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Theorem signstfvneq0 28712
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 751 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) )  ->  F  e.  (Word  RR  \  { (/) } ) )
21eldifad 3401 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) )  ->  F  e. Word  RR )
3 eldifsni 4070 . . . 4  |-  ( F  e.  (Word  RR  \  { (/) } )  ->  F  =/=  (/) )
43ad2antrr 723 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) )  ->  F  =/=  (/) )
5 simplr 753 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( F `  0
)  =/=  0 )
64, 5jca 530 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( F  =/=  (/)  /\  ( F `  0 )  =/=  0 ) )
7 simpr 459 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) )  ->  N  e.  ( 0..^ ( # `  F
) ) )
8 simprr 755 . . 3  |-  ( ( F  e. Word  RR  /\  ( ( F  =/=  (/)  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  N  e.  ( 0..^ ( # `  F
) ) )
9 neeq1 2663 . . . . . . . 8  |-  ( g  =  (/)  ->  ( g  =/=  (/)  <->  (/)  =/=  (/) ) )
10 fveq1 5773 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( g `
 0 )  =  ( (/) `  0 ) )
1110neeq1d 2659 . . . . . . . 8  |-  ( g  =  (/)  ->  ( ( g `  0 )  =/=  0  <->  ( (/) `  0
)  =/=  0 ) )
129, 11anbi12d 708 . . . . . . 7  |-  ( g  =  (/)  ->  ( ( g  =/=  (/)  /\  (
g `  0 )  =/=  0 )  <->  ( (/)  =/=  (/)  /\  ( (/) `  0 )  =/=  0 ) ) )
13 fveq2 5774 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( # `  g )  =  (
# `  (/) ) )
1413oveq2d 6212 . . . . . . . 8  |-  ( g  =  (/)  ->  ( 0..^ ( # `  g
) )  =  ( 0..^ ( # `  (/) ) ) )
15 fveq2 5774 . . . . . . . . . 10  |-  ( g  =  (/)  ->  ( T `
 g )  =  ( T `  (/) ) )
1615fveq1d 5776 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( ( T `  g ) `
 m )  =  ( ( T `  (/) ) `  m ) )
1716neeq1d 2659 . . . . . . . 8  |-  ( g  =  (/)  ->  ( ( ( T `  g
) `  m )  =/=  0  <->  ( ( T `
 (/) ) `  m
)  =/=  0 ) )
1814, 17raleqbidv 2993 . . . . . . 7  |-  ( g  =  (/)  ->  ( A. m  e.  ( 0..^ ( # `  g
) ) ( ( T `  g ) `
 m )  =/=  0  <->  A. m  e.  ( 0..^ ( # `  (/) ) ) ( ( T `  (/) ) `  m )  =/=  0 ) )
1912, 18imbi12d 318 . . . . . 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 2663 . . . . . . . 8  |-  ( g  =  e  ->  (
g  =/=  (/)  <->  e  =/=  (/) ) )
21 fveq1 5773 . . . . . . . . 9  |-  ( g  =  e  ->  (
g `  0 )  =  ( e ` 
0 ) )
2221neeq1d 2659 . . . . . . . 8  |-  ( g  =  e  ->  (
( g `  0
)  =/=  0  <->  (
e `  0 )  =/=  0 ) )
2320, 22anbi12d 708 . . . . . . 7  |-  ( g  =  e  ->  (
( g  =/=  (/)  /\  (
g `  0 )  =/=  0 )  <->  ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 ) ) )
24 fveq2 5774 . . . . . . . . 9  |-  ( g  =  e  ->  ( # `
 g )  =  ( # `  e
) )
2524oveq2d 6212 . . . . . . . 8  |-  ( g  =  e  ->  (
0..^ ( # `  g
) )  =  ( 0..^ ( # `  e
) ) )
26 fveq2 5774 . . . . . . . . . 10  |-  ( g  =  e  ->  ( T `  g )  =  ( T `  e ) )
2726fveq1d 5776 . . . . . . . . 9  |-  ( g  =  e  ->  (
( T `  g
) `  m )  =  ( ( T `
 e ) `  m ) )
2827neeq1d 2659 . . . . . . . 8  |-  ( g  =  e  ->  (
( ( T `  g ) `  m
)  =/=  0  <->  (
( T `  e
) `  m )  =/=  0 ) )
2925, 28raleqbidv 2993 . . . . . . 7  |-  ( g  =  e  ->  ( A. m  e.  (
0..^ ( # `  g
) ) ( ( T `  g ) `
 m )  =/=  0  <->  A. m  e.  ( 0..^ ( # `  e
) ) ( ( T `  e ) `
 m )  =/=  0 ) )
3023, 29imbi12d 318 . . . . . 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 2663 . . . . . . . 8  |-  ( g  =  ( e ++  <" k "> )  ->  ( g  =/=  (/)  <->  ( e ++  <" k "> )  =/=  (/) ) )
32 fveq1 5773 . . . . . . . . 9  |-  ( g  =  ( e ++  <" k "> )  ->  ( g `  0
)  =  ( ( e ++  <" k "> ) `  0
) )
3332neeq1d 2659 . . . . . . . 8  |-  ( g  =  ( e ++  <" k "> )  ->  ( ( g ` 
0 )  =/=  0  <->  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) )
3431, 33anbi12d 708 . . . . . . 7  |-  ( g  =  ( e ++  <" k "> )  ->  ( ( g  =/=  (/)  /\  ( g ` 
0 )  =/=  0
)  <->  ( ( e ++ 
<" k "> )  =/=  (/)  /\  ( ( e ++  <" k "> ) `  0
)  =/=  0 ) ) )
35 fveq2 5774 . . . . . . . . 9  |-  ( g  =  ( e ++  <" k "> )  ->  ( # `  g
)  =  ( # `  ( e ++  <" k "> ) ) )
3635oveq2d 6212 . . . . . . . 8  |-  ( g  =  ( e ++  <" k "> )  ->  ( 0..^ ( # `  g ) )  =  ( 0..^ ( # `  ( e ++  <" k "> ) ) ) )
37 fveq2 5774 . . . . . . . . . 10  |-  ( g  =  ( e ++  <" k "> )  ->  ( T `  g
)  =  ( T `
 ( e ++  <" k "> )
) )
3837fveq1d 5776 . . . . . . . . 9  |-  ( g  =  ( e ++  <" k "> )  ->  ( ( T `  g ) `  m
)  =  ( ( T `  ( e ++ 
<" k "> ) ) `  m
) )
3938neeq1d 2659 . . . . . . . 8  |-  ( g  =  ( e ++  <" k "> )  ->  ( ( ( T `
 g ) `  m )  =/=  0  <->  ( ( T `  (
e ++  <" k "> ) ) `  m )  =/=  0
) )
4036, 39raleqbidv 2993 . . . . . . 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 318 . . . . . 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 2663 . . . . . . . 8  |-  ( g  =  F  ->  (
g  =/=  (/)  <->  F  =/=  (/) ) )
43 fveq1 5773 . . . . . . . . 9  |-  ( g  =  F  ->  (
g `  0 )  =  ( F ` 
0 ) )
4443neeq1d 2659 . . . . . . . 8  |-  ( g  =  F  ->  (
( g `  0
)  =/=  0  <->  ( F `  0 )  =/=  0 ) )
4542, 44anbi12d 708 . . . . . . 7  |-  ( g  =  F  ->  (
( g  =/=  (/)  /\  (
g `  0 )  =/=  0 )  <->  ( F  =/=  (/)  /\  ( F `
 0 )  =/=  0 ) ) )
46 fveq2 5774 . . . . . . . . 9  |-  ( g  =  F  ->  ( # `
 g )  =  ( # `  F
) )
4746oveq2d 6212 . . . . . . . 8  |-  ( g  =  F  ->  (
0..^ ( # `  g
) )  =  ( 0..^ ( # `  F
) ) )
48 fveq2 5774 . . . . . . . . . 10  |-  ( g  =  F  ->  ( T `  g )  =  ( T `  F ) )
4948fveq1d 5776 . . . . . . . . 9  |-  ( g  =  F  ->  (
( T `  g
) `  m )  =  ( ( T `
 F ) `  m ) )
5049neeq1d 2659 . . . . . . . 8  |-  ( g  =  F  ->  (
( ( T `  g ) `  m
)  =/=  0  <->  (
( T `  F
) `  m )  =/=  0 ) )
5147, 50raleqbidv 2993 . . . . . . 7  |-  ( g  =  F  ->  ( A. m  e.  (
0..^ ( # `  g
) ) ( ( T `  g ) `
 m )  =/=  0  <->  A. m  e.  ( 0..^ ( # `  F
) ) ( ( T `  F ) `
 m )  =/=  0 ) )
5245, 51imbi12d 318 . . . . . 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 2586 . . . . . . . 8  |-  -.  (/)  =/=  (/)
5453intnanr 913 . . . . . . 7  |-  -.  ( (/) 
=/=  (/)  /\  ( (/) `  0 )  =/=  0
)
5554pm2.21i 131 . . . . . 6  |-  ( (
(/)  =/=  (/)  /\  ( (/) `  0 )  =/=  0 )  ->  A. m  e.  ( 0..^ ( # `  (/) ) ) ( ( T `  (/) ) `  m )  =/=  0
)
56 fveq2 5774 . . . . . . . . . . . 12  |-  ( n  =  m  ->  (
( T `  e
) `  n )  =  ( ( T `
 e ) `  m ) )
5756neeq1d 2659 . . . . . . . . . . 11  |-  ( n  =  m  ->  (
( ( T `  e ) `  n
)  =/=  0  <->  (
( T `  e
) `  m )  =/=  0 ) )
5857cbvralv 3009 . . . . . . . . . 10  |-  ( A. n  e.  ( 0..^ ( # `  e
) ) ( ( T `  e ) `
 n )  =/=  0  <->  A. m  e.  ( 0..^ ( # `  e
) ) ( ( T `  e ) `
 m )  =/=  0 )
5958imbi2i 310 . . . . . . . . 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 692 . . . . . . . 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 753 . . . . . . . . . . . 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 3715 . . . . . . . . . . . . . 14  |-  -.  m  e.  (/)
63 fveq2 5774 . . . . . . . . . . . . . . . . . 18  |-  ( e  =  (/)  ->  ( # `  e )  =  (
# `  (/) ) )
64 hash0 12340 . . . . . . . . . . . . . . . . . 18  |-  ( # `  (/) )  =  0
6563, 64syl6eq 2439 . . . . . . . . . . . . . . . . 17  |-  ( e  =  (/)  ->  ( # `  e )  =  0 )
6665oveq2d 6212 . . . . . . . . . . . . . . . 16  |-  ( e  =  (/)  ->  ( 0..^ ( # `  e
) )  =  ( 0..^ 0 ) )
67 fzo0 11744 . . . . . . . . . . . . . . . 16  |-  ( 0..^ 0 )  =  (/)
6866, 67syl6eq 2439 . . . . . . . . . . . . . . 15  |-  ( e  =  (/)  ->  ( 0..^ ( # `  e
) )  =  (/) )
6968eleq2d 2452 . . . . . . . . . . . . . 14  |-  ( e  =  (/)  ->  ( m  e.  ( 0..^ (
# `  e )
)  <->  m  e.  (/) ) )
7062, 69mtbiri 301 . . . . . . . . . . . . 13  |-  ( e  =  (/)  ->  -.  m  e.  ( 0..^ ( # `  e ) ) )
7170adantl 464 . . . . . . . . . . . 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 174 . . . . . . . . . . 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 769 . . . . . . . . . . . . 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 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  =/=  (/) )  -> 
k  e.  RR )
75 simplr 753 . . . . . . . . . . . . 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 28711 . . . . . . . . . . . . 13  |-  ( ( e  e. Word  RR  /\  k  e.  RR  /\  m  e.  ( 0..^ ( # `  e ) ) )  ->  ( ( T `
 ( e ++  <" k "> )
) `  m )  =  ( ( T `
 e ) `  m ) )
8173, 74, 75, 80syl3anc 1226 . . . . . . . . . . . 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 459 . . . . . . . . . . . . . 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 757 . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . 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 760 . . . . . . . . . . . . . . . 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 1216 . . . . . . . . . . . . . . 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 751 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  -> 
e  e. Word  RR )
88 simplr 753 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  -> 
k  e.  RR )
8988s1cld 12524 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  ->  <" k ">  e. Word  RR )
90 lennncl 12470 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  e. Word  RR  /\  e  =/=  (/) )  ->  ( # `
 e )  e.  NN )
9190adantlr 712 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  -> 
( # `  e )  e.  NN )
92 fzo0end 11803 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  e )  e.  NN  ->  ( ( # `
 e )  - 
1 )  e.  ( 0..^ ( # `  e
) ) )
93 elfzolt3b 11734 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( # `  e
)  -  1 )  e.  ( 0..^ (
# `  e )
)  ->  0  e.  ( 0..^ ( # `  e
) ) )
9491, 92, 933syl 20 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  -> 
0  e.  ( 0..^ ( # `  e
) ) )
95 ccatval1 12504 . . . . . . . . . . . . . . . . . 18  |-  ( ( e  e. Word  RR  /\  <" k ">  e. Word  RR  /\  0  e.  ( 0..^ ( # `  e ) ) )  ->  ( ( e ++ 
<" k "> ) `  0 )  =  ( e ` 
0 ) )
9687, 89, 94, 95syl3anc 1226 . . . . . . . . . . . . . . . . 17  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  -> 
( ( e ++  <" k "> ) `  0 )  =  ( e `  0
) )
9796neeq1d 2659 . . . . . . . . . . . . . . . 16  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  -> 
( ( ( e ++ 
<" k "> ) `  0 )  =/=  0  <->  ( e ` 
0 )  =/=  0
) )
9897biimpa 482 . . . . . . . . . . . . . . 15  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  /\  ( ( e ++  <" k "> ) `  0 )  =/=  0 )  ->  (
e `  0 )  =/=  0 )
9984, 82, 86, 98syl21anc 1225 . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . 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 677 . . . . . . . . . . . . 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 3133 . . . . . . . . . . . . 13  |-  ( ( m  e.  ( 0..^ ( # `  e
) )  /\  A. n  e.  ( 0..^ ( # `  e
) ) ( ( T `  e ) `
 n )  =/=  0 )  ->  (
( T `  e
) `  m )  =/=  0 )
10375, 101, 102syl2anc 659 . . . . . . . . . . . 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 2675 . . . . . . . . . . 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 2700 . . . . . . . . . 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 459 . . . . . . . . . . . 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 5778 . . . . . . . . . . 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 459 . . . . . . . . . . . . . . 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 766 . . . . . . . . . . . . . . 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 759 . . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . 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 6203 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( e  =  (/)  ->  ( e ++ 
<" k "> )  =  ( (/) ++  <" k "> ) )
113 s1cl 12523 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  e.  RR  ->  <" k ">  e. Word  RR )
114 ccatlid 12512 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <" k ">  e. Word  RR  ->  ( (/) ++  <" k "> )  =  <" k "> )
115113, 114syl 16 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  e.  RR  ->  ( (/) ++  <" k "> )  =  <" k "> )
116112, 115sylan9eq 2443 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( e  =  (/)  /\  k  e.  RR )  ->  (
e ++  <" k "> )  =  <" k "> )
117116fveq2d 5778 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  =  (/)  /\  k  e.  RR )  ->  ( T `  ( e ++  <" k "> ) )  =  ( T `  <" k "> ) )
118117adantr 463 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  ( T `  ( e ++  <" k "> ) )  =  ( T `  <" k "> ) )
119 simplr 753 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  k  e.  RR )
12076, 77, 78, 79signstf0 28708 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  e.  RR  ->  ( T `  <" k "> )  =  <" (sgn `  k ) "> )
121119, 120syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  ( T `  <" k "> )  =  <" (sgn `  k ) "> )
122118, 121eqtrd 2423 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  ( T `  ( e ++  <" k "> ) )  =  <" (sgn `  k ) "> )
12365ad2antrr 723 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  ( # `
 e )  =  0 )
124122, 123fveq12d 5780 . . . . . . . . . . . . . . . . 17  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  (
( T `  (
e ++  <" k "> ) ) `  ( # `  e ) )  =  ( <" (sgn `  k ) "> `  0 )
)
125 sgnclre 28661 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  RR  ->  (sgn `  k )  e.  RR )
126 s1fv 12528 . . . . . . . . . . . . . . . . . 18  |-  ( (sgn
`  k )  e.  RR  ->  ( <" (sgn `  k ) "> `  0 )  =  (sgn `  k )
)
127119, 125, 1263syl 20 . . . . . . . . . . . . . . . . 17  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  ( <" (sgn `  k
) "> `  0
)  =  (sgn `  k ) )
128124, 127eqtrd 2423 . . . . . . . . . . . . . . . 16  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  (
( T `  (
e ++  <" k "> ) ) `  ( # `  e ) )  =  (sgn `  k ) )
129116fveq1d 5776 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  =  (/)  /\  k  e.  RR )  ->  (
( e ++  <" k "> ) `  0
)  =  ( <" k "> `  0 ) )
130 s1fv 12528 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  e.  RR  ->  ( <" k "> `  0 )  =  k )
131130adantl 464 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  =  (/)  /\  k  e.  RR )  ->  ( <" k "> `  0 )  =  k )
132129, 131eqtrd 2423 . . . . . . . . . . . . . . . . . . 19  |-  ( ( e  =  (/)  /\  k  e.  RR )  ->  (
( e ++  <" k "> ) `  0
)  =  k )
133132neeq1d 2659 . . . . . . . . . . . . . . . . . 18  |-  ( ( e  =  (/)  /\  k  e.  RR )  ->  (
( ( e ++  <" k "> ) `  0 )  =/=  0  <->  k  =/=  0
) )
134133biimpa 482 . . . . . . . . . . . . . . . . 17  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  k  =/=  0 )
135 rexr 9550 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  e.  RR  ->  k  e.  RR* )
136 sgn0bi 28669 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  e.  RR*  ->  ( (sgn
`  k )  =  0  <->  k  =  0 ) )
137135, 136syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  RR  ->  (
(sgn `  k )  =  0  <->  k  = 
0 ) )
138137necon3bid 2640 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  RR  ->  (
(sgn `  k )  =/=  0  <->  k  =/=  0
) )
139138biimpar 483 . . . . . . . . . . . . . . . . 17  |-  ( ( k  e.  RR  /\  k  =/=  0 )  -> 
(sgn `  k )  =/=  0 )
140119, 134, 139syl2anc 659 . . . . . . . . . . . . . . . 16  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  (sgn `  k )  =/=  0
)
141128, 140eqnetrd 2675 . . . . . . . . . . . . . . 15  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e ++ 
<" k "> ) `  0 )  =/=  0 )  ->  (
( T `  (
e ++  <" k "> ) ) `  ( # `  e ) )  =/=  0 )
142108, 109, 111, 141syl21anc 1225 . . . . . . . . . . . . . 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 757 . . . . . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e ++  <" k "> )  =/=  (/)  /\  (
( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  ->  -.  e  =  (/) )
145 elsn 3958 . . . . . . . . . . . . . . . . . . 19  |-  ( e  e.  { (/) }  <->  e  =  (/) )
146144, 145sylnibr 303 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e ++  <" k "> )  =/=  (/)  /\  (
( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  ->  -.  e  e.  { (/) } )
147143, 146eldifd 3400 . . . . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . . . . . 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 28709 . . . . . . . . . . . . . . . . 17  |-  ( ( e  e.  (Word  RR  \  { (/) } )  /\  k  e.  RR )  ->  ( ( T `  ( e ++  <" k "> ) ) `  ( # `  e ) )  =  ( ( ( T `  e
) `  ( ( # `
 e )  - 
1 ) )  .+^  (sgn `  k ) ) )
150147, 148, 149syl2anc 659 . . . . . . . . . . . . . . . 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 716 . . . . . . . . . . . . . . 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 2585 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e ++  <" k "> )  =/=  (/)  /\  (
( e ++  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
e  =/=  (/) )
153143, 152, 90syl2anc 659 . . . . . . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . . . . . 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 28705 . . . . . . . . . . . . . . . . . 18  |-  ( ( e  e. Word  RR  /\  ( ( # `  e
)  -  1 )  e.  ( 0..^ (
# `  e )
) )  ->  (
( T `  e
) `  ( ( # `
 e )  - 
1 ) )  e. 
{ -u 1 ,  0 ,  1 } )
156143, 154, 155syl2anc 659 . . . . . . . . . . . . . . . . 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 716 . . . . . . . . . . . . . . . 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 9554 . . . . . . . . . . . . . . . . . 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 28660 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  RR*  ->  (sgn `  k )  e.  { -u 1 ,  0 ,  1 } )
160158, 159syl 16 . . . . . . . . . . . . . . . . 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 716 . . . . . . . . . . . . . . . 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 716 . . . . . . . . . . . . . . . . 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 716 . . . . . . . . . . . . . . . . . 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 757 . . . . . . . . . . . . . . . . . . 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 759 . . . . . . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . . . . . 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 1225 . . . . . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . . . . . . 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 677 . . . . . . . . . . . . . . . . 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 5774 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  ( ( # `  e )  -  1 )  ->  ( ( T `  e ) `  n )  =  ( ( T `  e
) `  ( ( # `
 e )  - 
1 ) ) )
171170neeq1d 2659 . . . . . . . . . . . . . . . . . 18  |-  ( n  =  ( ( # `  e )  -  1 )  ->  ( (
( T `  e
) `  n )  =/=  0  <->  ( ( T `
 e ) `  ( ( # `  e
)  -  1 ) )  =/=  0 ) )
172171rspcva 3133 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( # `  e
)  -  1 )  e.  ( 0..^ (
# `  e )
)  /\  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 )  ->  (
( T `  e
) `  ( ( # `
 e )  - 
1 ) )  =/=  0 )
173162, 169, 172syl2anc 659 . . . . . . . . . . . . . . . 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 28700 . . . . . . . . . . . . . . . 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 1225 . . . . . . . . . . . . . . 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 2675 . . . . . . . . . . . . . 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 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 "> ) ) ) ) )  ->  (
( T `  (
e ++  <" k "> ) ) `  ( # `  e ) )  =/=  0 )
178177anassrs 646 . . . . . . . . . . . 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 463 . . . . . . . . . . 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 2675 . . . . . . . . . 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 12469 . . . . . . . . . . . . 13  |-  ( e  e. Word  RR  ->  ( # `  e )  e.  NN0 )
182 nn0uz 11035 . . . . . . . . . . . . 13  |-  NN0  =  ( ZZ>= `  0 )
183181, 182syl6eleq 2480 . . . . . . . . . . . 12  |-  ( e  e. Word  RR  ->  ( # `  e )  e.  (
ZZ>= `  0 ) )
184183ad4antr 729 . . . . . . . . . . 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 459 . . . . . . . . . . . 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 12535 . . . . . . . . . . . . . . 15  |-  ( ( e  e. Word  RR  /\  k  e.  RR )  ->  ( # `  (
e ++  <" k "> ) )  =  ( ( # `  e
)  +  1 ) )
187186oveq2d 6212 . . . . . . . . . . . . . 14  |-  ( ( e  e. Word  RR  /\  k  e.  RR )  ->  ( 0..^ ( # `  ( e ++  <" k "> ) ) )  =  ( 0..^ ( ( # `  e
)  +  1 ) ) )
188187eleq2d 2452 . . . . . . . . . . . . 13  |-  ( ( e  e. Word  RR  /\  k  e.  RR )  ->  ( m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) )  <-> 
m  e.  ( 0..^ ( ( # `  e
)  +  1 ) ) ) )
189188biimpa 482 . . . . . . . . . . . 12  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  m  e.  ( 0..^ ( # `  (
e ++  <" k "> ) ) ) )  ->  m  e.  ( 0..^ ( ( # `  e )  +  1 ) ) )
19083, 185, 189syl2anc 659 . . . . . . . . . . 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 11819 . . . . . . . . . . . 12  |-  ( (
# `  e )  e.  ( ZZ>= `  0 )  ->  ( m  e.  ( 0..^ ( ( # `  e )  +  1 ) )  <->  ( m  e.  ( 0..^ ( # `  e ) )  \/  m  =  ( # `  e ) ) ) )
192191biimpa 482 . . . . . . . . . . 11  |-  ( ( ( # `  e
)  e.  ( ZZ>= ` 
0 )  /\  m  e.  ( 0..^ ( (
# `  e )  +  1 ) ) )  ->  ( m  e.  ( 0..^ ( # `  e ) )  \/  m  =  ( # `  e ) ) )
193184, 190, 192syl2anc 659 . . . . . . . . . 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 784 . . . . . . . . 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 2796 . . . . . . . 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 471 . . . . . . 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 602 . . . . . 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 12613 . . . . 5  |-  ( F  e. Word  RR  ->  ( ( F  =/=  (/)  /\  ( F `  0 )  =/=  0 )  ->  A. m  e.  ( 0..^ ( # `  F ) ) ( ( T `  F
) `  m )  =/=  0 ) )
199198imp 427 . . . 4  |-  ( ( F  e. Word  RR  /\  ( F  =/=  (/)  /\  ( F `  0 )  =/=  0 ) )  ->  A. m  e.  (
0..^ ( # `  F
) ) ( ( T `  F ) `
 m )  =/=  0 )
200199adantrr 714 . . 3  |-  ( ( F  e. Word  RR  /\  ( ( F  =/=  (/)  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  A. m  e.  ( 0..^ ( # `  F
) ) ( ( T `  F ) `
 m )  =/=  0 )
201 fveq2 5774 . . . . 5  |-  ( m  =  N  ->  (
( T `  F
) `  m )  =  ( ( T `
 F ) `  N ) )
202201neeq1d 2659 . . . 4  |-  ( m  =  N  ->  (
( ( T `  F ) `  m
)  =/=  0  <->  (
( T `  F
) `  N )  =/=  0 ) )
203202rspcva 3133 . . 3  |-  ( ( N  e.  ( 0..^ ( # `  F
) )  /\  A. m  e.  ( 0..^ ( # `  F
) ) ( ( T `  F ) `
 m )  =/=  0 )  ->  (
( T `  F
) `  N )  =/=  0 )
2048, 200, 203syl2anc 659 . 2  |-  ( ( F  e. Word  RR  /\  ( ( F  =/=  (/)  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( ( T `
 F ) `  N )  =/=  0
)
2052, 6, 7, 204syl12anc 1224 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 184    \/ wo 366    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   A.wral 2732    \ cdif 3386   (/)c0 3711   ifcif 3857   {csn 3944   {cpr 3946   {ctp 3948   <.cop 3950    |-> cmpt 4425   ` cfv 5496  (class class class)co 6196    |-> cmpt2 6198   RRcr 9402   0cc0 9403   1c1 9404    + caddc 9406   RR*cxr 9538    - cmin 9718   -ucneg 9719   NNcn 10452   NN0cn0 10712   ZZ>=cuz 11001   ...cfz 11593  ..^cfzo 11717   #chash 12307  Word cword 12438   ++ cconcat 12440   <"cs1 12441  sgncsgn 12921   sum_csu 13510   ndxcnx 14631   Basecbs 14634   +g cplusg 14702    gsumg cgsu 14848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-supp 6818  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-oi 7850  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-fzo 11718  df-seq 12011  df-hash 12308  df-word 12446  df-lsw 12447  df-concat 12448  df-s1 12449  df-substr 12450  df-sgn 12922  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-plusg 14715  df-0g 14849  df-gsum 14850  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-mulg 16177  df-cntz 16472
This theorem is referenced by:  signstfvcl  28713
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