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Theorem signstfvneq0 29533
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 768 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) )  ->  F  e.  (Word  RR  \  { (/) } ) )
21eldifad 3402 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) )  ->  F  e. Word  RR )
3 eldifsni 4089 . . . 4  |-  ( F  e.  (Word  RR  \  { (/) } )  ->  F  =/=  (/) )
43ad2antrr 740 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) )  ->  F  =/=  (/) )
5 simplr 770 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( F `  0
)  =/=  0 )
64, 5jca 541 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( F  =/=  (/)  /\  ( F `  0 )  =/=  0 ) )
7 simpr 468 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) )  ->  N  e.  ( 0..^ ( # `  F
) ) )
8 simprr 774 . . 3  |-  ( ( F  e. Word  RR  /\  ( ( F  =/=  (/)  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  N  e.  ( 0..^ ( # `  F
) ) )
9 neeq1 2705 . . . . . . . 8  |-  ( g  =  (/)  ->  ( g  =/=  (/)  <->  (/)  =/=  (/) ) )
10 fveq1 5878 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( g `
 0 )  =  ( (/) `  0 ) )
1110neeq1d 2702 . . . . . . . 8  |-  ( g  =  (/)  ->  ( ( g `  0 )  =/=  0  <->  ( (/) `  0
)  =/=  0 ) )
129, 11anbi12d 725 . . . . . . 7  |-  ( g  =  (/)  ->  ( ( g  =/=  (/)  /\  (
g `  0 )  =/=  0 )  <->  ( (/)  =/=  (/)  /\  ( (/) `  0 )  =/=  0 ) ) )
13 fveq2 5879 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( # `  g )  =  (
# `  (/) ) )
1413oveq2d 6324 . . . . . . . 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 2987 . . . . . . 7  |-  ( g  =  (/)  ->  ( A. m  e.  ( 0..^ ( # `  g
) ) ( ( T `  g ) `
 m )  =/=  0  <->  A. m  e.  ( 0..^ ( # `  (/) ) ) ( ( T `  (/) ) `  m )  =/=  0 ) )
1912, 18imbi12d 327 . . . . . 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 2705 . . . . . . . 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 725 . . . . . . 7  |-  ( g  =  e  ->  (
( g  =/=  (/)  /\  (
g `  0 )  =/=  0 )  <->  ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 ) ) )
24 fveq2 5879 . . . . . . . . 9  |-  ( g  =  e  ->  ( # `
 g )  =  ( # `  e
) )
2524oveq2d 6324 . . . . . . . 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 2987 . . . . . . 7  |-  ( g  =  e  ->  ( A. m  e.  (
0..^ ( # `  g
) ) ( ( T `  g ) `
 m )  =/=  0  <->  A. m  e.  ( 0..^ ( # `  e
) ) ( ( T `  e ) `
 m )  =/=  0 ) )
3023, 29imbi12d 327 . . . . . 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 2705 . . . . . . . 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 725 . . . . . . 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 6324 . . . . . . . 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 2987 . . . . . . 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 327 . . . . . 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 2705 . . . . . . . 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 725 . . . . . . 7  |-  ( g  =  F  ->  (
( g  =/=  (/)  /\  (
g `  0 )  =/=  0 )  <->  ( F  =/=  (/)  /\  ( F `
 0 )  =/=  0 ) ) )
46 fveq2 5879 . . . . . . . . 9  |-  ( g  =  F  ->  ( # `
 g )  =  ( # `  F
) )
4746oveq2d 6324 . . . . . . . 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 2987 . . . . . . 7  |-  ( g  =  F  ->  ( A. m  e.  (
0..^ ( # `  g
) ) ( ( T `  g ) `
 m )  =/=  0  <->  A. m  e.  ( 0..^ ( # `  F
) ) ( ( T `  F ) `
 m )  =/=  0 ) )
5245, 51imbi12d 327 . . . . . 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 2652 . . . . . . . 8  |-  -.  (/)  =/=  (/)
5453intnanr 929 . . . . . . 7  |-  -.  ( (/) 
=/=  (/)  /\  ( (/) `  0 )  =/=  0
)
5554pm2.21i 136 . . . . . 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 3005 . . . . . . . . . 10  |-  ( A. n  e.  ( 0..^ ( # `  e
) ) ( ( T `  e ) `
 n )  =/=  0  <->  A. m  e.  ( 0..^ ( # `  e
) ) ( ( T `  e ) `
 m )  =/=  0 )
5958imbi2i 319 . . . . . . . . 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 708 . . . . . . . 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 770 . . . . . . . . . . . 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 3726 . . . . . . . . . . . . . 14  |-  -.  m  e.  (/)
63 fveq2 5879 . . . . . . . . . . . . . . . . . 18  |-  ( e  =  (/)  ->  ( # `  e )  =  (
# `  (/) ) )
64 hash0 12586 . . . . . . . . . . . . . . . . . 18  |-  ( # `  (/) )  =  0
6563, 64syl6eq 2521 . . . . . . . . . . . . . . . . 17  |-  ( e  =  (/)  ->  ( # `  e )  =  0 )
6665oveq2d 6324 . . . . . . . . . . . . . . . 16  |-  ( e  =  (/)  ->  ( 0..^ ( # `  e
) )  =  ( 0..^ 0 ) )
67 fzo0 11969 . . . . . . . . . . . . . . . 16  |-  ( 0..^ 0 )  =  (/)
6866, 67syl6eq 2521 . . . . . . . . . . . . . . 15  |-  ( e  =  (/)  ->  ( 0..^ ( # `  e
) )  =  (/) )
6968eleq2d 2534 . . . . . . . . . . . . . 14  |-  ( e  =  (/)  ->  ( m  e.  ( 0..^ (
# `  e )
)  <->  m  e.  (/) ) )
7062, 69mtbiri 310 . . . . . . . . . . . . 13  |-  ( e  =  (/)  ->  -.  m  e.  ( 0..^ ( # `  e ) ) )
7170adantl 473 . . . . . . . . . . . 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 179 . . . . . . . . . . 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 788 . . . . . . . . . . . . 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 789 . . . . . . . . . . . . 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 770 . . . . . . . . . . . . 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 29532 . . . . . . . . . . . . 13  |-  ( ( e  e. Word  RR  /\  k  e.  RR  /\  m  e.  ( 0..^ ( # `  e ) ) )  ->  ( ( T `
 ( e ++  <" k "> )
) `  m )  =  ( ( T `
 e ) `  m ) )
8173, 74, 75, 80syl3anc 1292 . . . . . . . . . . . 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 468 . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . . 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 740 . . . . . . . . . . . . . . 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 779 . . . . . . . . . . . . . . . 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 1256 . . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  -> 
e  e. Word  RR )
88 simplr 770 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  -> 
k  e.  RR )
8988s1cld 12795 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  ->  <" k ">  e. Word  RR )
90 lennncl 12738 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  e. Word  RR  /\  e  =/=  (/) )  ->  ( # `
 e )  e.  NN )
9190adantlr 729 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  -> 
( # `  e )  e.  NN )
92 fzo0end 12032 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  e )  e.  NN  ->  ( ( # `
 e )  - 
1 )  e.  ( 0..^ ( # `  e
) ) )
93 elfzolt3b 11959 . . . . . . . . . . . . . . . . . . 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 12773 . . . . . . . . . . . . . . . . . 18  |-  ( ( e  e. Word  RR  /\  <" k ">  e. Word  RR  /\  0  e.  ( 0..^ ( # `  e ) ) )  ->  ( ( e ++ 
<" k "> ) `  0 )  =  ( e ` 
0 ) )
9687, 89, 94, 95syl3anc 1292 . . . . . . . . . . . . . . . . 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 492 . . . . . . . . . . . . . . 15  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  /\  ( ( e ++  <" k "> ) `  0 )  =/=  0 )  ->  (
e `  0 )  =/=  0 )
9984, 82, 86, 98syl21anc 1291 . . . . . . . . . . . . . 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 787 . . . . . . . . . . . . . 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 693 . . . . . . . . . . . . 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 3134 . . . . . . . . . . . . 13  |-  ( ( m  e.  ( 0..^ ( # `  e
) )  /\  A. n  e.  ( 0..^ ( # `  e
) ) ( ( T `  e ) `
 n )  =/=  0 )  ->  (
( T `  e
) `  m )  =/=  0 )
10375, 101, 102syl2anc 673 . . . . . . . . . . . 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 2710 . . . . . . . . . . 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 2730 . . . . . . . . . 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 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
) )  ->  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 468 . . . . . . . . . . . . . . 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 785 . . . . . . . . . . . . . . 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 778 . . . . . . . . . . . . . . . 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 470 . . . . . . . . . . . . . . 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 6315 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( e  =  (/)  ->  ( e ++ 
<" k "> )  =  ( (/) ++  <" k "> ) )
113 s1cl 12794 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  e.  RR  ->  <" k ">  e. Word  RR )
114 ccatlid 12781 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <" k ">  e. Word  RR  ->  ( (/) ++  <" k "> )  =  <" k "> )
115113, 114syl 17 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  e.  RR  ->  ( (/) ++  <" k "> )  =  <" k "> )
116112, 115sylan9eq 2525 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( e  =  (/)  /\  k  e.  RR )  ->  (
e ++  <" k "> )  =  <" k "> )
117116fveq2d 5883 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  =  (/)  /\  k  e.  RR )  ->  ( T `  ( e ++  <" k "> ) )  =  ( T `  <" k "> ) )
118117adantr 472 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  ( T `  ( e ++  <" k "> ) )  =  ( T `  <" k "> ) )
119 simplr 770 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  k  e.  RR )
12076, 77, 78, 79signstf0 29529 . . . . . . . . . . . . . . . . . . . 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 2505 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  ( T `  ( e ++  <" k "> ) )  =  <" (sgn `  k ) "> )
12365ad2antrr 740 . . . . . . . . . . . . . . . . . 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 29483 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  RR  ->  (sgn `  k )  e.  RR )
126 s1fv 12801 . . . . . . . . . . . . . . . . . 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 2505 . . . . . . . . . . . . . . . 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 12801 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  e.  RR  ->  ( <" k "> `  0 )  =  k )
131130adantl 473 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  =  (/)  /\  k  e.  RR )  ->  ( <" k "> `  0 )  =  k )
132129, 131eqtrd 2505 . . . . . . . . . . . . . . . . . . 19  |-  ( ( e  =  (/)  /\  k  e.  RR )  ->  (
( e ++  <" k "> ) `  0
)  =  k )
133132neeq1d 2702 . . . . . . . . . . . . . . . . . 18  |-  ( ( e  =  (/)  /\  k  e.  RR )  ->  (
( ( e ++  <" k "> ) `  0 )  =/=  0  <->  k  =/=  0
) )
134133biimpa 492 . . . . . . . . . . . . . . . . 17  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  k  =/=  0 )
135 rexr 9704 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  e.  RR  ->  k  e.  RR* )
136 sgn0bi 29491 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  e.  RR*  ->  ( (sgn
`  k )  =  0  <->  k  =  0 ) )
137135, 136syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  RR  ->  (
(sgn `  k )  =  0  <->  k  = 
0 ) )
138137necon3bid 2687 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  RR  ->  (
(sgn `  k )  =/=  0  <->  k  =/=  0
) )
139138biimpar 493 . . . . . . . . . . . . . . . . 17  |-  ( ( k  e.  RR  /\  k  =/=  0 )  -> 
(sgn `  k )  =/=  0 )
140119, 134, 139syl2anc 673 . . . . . . . . . . . . . . . 16  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  (sgn `  k )  =/=  0
)
141128, 140eqnetrd 2710 . . . . . . . . . . . . . . 15  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  (
( T `  (
e ++  <" k "> ) ) `  ( # `  e ) )  =/=  0 )
142108, 109, 111, 141syl21anc 1291 . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e ++  <" k "> )  =/=  (/)  /\  (
( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  ->  -.  e  =  (/) )
145 elsn 3973 . . . . . . . . . . . . . . . . . . 19  |-  ( e  e.  { (/) }  <->  e  =  (/) )
146144, 145sylnibr 312 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e ++  <" k "> )  =/=  (/)  /\  (
( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  ->  -.  e  e.  { (/) } )
147143, 146eldifd 3401 . . . . . . . . . . . . . . . . 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 777 . . . . . . . . . . . . . . . . 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 29530 . . . . . . . . . . . . . . . . 17  |-  ( ( e  e.  (Word  RR  \  { (/) } )  /\  k  e.  RR )  ->  ( ( T `  ( e ++  <" k "> ) ) `  ( # `  e ) )  =  ( ( ( T `  e
) `  ( ( # `
 e )  - 
1 ) )  .+^  (sgn `  k ) ) )
150147, 148, 149syl2anc 673 . . . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . . . 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 2650 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e ++  <" k "> )  =/=  (/)  /\  (
( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
e  =/=  (/) )
153143, 152, 90syl2anc 673 . . . . . . . . . . . . . . . . . . 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 29526 . . . . . . . . . . . . . . . . . 18  |-  ( ( e  e. Word  RR  /\  ( ( # `  e
)  -  1 )  e.  ( 0..^ (
# `  e )
) )  ->  (
( T `  e
) `  ( ( # `
 e )  - 
1 ) )  e. 
{ -u 1 ,  0 ,  1 } )
156143, 154, 155syl2anc 673 . . . . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . . . . 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 9708 . . . . . . . . . . . . . . . . . 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 29482 . . . . . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . . . . . 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 778 . . . . . . . . . . . . . . . . . . . 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 470 . . . . . . . . . . . . . . . . . . 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 1291 . . . . . . . . . . . . . . . . . 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 777 . . . . . . . . . . . . . . . . . 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 693 . . . . . . . . . . . . . . . . 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 3134 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( # `  e
)  -  1 )  e.  ( 0..^ (
# `  e )
)  /\  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 )  ->  (
( T `  e
) `  ( ( # `
 e )  - 
1 ) )  =/=  0 )
173162, 169, 172syl2anc 673 . . . . . . . . . . . . . . . 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 29521 . . . . . . . . . . . . . . . 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 1291 . . . . . . . . . . . . . . 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 2710 . . . . . . . . . . . . . 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 808 . . . . . . . . . . . . 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 660 . . . . . . . . . . . 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 472 . . . . . . . . . . 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 2710 . . . . . . . . . 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 12737 . . . . . . . . . . . . 13  |-  ( e  e. Word  RR  ->  ( # `  e )  e.  NN0 )
182 nn0uz 11217 . . . . . . . . . . . . 13  |-  NN0  =  ( ZZ>= `  0 )
183181, 182syl6eleq 2559 . . . . . . . . . . . 12  |-  ( e  e. Word  RR  ->  ( # `  e )  e.  (
ZZ>= `  0 ) )
184183ad4antr 746 . . . . . . . . . . 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 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 ++  <" k "> ) ) ) )
186 ccatws1len 12809 . . . . . . . . . . . . . . 15  |-  ( ( e  e. Word  RR  /\  k  e.  RR )  ->  ( # `  (
e ++  <" k "> ) )  =  ( ( # `  e
)  +  1 ) )
187186oveq2d 6324 . . . . . . . . . . . . . 14  |-  ( ( e  e. Word  RR  /\  k  e.  RR )  ->  ( 0..^ ( # `  ( e ++  <" k "> ) ) )  =  ( 0..^ ( ( # `  e
)  +  1 ) ) )
188187eleq2d 2534 . . . . . . . . . . . . 13  |-  ( ( e  e. Word  RR  /\  k  e.  RR )  ->  ( m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) )  <-> 
m  e.  ( 0..^ ( ( # `  e
)  +  1 ) ) ) )
189188biimpa 492 . . . . . . . . . . . 12  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) )  ->  m  e.  ( 0..^ ( ( # `  e )  +  1 ) ) )
19083, 185, 189syl2anc 673 . . . . . . . . . . 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 12050 . . . . . . . . . . . 12  |-  ( (
# `  e )  e.  ( ZZ>= `  0 )  ->  ( m  e.  ( 0..^ ( ( # `  e )  +  1 ) )  <->  ( m  e.  ( 0..^ ( # `  e ) )  \/  m  =  ( # `  e ) ) ) )
192191biimpa 492 . . . . . . . . . . 11  |-  ( ( ( # `  e
)  e.  ( ZZ>= ` 
0 )  /\  m  e.  ( 0..^ ( (
# `  e )  +  1 ) ) )  ->  ( m  e.  ( 0..^ ( # `  e ) )  \/  m  =  ( # `  e ) ) )
193184, 190, 192syl2anc 673 . . . . . . . . . 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 803 . . . . . . . . 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 2809 . . . . . . . 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 481 . . . . . . 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 615 . . . . . 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 12887 . . . . 5  |-  ( F  e. Word  RR  ->  ( ( F  =/=  (/)  /\  ( F `  0 )  =/=  0 )  ->  A. m  e.  ( 0..^ ( # `  F ) ) ( ( T `  F
) `  m )  =/=  0 ) )
199198imp 436 . . . 4  |-  ( ( F  e. Word  RR  /\  ( F  =/=  (/)  /\  ( F `  0 )  =/=  0 ) )  ->  A. m  e.  (
0..^ ( # `  F
) ) ( ( T `  F ) `
 m )  =/=  0 )
200199adantrr 731 . . 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 3134 . . 3  |-  ( ( N  e.  ( 0..^ ( # `  F
) )  /\  A. m  e.  ( 0..^ ( # `  F
) ) ( ( T `  F ) `
 m )  =/=  0 )  ->  (
( T `  F
) `  N )  =/=  0 )
2048, 200, 203syl2anc 673 . 2  |-  ( ( F  e. Word  RR  /\  ( ( F  =/=  (/)  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( ( T `
 F ) `  N )  =/=  0
)
2052, 6, 7, 204syl12anc 1290 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 189    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756    \ cdif 3387   (/)c0 3722   ifcif 3872   {csn 3959   {cpr 3961   {ctp 3963   <.cop 3965    |-> cmpt 4454   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560   RR*cxr 9692    - cmin 9880   -ucneg 9881   NNcn 10631   NN0cn0 10893   ZZ>=cuz 11182   ...cfz 11810  ..^cfzo 11942   #chash 12553  Word cword 12703   ++ cconcat 12705   <"cs1 12706  sgncsgn 13226   sum_csu 13829   ndxcnx 15196   Basecbs 15199   +g cplusg 15268    gsumg cgsu 15417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-seq 12252  df-hash 12554  df-word 12711  df-lsw 12712  df-concat 12713  df-s1 12714  df-substr 12715  df-sgn 13227  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-plusg 15281  df-0g 15418  df-gsum 15419  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mulg 16754  df-cntz 17049
This theorem is referenced by:  signstfvcl  29534
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