    f, W, i, n    i, N, n    n, a, T, b
Allowed substitution hints:    .+^ ( f, i,
j, n)    T( f,
i, j)    F( j,
a, b)    N( f,
j, a, b)    V( f, i, j, n, a, b)    W( j, a, b)

Proof of Theorem signstfvneq0
Dummy variables  e 
k  m  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 759 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) )  ->  F  e.  (Word  RR  \  { (/) } ) )
21eldifad 3449 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) )  ->  F  e. Word  RR )
3 eldifsni 4124 . . . 4  |-  ( F  e.  (Word  RR  \  { (/) } )  ->  F  =/=  (/) )
43ad2antrr 731 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) )  ->  F  =/=  (/) )
5 simplr 761 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( F `  0
)  =/=  0 )
64, 5jca 535 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( F  =/=  (/)  /\  ( F `  0 )  =/=  0 ) )
7 simpr 463 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) )  ->  N  e.  ( 0..^ ( # `  F
) ) )
8 simprr 765 . . 3  |-  ( ( F  e. Word  RR  /\  ( ( F  =/=  (/)  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  N  e.  ( 0..^ ( # `  F
) ) )
9 neeq1 2706 . . . . . . . 8  |-  ( g  =  (/)  ->  ( g  =/=  (/)  <->  (/)  =/=  (/) ) )
10 fveq1 5878 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( g `
 0 )  =  ( (/) `  0 ) )
1110neeq1d 2702 . . . . . . . 8  |-  ( g  =  (/)  ->  ( ( g `  0 )  =/=  0  <->  ( (/) `  0
)  =/=  0 ) )
129, 11anbi12d 716 . . . . . . 7  |-  ( g  =  (/)  ->  ( ( g  =/=  (/)  /\  (
g `  0 )  =/=  0 )  <->  ( (/)  =/=  (/)  /\  ( (/) `  0 )  =/=  0 ) ) )
13 fveq2 5879 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( # `  g )  =  (
# `  (/) ) )
1413oveq2d 6319 . . . . . . . 8  |-  ( g  =  (/)  ->  ( 0..^ ( # `  g
) )  =  ( 0..^ ( # `  (/) ) ) )
15 fveq2 5879 . . . . . . . . . 10  |-  ( g  =  (/)  ->  ( T `
 g )  =  ( T `  (/) ) )
1615fveq1d 5881 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( ( T `  g ) `
 m )  =  ( ( T `  (/) ) `  m ) )
1716neeq1d 2702 . . . . . . . 8  |-  ( g  =  (/)  ->  ( ( ( T `  g
) `  m )  =/=  0  <->  ( ( T `
 (/) ) `  m
)  =/=  0 ) )
1814, 17raleqbidv 3040 . . . . . . 7  |-  ( g  =  (/)  ->  ( A. m  e.  ( 0..^ ( # `  g
) ) ( ( T `  g ) `
 m )  =/=  0  <->  A. m  e.  ( 0..^ ( # `  (/) ) ) ( ( T `  (/) ) `  m )  =/=  0 ) )
1912, 18imbi12d 322 . . . . . 6  |-  ( g  =  (/)  ->  ( ( ( g  =/=  (/)  /\  (
g `  0 )  =/=  0 )  ->  A. m  e.  ( 0..^ ( # `  g ) ) ( ( T `  g
) `  m )  =/=  0 )  <->  ( ( (/) 
=/=  (/)  /\  ( (/) `  0 )  =/=  0
)  ->  A. m  e.  ( 0..^ ( # `  (/) ) ) ( ( T `  (/) ) `  m )  =/=  0
) ) )
20 neeq1 2706 . . . . . . . 8  |-  ( g  =  e  ->  (
g  =/=  (/)  <->  e  =/=  (/) ) )
21 fveq1 5878 . . . . . . . . 9  |-  ( g  =  e  ->  (
g `  0 )  =  ( e ` 
0 ) )
2221neeq1d 2702 . . . . . . . 8  |-  ( g  =  e  ->  (
( g `  0
)  =/=  0  <->  (
e `  0 )  =/=  0 ) )
2320, 22anbi12d 716 . . . . . . 7  |-  ( g  =  e  ->  (
( g  =/=  (/)  /\  (
g `  0 )  =/=  0 )  <->  ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 ) ) )
24 fveq2 5879 . . . . . . . . 9  |-  ( g  =  e  ->  ( # `
 g )  =  ( # `  e
) )
2524oveq2d 6319 . . . . . . . 8  |-  ( g  =  e  ->  (
0..^ ( # `  g
) )  =  ( 0..^ ( # `  e
) ) )
26 fveq2 5879 . . . . . . . . . 10  |-  ( g  =  e  ->  ( T `  g )  =  ( T `  e ) )
2726fveq1d 5881 . . . . . . . . 9  |-  ( g  =  e  ->  (
( T `  g
) `  m )  =  ( ( T `
 e ) `  m ) )
2827neeq1d 2702 . . . . . . . 8  |-  ( g  =  e  ->  (
( ( T `  g ) `  m
)  =/=  0  <->  (
( T `  e
) `  m )  =/=  0 ) )
2925, 28raleqbidv 3040 . . . . . . 7  |-  ( g  =  e  ->  ( A. m  e.  (
0..^ ( # `  g
) ) ( ( T `  g ) `
 m )  =/=  0  <->  A. m  e.  ( 0..^ ( # `  e
) ) ( ( T `  e ) `
 m )  =/=  0 ) )
3023, 29imbi12d 322 . . . . . 6  |-  ( g  =  e  ->  (
( ( g  =/=  (/)  /\  ( g ` 
0 )  =/=  0
)  ->  A. m  e.  ( 0..^ ( # `  g ) ) ( ( T `  g
) `  m )  =/=  0 )  <->  ( (
e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. m  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  m )  =/=  0 ) ) )
31 neeq1 2706 . . . . . . . 8  |-  ( g  =  ( e ++  <" k "> )  ->  ( g  =/=  (/)  <->  ( e ++  <" k "> )  =/=  (/) ) )
32 fveq1 5878 . . . . . . . . 9  |-  ( g  =  ( e ++  <" k "> )  ->  ( g `  0
)  =  ( ( e ++  <" k "> ) `  0
) )
3332neeq1d 2702 . . . . . . . 8  |-  ( g  =  ( e ++  <" k "> )  ->  ( ( g ` 
0 )  =/=  0  <->  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )
3431, 33anbi12d 716 . . . . . . 7  |-  ( g  =  ( e ++  <" k "> )  ->  ( ( g  =/=  (/)  /\  ( g ` 
0 )  =/=  0
)  <->  ( ( e ++ 
<" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) ) )
35 fveq2 5879 . . . . . . . . 9  |-  ( g  =  ( e ++  <" k "> )  ->  ( # `  g
)  =  ( # `  ( e ++  <" k "> ) ) )
3635oveq2d 6319 . . . . . . . 8  |-  ( g  =  ( e ++  <" k "> )  ->  ( 0..^ ( # `  g ) )  =  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )
37 fveq2 5879 . . . . . . . . . 10  |-  ( g  =  ( e ++  <" k "> )  ->  ( T `  g
)  =  ( T `
 ( e ++  <" k "> )
) )
3837fveq1d 5881 . . . . . . . . 9  |-  ( g  =  ( e ++  <" k "> )  ->  ( ( T `  g ) `  m
)  =  ( ( T `  ( e ++ 
<" k "> ) ) `  m
) )
3938neeq1d 2702 . . . . . . . 8  |-  ( g  =  ( e ++  <" k "> )  ->  ( ( ( T `
 g ) `  m )  =/=  0  <->  ( ( T `  (
e ++  <" k "> ) ) `  m )  =/=  0
) )
4036, 39raleqbidv 3040 . . . . . . 7  |-  ( g  =  ( e ++  <" k "> )  ->  ( A. m  e.  ( 0..^ ( # `  g ) ) ( ( T `  g
) `  m )  =/=  0  <->  A. m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ( ( T `  ( e ++  <" k "> ) ) `  m )  =/=  0
) )
4134, 40imbi12d 322 . . . . . 6  |-  ( g  =  ( e ++  <" k "> )  ->  ( ( ( g  =/=  (/)  /\  ( g `
 0 )  =/=  0 )  ->  A. m  e.  ( 0..^ ( # `  g ) ) ( ( T `  g
) `  m )  =/=  0 )  <->  ( (
( e ++  <" k "> )  =/=  (/)  /\  (
( e ++  <" k "> ) `  0
)  =/=  0 )  ->  A. m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ( ( T `  ( e ++  <" k "> ) ) `  m )  =/=  0
) ) )
42 neeq1 2706 . . . . . . . 8  |-  ( g  =  F  ->  (
g  =/=  (/)  <->  F  =/=  (/) ) )
43 fveq1 5878 . . . . . . . . 9  |-  ( g  =  F  ->  (
g `  0 )  =  ( F ` 
0 ) )
4443neeq1d 2702 . . . . . . . 8  |-  ( g  =  F  ->  (
( g `  0
)  =/=  0  <->  ( F `  0 )  =/=  0 ) )
4542, 44anbi12d 716 . . . . . . 7  |-  ( g  =  F  ->  (
( g  =/=  (/)  /\  (
g `  0 )  =/=  0 )  <->  ( F  =/=  (/)  /\  ( F `
 0 )  =/=  0 ) ) )
46 fveq2 5879 . . . . . . . . 9  |-  ( g  =  F  ->  ( # `
 g )  =  ( # `  F
) )
4746oveq2d 6319 . . . . . . . 8  |-  ( g  =  F  ->  (
0..^ ( # `  g
) )  =  ( 0..^ ( # `  F
) ) )
48 fveq2 5879 . . . . . . . . . 10  |-  ( g  =  F  ->  ( T `  g )  =  ( T `  F ) )
4948fveq1d 5881 . . . . . . . . 9  |-  ( g  =  F  ->  (
( T `  g
) `  m )  =  ( ( T `
 F ) `  m ) )
5049neeq1d 2702 . . . . . . . 8  |-  ( g  =  F  ->  (
( ( T `  g ) `  m
)  =/=  0  <->  (
( T `  F
) `  m )  =/=  0 ) )
5147, 50raleqbidv 3040 . . . . . . 7  |-  ( g  =  F  ->  ( A. m  e.  (
0..^ ( # `  g
) ) ( ( T `  g ) `
 m )  =/=  0  <->  A. m  e.  ( 0..^ ( # `  F
) ) ( ( T `  F ) `
 m )  =/=  0 ) )
5245, 51imbi12d 322 . . . . . 6  |-  ( g  =  F  ->  (
( ( g  =/=  (/)  /\  ( g ` 
0 )  =/=  0
)  ->  A. m  e.  ( 0..^ ( # `  g ) ) ( ( T `  g
) `  m )  =/=  0 )  <->  ( ( F  =/=  (/)  /\  ( F `
 0 )  =/=  0 )  ->  A. m  e.  ( 0..^ ( # `  F ) ) ( ( T `  F
) `  m )  =/=  0 ) ) )
53 neirr 2629 . . . . . . . 8  |-  -.  (/)  =/=  (/)
5453intnanr 924 . . . . . . 7  |-  -.  ( (/) 
=/=  (/)  /\  ( (/) `  0 )  =/=  0
)
5554pm2.21i 135 . . . . . 6  |-  ( (
(/)  =/=  (/)  /\  ( (/) `  0 )  =/=  0 )  ->  A. m  e.  ( 0..^ ( # `  (/) ) ) ( ( T `  (/) ) `  m )  =/=  0
)
56 fveq2 5879 . . . . . . . . . . . 12  |-  ( n  =  m  ->  (
( T `  e
) `  n )  =  ( ( T `
 e ) `  m ) )
5756neeq1d 2702 . . . . . . . . . . 11  |-  ( n  =  m  ->  (
( ( T `  e ) `  n
)  =/=  0  <->  (
( T `  e
) `  m )  =/=  0 ) )
5857cbvralv 3056 . . . . . . . . . 10  |-  ( A. n  e.  ( 0..^ ( # `  e
) ) ( ( T `  e ) `
 n )  =/=  0  <->  A. m  e.  ( 0..^ ( # `  e
) ) ( ( T `  e ) `
 m )  =/=  0 )
5958imbi2i 314 . . . . . . . . 9  |-  ( ( ( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 )  <->  ( (
e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. m  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  m )  =/=  0 ) )
6059anbi2i 699 . . . . . . . 8  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  <->  ( (
e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. m  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  m )  =/=  0 ) ) )
61 simplr 761 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =  (/) )  ->  m  e.  ( 0..^ ( # `  e
) ) )
62 noel 3766 . . . . . . . . . . . . . 14  |-  -.  m  e.  (/)
63 fveq2 5879 . . . . . . . . . . . . . . . . . 18  |-  ( e  =  (/)  ->  ( # `  e )  =  (
# `  (/) ) )
64 hash0 12549 . . . . . . . . . . . . . . . . . 18  |-  ( # `  (/) )  =  0
6563, 64syl6eq 2480 . . . . . . . . . . . . . . . . 17  |-  ( e  =  (/)  ->  ( # `  e )  =  0 )
6665oveq2d 6319 . . . . . . . . . . . . . . . 16  |-  ( e  =  (/)  ->  ( 0..^ ( # `  e
) )  =  ( 0..^ 0 ) )
67 fzo0 11944 . . . . . . . . . . . . . . . 16  |-  ( 0..^ 0 )  =  (/)
6866, 67syl6eq 2480 . . . . . . . . . . . . . . 15  |-  ( e  =  (/)  ->  ( 0..^ ( # `  e
) )  =  (/) )
6968eleq2d 2493 . . . . . . . . . . . . . 14  |-  ( e  =  (/)  ->  ( m  e.  ( 0..^ (
# `  e )
)  <->  m  e.  (/) ) )
7062, 69mtbiri 305 . . . . . . . . . . . . 13  |-  ( e  =  (/)  ->  -.  m  e.  ( 0..^ ( # `  e ) ) )
7170adantl 468 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =  (/) )  ->  -.  m  e.  ( 0..^ ( # `  e
) ) )
7261, 71pm2.21dd 178 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =  (/) )  ->  ( ( T `
 ( e ++  <" k "> )
) `  m )  =/=  0 )
73 simp-6l 779 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =/=  (/) )  -> 
e  e. Word  RR )
74 simp-6r 780 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =/=  (/) )  -> 
k  e.  RR )
75 simplr 761 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =/=  (/) )  ->  m  e.  ( 0..^ ( # `  e
) ) )
76 signsv.p . . . . . . . . . . . . . 14  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
77 signsv.w . . . . . . . . . . . . . 14  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
78 signsv.t . . . . . . . . . . . . . 14  |-  T  =  ( f  e. Word  RR  |->  ( n  e.  (
0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) ) )
79 signsv.v . . . . . . . . . . . . . 14  |-  V  =  ( f  e. Word  RR  |->  sum_ j  e.  ( 1..^ ( # `  f
) ) if ( ( ( T `  f ) `  j
)  =/=  ( ( T `  f ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
8076, 77, 78, 79signstfvp 29462 . . . . . . . . . . . . 13  |-  ( ( e  e. Word  RR  /\  k  e.  RR  /\  m  e.  ( 0..^ ( # `  e ) ) )  ->  ( ( T `
 ( e ++  <" k "> )
) `  m )  =  ( ( T `
 e ) `  m ) )
8173, 74, 75, 80syl3anc 1265 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =/=  (/) )  -> 
( ( T `  ( e ++  <" k "> ) ) `  m )  =  ( ( T `  e
) `  m )
)
82 simpr 463 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =/=  (/) )  -> 
e  =/=  (/) )
83 simplll 767 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  ->  ( e  e. Word  RR  /\  k  e.  RR ) )
8483ad2antrr 731 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =/=  (/) )  -> 
( e  e. Word  RR  /\  k  e.  RR ) )
85 simplrr 770 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  ( m  e.  ( 0..^ (
# `  ( e ++  <" k "> ) ) )  /\  m  e.  ( 0..^ ( # `  e
) )  /\  e  =/=  (/) ) )  -> 
( ( e ++  <" k "> ) `  0 )  =/=  0 )
86853anassrs 1229 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =/=  (/) )  -> 
( ( e ++  <" k "> ) `  0 )  =/=  0 )
87 simpll 759 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  -> 
e  e. Word  RR )
88 simplr 761 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  -> 
k  e.  RR )
8988s1cld 12740 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  ->  <" k ">  e. Word  RR )
90 lennncl 12686 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  e. Word  RR  /\  e  =/=  (/) )  ->  ( # `
 e )  e.  NN )
9190adantlr 720 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  -> 
( # `  e )  e.  NN )
92 fzo0end 12004 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  e )  e.  NN  ->  ( ( # `
 e )  - 
1 )  e.  ( 0..^ ( # `  e
) ) )
93 elfzolt3b 11934 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( # `  e
)  -  1 )  e.  ( 0..^ (
# `  e )
)  ->  0  e.  ( 0..^ ( # `  e
) ) )
9491, 92, 933syl 18 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  -> 
0  e.  ( 0..^ ( # `  e
) ) )
95 ccatval1 12720 . . . . . . . . . . . . . . . . . 18  |-  ( ( e  e. Word  RR  /\  <" k ">  e. Word  RR  /\  0  e.  ( 0..^ ( # `  e ) ) )  ->  ( ( e ++ 
<" k "> ) `  0 )  =  ( e ` 
0 ) )
9687, 89, 94, 95syl3anc 1265 . . . . . . . . . . . . . . . . 17  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  -> 
( ( e ++  <" k "> ) `  0 )  =  ( e `  0
) )
9796neeq1d 2702 . . . . . . . . . . . . . . . 16  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  -> 
( ( ( e ++ 
<" k "> ) `  0 )  =/=  0  <->  ( e ` 
0 )  =/=  0
) )
9897biimpa 487 . . . . . . . . . . . . . . 15  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  /\  ( ( e ++  <" k "> ) `  0 )  =/=  0 )  ->  (
e `  0 )  =/=  0 )
9984, 82, 86, 98syl21anc 1264 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =/=  (/) )  -> 
( e `  0
)  =/=  0 )
100 simp-5r 778 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =/=  (/) )  -> 
( ( e  =/=  (/)  /\  ( e ` 
0 )  =/=  0
)  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )
10182, 99, 100mp2and 684 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =/=  (/) )  ->  A. n  e.  (
0..^ ( # `  e
) ) ( ( T `  e ) `
 n )  =/=  0 )
10257rspcva 3181 . . . . . . . . . . . . 13  |-  ( ( m  e.  ( 0..^ ( # `  e
) )  /\  A. n  e.  ( 0..^ ( # `  e
) ) ( ( T `  e ) `
 n )  =/=  0 )  ->  (
( T `  e
) `  m )  =/=  0 )
10375, 101, 102syl2anc 666 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =/=  (/) )  -> 
( ( T `  e ) `  m
)  =/=  0 )
10481, 103eqnetrd 2718 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =/=  (/) )  -> 
( ( T `  ( e ++  <" k "> ) ) `  m )  =/=  0
)
10572, 104pm2.61dane 2743 . . . . . . . . . 10  |-  ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e ` 
0 )  =/=  0
)  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  ->  ( ( T `
 ( e ++  <" k "> )
) `  m )  =/=  0 )
106 simpr 463 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e ` 
0 )  =/=  0
)  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  /\  m  =  ( # `  e
) )  ->  m  =  ( # `  e
) )
107106fveq2d 5883 . . . . . . . . . . 11  |-  ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e ` 
0 )  =/=  0
)  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  /\  m  =  ( # `  e
) )  ->  (
( T `  (
e ++  <" k "> ) ) `  m )  =  ( ( T `  (
e ++  <" k "> ) ) `  ( # `  e ) ) )
108 simpr 463 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e ++ 
<" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  e  =  (/) )  ->  e  =  (/) )
109 simp-4r 776 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e ++ 
<" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  e  =  (/) )  ->  k  e.  RR )
110 simplrl 769 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e ++ 
<" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  e  =  (/) )  ->  (
( e ++  <" k "> )  =/=  (/)  /\  (
( e ++  <" k "> ) `  0
)  =/=  0 ) )
111110simprd 465 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e ++ 
<" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  e  =  (/) )  ->  (
( e ++  <" k "> ) `  0
)  =/=  0 )
112 oveq1 6310 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( e  =  (/)  ->  ( e ++ 
<" k "> )  =  ( (/) ++  <" k "> ) )
113 s1cl 12739 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  e.  RR  ->  <" k ">  e. Word  RR )
114 ccatlid 12728 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <" k ">  e. Word  RR  ->  ( (/) ++  <" k "> )  =  <" k "> )
115113, 114syl 17 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  e.  RR  ->  ( (/) ++  <" k "> )  =  <" k "> )
116112, 115sylan9eq 2484 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( e  =  (/)  /\  k  e.  RR )  ->  (
e ++  <" k "> )  =  <" k "> )
117116fveq2d 5883 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  =  (/)  /\  k  e.  RR )  ->  ( T `  ( e ++  <" k "> ) )  =  ( T `  <" k "> ) )
118117adantr 467 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  ( T `  ( e ++  <" k "> ) )  =  ( T `  <" k "> ) )
119 simplr 761 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  k  e.  RR )
12076, 77, 78, 79signstf0 29459 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  e.  RR  ->  ( T `  <" k "> )  =  <" (sgn `  k ) "> )
121119, 120syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  ( T `  <" k "> )  =  <" (sgn `  k ) "> )
122118, 121eqtrd 2464 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  ( T `  ( e ++  <" k "> ) )  =  <" (sgn `  k ) "> )
12365ad2antrr 731 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  ( # `
 e )  =  0 )
124122, 123fveq12d 5885 . . . . . . . . . . . . . . . . 17  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  (
( T `  (
e ++  <" k "> ) ) `  ( # `  e ) )  =  ( <" (sgn `  k ) "> `  0 )
)
125 sgnclre 29412 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  RR  ->  (sgn `  k )  e.  RR )
126 s1fv 12744 . . . . . . . . . . . . . . . . . 18  |-  ( (sgn
`  k )  e.  RR  ->  ( <" (sgn `  k ) "> `  0 )  =  (sgn `  k )
)
127119, 125, 1263syl 18 . . . . . . . . . . . . . . . . 17  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  ( <" (sgn `  k
) "> `  0
)  =  (sgn `  k ) )
128124, 127eqtrd 2464 . . . . . . . . . . . . . . . 16  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  (
( T `  (
e ++  <" k "> ) ) `  ( # `  e ) )  =  (sgn `  k ) )
129116fveq1d 5881 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  =  (/)  /\  k  e.  RR )  ->  (
( e ++  <" k "> ) `  0
)  =  ( <" k "> `  0 ) )
130 s1fv 12744 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  e.  RR  ->  ( <" k "> `  0 )  =  k )
131130adantl 468 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  =  (/)  /\  k  e.  RR )  ->  ( <" k "> `  0 )  =  k )
132129, 131eqtrd 2464 . . . . . . . . . . . . . . . . . . 19  |-  ( ( e  =  (/)  /\  k  e.  RR )  ->  (
( e ++  <" k "> ) `  0
)  =  k )
133132neeq1d 2702 . . . . . . . . . . . . . . . . . 18  |-  ( ( e  =  (/)  /\  k  e.  RR )  ->  (
( ( e ++  <" k "> ) `  0 )  =/=  0  <->  k  =/=  0
) )
134133biimpa 487 . . . . . . . . . . . . . . . . 17  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  k  =/=  0 )
135 rexr 9688 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  e.  RR  ->  k  e.  RR* )
136 sgn0bi 29420 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  e.  RR*  ->  ( (sgn
`  k )  =  0  <->  k  =  0 ) )
137135, 136syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  RR  ->  (
(sgn `  k )  =  0  <->  k  = 
0 ) )
138137necon3bid 2683 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  RR  ->  (
(sgn `  k )  =/=  0  <->  k  =/=  0
) )
139138biimpar 488 . . . . . . . . . . . . . . . . 17  |-  ( ( k  e.  RR  /\  k  =/=  0 )  -> 
(sgn `  k )  =/=  0 )
140119, 134, 139syl2anc 666 . . . . . . . . . . . . . . . 16  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  (sgn `  k )  =/=  0
)
141128, 140eqnetrd 2718 . . . . . . . . . . . . . . 15  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  (
( T `  (
e ++  <" k "> ) ) `  ( # `  e ) )  =/=  0 )
142108, 109, 111, 141syl21anc 1264 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e ++ 
<" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  e  =  (/) )  ->  (
( T `  (
e ++  <" k "> ) ) `  ( # `  e ) )  =/=  0 )
143 simplll 767 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e ++  <" k "> )  =/=  (/)  /\  (
( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
e  e. Word  RR )
144 simpr 463 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e ++  <" k "> )  =/=  (/)  /\  (
( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  ->  -.  e  =  (/) )
145 elsn 4011 . . . . . . . . . . . . . . . . . . 19  |-  ( e  e.  { (/) }  <->  e  =  (/) )
146144, 145sylnibr 307 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e ++  <" k "> )  =/=  (/)  /\  (
( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  ->  -.  e  e.  { (/) } )
147143, 146eldifd 3448 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e ++  <" k "> )  =/=  (/)  /\  (
( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
e  e.  (Word  RR  \  { (/) } ) )
148 simpllr 768 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e ++  <" k "> )  =/=  (/)  /\  (
( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
k  e.  RR )
14976, 77, 78, 79signstfvn 29460 . . . . . . . . . . . . . . . . 17  |-  ( ( e  e.  (Word  RR  \  { (/) } )  /\  k  e.  RR )  ->  ( ( T `  ( e ++  <" k "> ) ) `  ( # `  e ) )  =  ( ( ( T `  e
) `  ( ( # `
 e )  - 
1 ) )  .+^  (sgn `  k ) ) )
150147, 148, 149syl2anc 666 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e ++  <" k "> )  =/=  (/)  /\  (
( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( ( T `  ( e ++  <" k "> ) ) `  ( # `  e ) )  =  ( ( ( T `  e
) `  ( ( # `
 e )  - 
1 ) )  .+^  (sgn `  k ) ) )
151150adantllr 724 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e ++ 
<" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( ( T `  ( e ++  <" k "> ) ) `  ( # `  e ) )  =  ( ( ( T `  e
) `  ( ( # `
 e )  - 
1 ) )  .+^  (sgn `  k ) ) )
152144neqned 2628 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e ++  <" k "> )  =/=  (/)  /\  (
( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
e  =/=  (/) )
153143, 152, 90syl2anc 666 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e ++  <" k "> )  =/=  (/)  /\  (
( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( # `  e )  e.  NN )
154153, 92syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e ++  <" k "> )  =/=  (/)  /\  (
( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( ( # `  e
)  -  1 )  e.  ( 0..^ (
# `  e )
) )
15576, 77, 78, 79signstcl 29456 . . . . . . . . . . . . . . . . . 18  |-  ( ( e  e. Word  RR  /\  ( ( # `  e
)  -  1 )  e.  ( 0..^ (
# `  e )
) )  ->  (
( T `  e
) `  ( ( # `
 e )  - 
1 ) )  e. 
{ -u 1 ,  0 ,  1 } )
156143, 154, 155syl2anc 666 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e ++  <" k "> )  =/=  (/)  /\  (
( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( ( T `  e ) `  (
( # `  e )  -  1 ) )  e.  { -u 1 ,  0 ,  1 } )
157156adantllr 724 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e ++ 
<" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( ( T `  e ) `  (
( # `  e )  -  1 ) )  e.  { -u 1 ,  0 ,  1 } )
158148rexrd 9692 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e ++  <" k "> )  =/=  (/)  /\  (
( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
k  e.  RR* )
159 sgncl 29411 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  RR*  ->  (sgn `  k )  e.  { -u 1 ,  0 ,  1 } )
160158, 159syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e ++  <" k "> )  =/=  (/)  /\  (
( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
(sgn `  k )  e.  { -u 1 ,  0 ,  1 } )
161160adantllr 724 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e ++ 
<" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
(sgn `  k )  e.  { -u 1 ,  0 ,  1 } )
162154adantllr 724 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e ++ 
<" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( ( # `  e
)  -  1 )  e.  ( 0..^ (
# `  e )
) )
163152adantllr 724 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e ++ 
<" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
e  =/=  (/) )
164 simplll 767 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e ++ 
<" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( e  e. Word  RR  /\  k  e.  RR ) )
165 simplrl 769 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e ++ 
<" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )
166165simprd 465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e ++ 
<" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( ( e ++  <" k "> ) `  0 )  =/=  0 )
167164, 163, 166, 98syl21anc 1264 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e ++ 
<" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( e `  0
)  =/=  0 )
168 simpllr 768 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e ++ 
<" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( ( e  =/=  (/)  /\  ( e ` 
0 )  =/=  0
)  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )
169163, 167, 168mp2and 684 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e ++ 
<" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  ->  A. n  e.  (
0..^ ( # `  e
) ) ( ( T `  e ) `
 n )  =/=  0 )
170 fveq2 5879 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  ( ( # `  e )  -  1 )  ->  ( ( T `  e ) `  n )  =  ( ( T `  e
) `  ( ( # `
 e )  - 
1 ) ) )
171170neeq1d 2702 . . . . . . . . . . . . . . . . . 18  |-  ( n  =  ( ( # `  e )  -  1 )  ->  ( (
( T `  e
) `  n )  =/=  0  <->  ( ( T `
 e ) `  ( ( # `  e
)  -  1 ) )  =/=  0 ) )
172171rspcva 3181 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( # `  e
)  -  1 )  e.  ( 0..^ (
# `  e )
)  /\  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 )  ->  (
( T `  e
) `  ( ( # `
 e )  - 
1 ) )  =/=  0 )
173162, 169, 172syl2anc 666 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e ++ 
<" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( ( T `  e ) `  (
( # `  e )  -  1 ) )  =/=  0 )
17476, 77signswn0 29451 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( T `
 e ) `  ( ( # `  e
)  -  1 ) )  e.  { -u
1 ,  0 ,  1 }  /\  (sgn `  k )  e.  { -u 1 ,  0 ,  1 } )  /\  ( ( T `  e ) `  (
( # `  e )  -  1 ) )  =/=  0 )  -> 
( ( ( T `
 e ) `  ( ( # `  e
)  -  1 ) )  .+^  (sgn `  k
) )  =/=  0
)
175157, 161, 173, 174syl21anc 1264 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e ++ 
<" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( ( ( T `
 e ) `  ( ( # `  e
)  -  1 ) )  .+^  (sgn `  k
) )  =/=  0
)
176151, 175eqnetrd 2718 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e ++ 
<" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( ( T `  ( e ++  <" k "> ) ) `  ( # `  e ) )  =/=  0 )
177142, 176pm2.61dan 799 . . . . . . . . . . . . 13  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e ++ 
<" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  ->  (
( T `  (
e ++  <" k "> ) ) `  ( # `  e ) )  =/=  0 )
178177anassrs 653 . . . . . . . . . . . 12  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  ->  ( ( T `  ( e ++  <" k "> ) ) `  ( # `
 e ) )  =/=  0 )
179178adantr 467 . . . . . . . . . . 11  |-  ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e ` 
0 )  =/=  0
)  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  /\  m  =  ( # `  e
) )  ->  (
( T `  (
e ++  <" k "> ) ) `  ( # `  e ) )  =/=  0 )
180107, 179eqnetrd 2718 . . . . . . . . . 10  |-  ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e ` 
0 )  =/=  0
)  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  /\  m  =  ( # `  e
) )  ->  (
( T `  (
e ++  <" k "> ) ) `  m )  =/=  0
)
181 lencl 12685 . . . . . . . . . . . . 13  |-  ( e  e. Word  RR  ->  ( # `  e )  e.  NN0 )
182 nn0uz 11195 . . . . . . . . . . . . 13  |-  NN0  =  ( ZZ>= `  0 )
183181, 182syl6eleq 2521 . . . . . . . . . . . 12  |-  ( e  e. Word  RR  ->  ( # `  e )  e.  (
ZZ>= `  0 ) )
184183ad4antr 737 . . . . . . . . . . 11  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  ->  ( # `  e
)  e.  ( ZZ>= ` 
0 ) )
185 simpr 463 . . . . . . . . . . . 12  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  ->  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) )
186 ccatws1len 12751 . . . . . . . . . . . . . . 15  |-  ( ( e  e. Word  RR  /\  k  e.  RR )  ->  ( # `  (
e ++  <" k "> ) )  =  ( ( # `  e
)  +  1 ) )
187186oveq2d 6319 . . . . . . . . . . . . . 14  |-  ( ( e  e. Word  RR  /\  k  e.  RR )  ->  ( 0..^ ( # `  ( e ++  <" k "> ) ) )  =  ( 0..^ ( ( # `  e
)  +  1 ) ) )
188187eleq2d 2493 . . . . . . . . . . . . 13  |-  ( ( e  e. Word  RR  /\  k  e.  RR )  ->  ( m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) )  <-> 
m  e.  ( 0..^ ( ( # `  e
)  +  1 ) ) ) )
189188biimpa 487 . . . . . . . . . . . 12  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) )  ->  m  e.  ( 0..^ ( ( # `  e )  +  1 ) ) )
19083, 185, 189syl2anc 666 . . . . . . . . . . 11  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  ->  m  e.  ( 0..^ ( ( # `  e )  +  1 ) ) )
191 fzosplitsni 12020 . . . . . . . . . . . 12  |-  ( (
# `  e )  e.  ( ZZ>= `  0 )  ->  ( m  e.  ( 0..^ ( ( # `  e )  +  1 ) )  <->  ( m  e.  ( 0..^ ( # `  e ) )  \/  m  =  ( # `  e ) ) ) )
192191biimpa 487 . . . . . . . . . . 11  |-  ( ( ( # `  e
)  e.  ( ZZ>= ` 
0 )  /\  m  e.  ( 0..^ ( (
# `  e )  +  1 ) ) )  ->  ( m  e.  ( 0..^ ( # `  e ) )  \/  m  =  ( # `  e ) ) )
193184, 190, 192syl2anc 666 . . . . . . . . . 10  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  ->  ( m  e.  ( 0..^ ( # `  e ) )  \/  m  =  ( # `  e ) ) )
194105, 180, 193mpjaodan 794 . . . . . . . . 9  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )  ->  ( ( T `  ( e ++  <" k "> ) ) `  m
)  =/=  0 )
195194ralrimiva 2840 . . . . . . . 8  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  ->  A. m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) ( ( T `  ( e ++  <" k "> ) ) `  m )  =/=  0
)
19660, 195sylanbr 476 . . . . . . 7  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. m  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  m )  =/=  0 ) )  /\  ( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )  ->  A. m  e.  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) ( ( T `  ( e ++  <" k "> ) ) `  m )  =/=  0
)
197196exp31 608 . . . . . 6  |-  ( ( e  e. Word  RR  /\  k  e.  RR )  ->  ( ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. m  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  m )  =/=  0 )  ->  (
( ( e ++  <" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 )  ->  A. m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ( ( T `  ( e ++  <" k "> ) ) `  m )  =/=  0
) ) )
19819, 30, 41, 52, 55, 197wrdind 12829 . . . . 5  |-  ( F  e. Word  RR  ->  ( ( F  =/=  (/)  /\  ( F `  0 )  =/=  0 )  ->  A. m  e.  ( 0..^ ( # `  F ) ) ( ( T `  F
) `  m )  =/=  0 ) )
199198imp 431 . . . 4  |-  ( ( F  e. Word  RR  /\  ( F  =/=  (/)  /\  ( F `  0 )  =/=  0 ) )  ->  A. m  e.  (
0..^ ( # `  F
) ) ( ( T `  F ) `
 m )  =/=  0 )
200199adantrr 722 . . 3  |-  ( ( F  e. Word  RR  /\  ( ( F  =/=  (/)  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  A. m  e.  ( 0..^ ( # `  F
) ) ( ( T `  F ) `
 m )  =/=  0 )
201 fveq2 5879 . . . . 5  |-  ( m  =  N  ->  (
( T `  F
) `  m )  =  ( ( T `
 F ) `  N ) )
202201neeq1d 2702 . . . 4  |-  ( m  =  N  ->  (
( ( T `  F ) `  m
)  =/=  0  <->  (
( T `  F
) `  N )  =/=  0 ) )
203202rspcva 3181 . . 3  |-  ( ( N  e.  ( 0..^ ( # `  F
) )  /\  A. m  e.  ( 0..^ ( # `  F
) ) ( ( T `  F ) `
 m )  =/=  0 )  ->  (
( T `  F
) `  N )  =/=  0 )
2048, 200, 203syl2anc 666 . 2  |-  ( ( F  e. Word  RR  /\  ( ( F  =/=  (/)  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( ( T `
 F ) `  N )  =/=  0
)
2052, 6, 7, 204syl12anc 1263 1  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( ( T `  F ) `  N
)  =/=  0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869    =/= wne 2619   A.wral 2776    \ cdif 3434   (/)c0 3762   ifcif 3910   {csn 3997   {cpr 3999   {ctp 4001   <.cop 4003    |-> cmpt 4480   ` cfv 5599  (class class class)co 6303    |-> cmpt2 6305   RRcr 9540   0cc0 9541   1c1 9542    + caddc 9544   RR*cxr 9676    - cmin 9862   -ucneg 9863   NNcn 10611   NN0cn0 10871   ZZ>=cuz 11161   ...cfz 11786  ..^cfzo 11917   #chash 12516  Word cword 12654   ++ cconcat 12656   <"cs1 12657  sgncsgn 13143   sum_csu 13745   ndxcnx 15111   Basecbs 15114   +g cplusg 15183    gsumg cgsu 15332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-inf2 8150  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-supp 6924  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-oadd 7192  df-er 7369  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-oi 8029  df-card 8376  df-cda 8600  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-nn 10612  df-2 10670  df-n0 10872  df-z 10940  df-uz 11162  df-fz 11787  df-fzo 11918  df-seq 12215  df-hash 12517  df-word 12662  df-lsw 12663  df-concat 12664  df-s1 12665  df-substr 12666  df-sgn 13144  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-plusg 15196  df-0g 15333  df-gsum 15334  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-mulg 16669  df-cntz 16964
This theorem is referenced by:  signstfvcl  29464
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