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Theorem signstfvneq0 26973
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 753 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) )  ->  F  e.  (Word  RR  \  { (/) } ) )
21eldifad 3340 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) )  ->  F  e. Word  RR )
3 eldifsn 4000 . . . . 5  |-  ( F  e.  (Word  RR  \  { (/) } )  <->  ( F  e. Word  RR  /\  F  =/=  (/) ) )
43simprbi 464 . . . 4  |-  ( F  e.  (Word  RR  \  { (/) } )  ->  F  =/=  (/) )
51, 4syl 16 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) )  ->  F  =/=  (/) )
6 simplr 754 . . 3  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( F `  0
)  =/=  0 )
75, 6jca 532 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( F  =/=  (/)  /\  ( F `  0 )  =/=  0 ) )
8 simpr 461 . 2  |-  ( ( ( F  e.  (Word 
RR  \  { (/) } )  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) )  ->  N  e.  ( 0..^ ( # `  F
) ) )
9 simprr 756 . . 3  |-  ( ( F  e. Word  RR  /\  ( ( F  =/=  (/)  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  N  e.  ( 0..^ ( # `  F
) ) )
10 neeq1 2616 . . . . . . . 8  |-  ( g  =  (/)  ->  ( g  =/=  (/)  <->  (/)  =/=  (/) ) )
11 fveq1 5690 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( g `
 0 )  =  ( (/) `  0 ) )
1211neeq1d 2621 . . . . . . . 8  |-  ( g  =  (/)  ->  ( ( g `  0 )  =/=  0  <->  ( (/) `  0
)  =/=  0 ) )
1310, 12anbi12d 710 . . . . . . 7  |-  ( g  =  (/)  ->  ( ( g  =/=  (/)  /\  (
g `  0 )  =/=  0 )  <->  ( (/)  =/=  (/)  /\  ( (/) `  0 )  =/=  0 ) ) )
14 fveq2 5691 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( # `  g )  =  (
# `  (/) ) )
1514oveq2d 6107 . . . . . . . 8  |-  ( g  =  (/)  ->  ( 0..^ ( # `  g
) )  =  ( 0..^ ( # `  (/) ) ) )
16 fveq2 5691 . . . . . . . . . 10  |-  ( g  =  (/)  ->  ( T `
 g )  =  ( T `  (/) ) )
1716fveq1d 5693 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( ( T `  g ) `
 m )  =  ( ( T `  (/) ) `  m ) )
1817neeq1d 2621 . . . . . . . 8  |-  ( g  =  (/)  ->  ( ( ( T `  g
) `  m )  =/=  0  <->  ( ( T `
 (/) ) `  m
)  =/=  0 ) )
1915, 18raleqbidv 2931 . . . . . . 7  |-  ( g  =  (/)  ->  ( A. m  e.  ( 0..^ ( # `  g
) ) ( ( T `  g ) `
 m )  =/=  0  <->  A. m  e.  ( 0..^ ( # `  (/) ) ) ( ( T `  (/) ) `  m )  =/=  0 ) )
2013, 19imbi12d 320 . . . . . 6  |-  ( g  =  (/)  ->  ( ( ( g  =/=  (/)  /\  (
g `  0 )  =/=  0 )  ->  A. m  e.  ( 0..^ ( # `  g ) ) ( ( T `  g
) `  m )  =/=  0 )  <->  ( ( (/) 
=/=  (/)  /\  ( (/) `  0 )  =/=  0
)  ->  A. m  e.  ( 0..^ ( # `  (/) ) ) ( ( T `  (/) ) `  m )  =/=  0
) ) )
21 neeq1 2616 . . . . . . . 8  |-  ( g  =  e  ->  (
g  =/=  (/)  <->  e  =/=  (/) ) )
22 fveq1 5690 . . . . . . . . 9  |-  ( g  =  e  ->  (
g `  0 )  =  ( e ` 
0 ) )
2322neeq1d 2621 . . . . . . . 8  |-  ( g  =  e  ->  (
( g `  0
)  =/=  0  <->  (
e `  0 )  =/=  0 ) )
2421, 23anbi12d 710 . . . . . . 7  |-  ( g  =  e  ->  (
( g  =/=  (/)  /\  (
g `  0 )  =/=  0 )  <->  ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 ) ) )
25 fveq2 5691 . . . . . . . . 9  |-  ( g  =  e  ->  ( # `
 g )  =  ( # `  e
) )
2625oveq2d 6107 . . . . . . . 8  |-  ( g  =  e  ->  (
0..^ ( # `  g
) )  =  ( 0..^ ( # `  e
) ) )
27 fveq2 5691 . . . . . . . . . 10  |-  ( g  =  e  ->  ( T `  g )  =  ( T `  e ) )
2827fveq1d 5693 . . . . . . . . 9  |-  ( g  =  e  ->  (
( T `  g
) `  m )  =  ( ( T `
 e ) `  m ) )
2928neeq1d 2621 . . . . . . . 8  |-  ( g  =  e  ->  (
( ( T `  g ) `  m
)  =/=  0  <->  (
( T `  e
) `  m )  =/=  0 ) )
3026, 29raleqbidv 2931 . . . . . . 7  |-  ( g  =  e  ->  ( A. m  e.  (
0..^ ( # `  g
) ) ( ( T `  g ) `
 m )  =/=  0  <->  A. m  e.  ( 0..^ ( # `  e
) ) ( ( T `  e ) `
 m )  =/=  0 ) )
3124, 30imbi12d 320 . . . . . 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 ) ) )
32 neeq1 2616 . . . . . . . 8  |-  ( g  =  ( e concat  <" k "> )  ->  ( g  =/=  (/)  <->  ( e concat  <" k "> )  =/=  (/) ) )
33 fveq1 5690 . . . . . . . . 9  |-  ( g  =  ( e concat  <" k "> )  ->  ( g `  0
)  =  ( ( e concat  <" k "> ) `  0
) )
3433neeq1d 2621 . . . . . . . 8  |-  ( g  =  ( e concat  <" k "> )  ->  ( ( g ` 
0 )  =/=  0  <->  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )
3532, 34anbi12d 710 . . . . . . 7  |-  ( g  =  ( e concat  <" k "> )  ->  ( ( g  =/=  (/)  /\  ( g ` 
0 )  =/=  0
)  <->  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) ) )
36 fveq2 5691 . . . . . . . . 9  |-  ( g  =  ( e concat  <" k "> )  ->  ( # `  g
)  =  ( # `  ( e concat  <" k "> ) ) )
3736oveq2d 6107 . . . . . . . 8  |-  ( g  =  ( e concat  <" k "> )  ->  ( 0..^ ( # `  g ) )  =  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )
38 fveq2 5691 . . . . . . . . . 10  |-  ( g  =  ( e concat  <" k "> )  ->  ( T `  g
)  =  ( T `
 ( e concat  <" k "> )
) )
3938fveq1d 5693 . . . . . . . . 9  |-  ( g  =  ( e concat  <" k "> )  ->  ( ( T `  g ) `  m
)  =  ( ( T `  ( e concat  <" k "> ) ) `  m
) )
4039neeq1d 2621 . . . . . . . 8  |-  ( g  =  ( e concat  <" k "> )  ->  ( ( ( T `
 g ) `  m )  =/=  0  <->  ( ( T `  (
e concat  <" k "> ) ) `  m )  =/=  0
) )
4137, 40raleqbidv 2931 . . . . . . 7  |-  ( g  =  ( e concat  <" k "> )  ->  ( A. m  e.  ( 0..^ ( # `  g ) ) ( ( T `  g
) `  m )  =/=  0  <->  A. m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ( ( T `  ( e concat  <" k "> ) ) `  m )  =/=  0
) )
4235, 41imbi12d 320 . . . . . 6  |-  ( g  =  ( e concat  <" k "> )  ->  ( ( ( g  =/=  (/)  /\  ( g `
 0 )  =/=  0 )  ->  A. m  e.  ( 0..^ ( # `  g ) ) ( ( T `  g
) `  m )  =/=  0 )  <->  ( (
( e concat  <" k "> )  =/=  (/)  /\  (
( e concat  <" k "> ) `  0
)  =/=  0 )  ->  A. m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ( ( T `  ( e concat  <" k "> ) ) `  m )  =/=  0
) ) )
43 neeq1 2616 . . . . . . . 8  |-  ( g  =  F  ->  (
g  =/=  (/)  <->  F  =/=  (/) ) )
44 fveq1 5690 . . . . . . . . 9  |-  ( g  =  F  ->  (
g `  0 )  =  ( F ` 
0 ) )
4544neeq1d 2621 . . . . . . . 8  |-  ( g  =  F  ->  (
( g `  0
)  =/=  0  <->  ( F `  0 )  =/=  0 ) )
4643, 45anbi12d 710 . . . . . . 7  |-  ( g  =  F  ->  (
( g  =/=  (/)  /\  (
g `  0 )  =/=  0 )  <->  ( F  =/=  (/)  /\  ( F `
 0 )  =/=  0 ) ) )
47 fveq2 5691 . . . . . . . . 9  |-  ( g  =  F  ->  ( # `
 g )  =  ( # `  F
) )
4847oveq2d 6107 . . . . . . . 8  |-  ( g  =  F  ->  (
0..^ ( # `  g
) )  =  ( 0..^ ( # `  F
) ) )
49 fveq2 5691 . . . . . . . . . 10  |-  ( g  =  F  ->  ( T `  g )  =  ( T `  F ) )
5049fveq1d 5693 . . . . . . . . 9  |-  ( g  =  F  ->  (
( T `  g
) `  m )  =  ( ( T `
 F ) `  m ) )
5150neeq1d 2621 . . . . . . . 8  |-  ( g  =  F  ->  (
( ( T `  g ) `  m
)  =/=  0  <->  (
( T `  F
) `  m )  =/=  0 ) )
5248, 51raleqbidv 2931 . . . . . . 7  |-  ( g  =  F  ->  ( A. m  e.  (
0..^ ( # `  g
) ) ( ( T `  g ) `
 m )  =/=  0  <->  A. m  e.  ( 0..^ ( # `  F
) ) ( ( T `  F ) `
 m )  =/=  0 ) )
5346, 52imbi12d 320 . . . . . 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 ) ) )
54 neirr 2613 . . . . . . . 8  |-  -.  (/)  =/=  (/)
5554intnanr 906 . . . . . . 7  |-  -.  ( (/) 
=/=  (/)  /\  ( (/) `  0 )  =/=  0
)
5655pm2.21i 131 . . . . . 6  |-  ( (
(/)  =/=  (/)  /\  ( (/) `  0 )  =/=  0 )  ->  A. m  e.  ( 0..^ ( # `  (/) ) ) ( ( T `  (/) ) `  m )  =/=  0
)
57 fveq2 5691 . . . . . . . . . . . . 13  |-  ( n  =  m  ->  (
( T `  e
) `  n )  =  ( ( T `
 e ) `  m ) )
5857neeq1d 2621 . . . . . . . . . . . 12  |-  ( n  =  m  ->  (
( ( T `  e ) `  n
)  =/=  0  <->  (
( T `  e
) `  m )  =/=  0 ) )
5958cbvralv 2947 . . . . . . . . . . 11  |-  ( A. n  e.  ( 0..^ ( # `  e
) ) ( ( T `  e ) `
 n )  =/=  0  <->  A. m  e.  ( 0..^ ( # `  e
) ) ( ( T `  e ) `
 m )  =/=  0 )
6059imbi2i 312 . . . . . . . . . 10  |-  ( ( ( 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 ) )
6160anbi2i 694 . . . . . . . . 9  |-  ( ( ( 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 ) ) )
62 simplr 754 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =  (/) )  ->  m  e.  ( 0..^ ( # `  e
) ) )
63 noel 3641 . . . . . . . . . . . . . . 15  |-  -.  m  e.  (/)
64 fveq2 5691 . . . . . . . . . . . . . . . . . . 19  |-  ( e  =  (/)  ->  ( # `  e )  =  (
# `  (/) ) )
65 hash0 12135 . . . . . . . . . . . . . . . . . . 19  |-  ( # `  (/) )  =  0
6664, 65syl6eq 2491 . . . . . . . . . . . . . . . . . 18  |-  ( e  =  (/)  ->  ( # `  e )  =  0 )
6766oveq2d 6107 . . . . . . . . . . . . . . . . 17  |-  ( e  =  (/)  ->  ( 0..^ ( # `  e
) )  =  ( 0..^ 0 ) )
68 fzo0 11573 . . . . . . . . . . . . . . . . 17  |-  ( 0..^ 0 )  =  (/)
6967, 68syl6eq 2491 . . . . . . . . . . . . . . . 16  |-  ( e  =  (/)  ->  ( 0..^ ( # `  e
) )  =  (/) )
7069eleq2d 2510 . . . . . . . . . . . . . . 15  |-  ( e  =  (/)  ->  ( m  e.  ( 0..^ (
# `  e )
)  <->  m  e.  (/) ) )
7163, 70mtbiri 303 . . . . . . . . . . . . . 14  |-  ( e  =  (/)  ->  -.  m  e.  ( 0..^ ( # `  e ) ) )
7271adantl 466 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =  (/) )  ->  -.  m  e.  ( 0..^ ( # `  e
) ) )
7362, 72pm2.21dd 174 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =  (/) )  ->  ( ( T `
 ( e concat  <" k "> )
) `  m )  =/=  0 )
74 simp-6l 769 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =/=  (/) )  -> 
e  e. Word  RR )
75 simp-6r 770 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =/=  (/) )  -> 
k  e.  RR )
76 simplr 754 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =/=  (/) )  ->  m  e.  ( 0..^ ( # `  e
) ) )
77 signsv.p . . . . . . . . . . . . . . 15  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
78 signsv.w . . . . . . . . . . . . . . 15  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
79 signsv.t . . . . . . . . . . . . . . 15  |-  T  =  ( f  e. Word  RR  |->  ( n  e.  (
0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) ) )
80 signsv.v . . . . . . . . . . . . . . 15  |-  V  =  ( f  e. Word  RR  |->  sum_ j  e.  ( 1..^ ( # `  f
) ) if ( ( ( T `  f ) `  j
)  =/=  ( ( T `  f ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
8177, 78, 79, 80signstfvp 26972 . . . . . . . . . . . . . 14  |-  ( ( e  e. Word  RR  /\  k  e.  RR  /\  m  e.  ( 0..^ ( # `  e ) ) )  ->  ( ( T `
 ( e concat  <" k "> )
) `  m )  =  ( ( T `
 e ) `  m ) )
8274, 75, 76, 81syl3anc 1218 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =/=  (/) )  -> 
( ( T `  ( e concat  <" k "> ) ) `  m )  =  ( ( T `  e
) `  m )
)
83 simpr 461 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =/=  (/) )  -> 
e  =/=  (/) )
84 simplll 757 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  ->  ( e  e. Word  RR  /\  k  e.  RR ) )
8584ad2antrr 725 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =/=  (/) )  -> 
( e  e. Word  RR  /\  k  e.  RR ) )
86 simplrr 760 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  ( m  e.  ( 0..^ (
# `  ( e concat  <" k "> ) ) )  /\  m  e.  ( 0..^ ( # `  e
) )  /\  e  =/=  (/) ) )  -> 
( ( e concat  <" k "> ) `  0 )  =/=  0 )
87863anassrs 1209 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =/=  (/) )  -> 
( ( e concat  <" k "> ) `  0 )  =/=  0 )
88 simpll 753 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  -> 
e  e. Word  RR )
89 simplr 754 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  -> 
k  e.  RR )
9089s1cld 12294 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  ->  <" k ">  e. Word  RR )
91 lennncl 12250 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( e  e. Word  RR  /\  e  =/=  (/) )  ->  ( # `
 e )  e.  NN )
9288, 91sylancom 667 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  -> 
( # `  e )  e.  NN )
93 fzo0end 11619 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  e )  e.  NN  ->  ( ( # `
 e )  - 
1 )  e.  ( 0..^ ( # `  e
) ) )
94 elfzolt3b 11564 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( # `  e
)  -  1 )  e.  ( 0..^ (
# `  e )
)  ->  0  e.  ( 0..^ ( # `  e
) ) )
9592, 93, 943syl 20 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  -> 
0  e.  ( 0..^ ( # `  e
) ) )
96 ccatval1 12276 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  e. Word  RR  /\  <" k ">  e. Word  RR  /\  0  e.  ( 0..^ ( # `  e ) ) )  ->  ( ( e concat  <" k "> ) `  0 )  =  ( e ` 
0 ) )
9788, 90, 95, 96syl3anc 1218 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  -> 
( ( e concat  <" k "> ) `  0 )  =  ( e `  0
) )
9897neeq1d 2621 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  -> 
( ( ( e concat  <" k "> ) `  0 )  =/=  0  <->  ( e ` 
0 )  =/=  0
) )
9998biimpa 484 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  e  =/=  (/) )  /\  ( ( e concat  <" k "> ) `  0 )  =/=  0 )  ->  (
e `  0 )  =/=  0 )
10085, 83, 87, 99syl21anc 1217 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =/=  (/) )  -> 
( e `  0
)  =/=  0 )
10183, 100jca 532 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =/=  (/) )  -> 
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 ) )
102 simp-5r 768 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =/=  (/) )  -> 
( ( e  =/=  (/)  /\  ( e ` 
0 )  =/=  0
)  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )
103101, 102mpd 15 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =/=  (/) )  ->  A. n  e.  (
0..^ ( # `  e
) ) ( ( T `  e ) `
 n )  =/=  0 )
10458rspcv 3069 . . . . . . . . . . . . . . 15  |-  ( m  e.  ( 0..^ (
# `  e )
)  ->  ( A. n  e.  ( 0..^ ( # `  e
) ) ( ( T `  e ) `
 n )  =/=  0  ->  ( ( T `  e ) `  m )  =/=  0
) )
105104imp 429 . . . . . . . . . . . . . 14  |-  ( ( m  e.  ( 0..^ ( # `  e
) )  /\  A. n  e.  ( 0..^ ( # `  e
) ) ( ( T `  e ) `
 n )  =/=  0 )  ->  (
( T `  e
) `  m )  =/=  0 )
10676, 103, 105syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =/=  (/) )  -> 
( ( T `  e ) `  m
)  =/=  0 )
10782, 106eqnetrd 2626 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  /\  e  =/=  (/) )  -> 
( ( T `  ( e concat  <" k "> ) ) `  m )  =/=  0
)
108 exmidne 2614 . . . . . . . . . . . . 13  |-  ( e  =  (/)  \/  e  =/=  (/) )
109108a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e ` 
0 )  =/=  0
)  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  ->  ( e  =  (/)  \/  e  =/=  (/) ) )
11073, 107, 109mpjaodan 784 . . . . . . . . . . 11  |-  ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e ` 
0 )  =/=  0
)  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  /\  m  e.  ( 0..^ ( # `  e ) ) )  ->  ( ( T `
 ( e concat  <" k "> )
) `  m )  =/=  0 )
111 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e ` 
0 )  =/=  0
)  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  /\  m  =  ( # `  e
) )  ->  m  =  ( # `  e
) )
112111fveq2d 5695 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e ` 
0 )  =/=  0
)  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  /\  m  =  ( # `  e
) )  ->  (
( T `  (
e concat  <" k "> ) ) `  m )  =  ( ( T `  (
e concat  <" k "> ) ) `  ( # `  e ) ) )
113 simpr 461 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  e  =  (/) )  ->  e  =  (/) )
114 simp-4r 766 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  e  =  (/) )  ->  k  e.  RR )
115 simplrl 759 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  e  =  (/) )  ->  (
( e concat  <" k "> )  =/=  (/)  /\  (
( e concat  <" k "> ) `  0
)  =/=  0 ) )
116115simprd 463 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  e  =  (/) )  ->  (
( e concat  <" k "> ) `  0
)  =/=  0 )
117 oveq1 6098 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( e  =  (/)  ->  ( e concat  <" k "> )  =  ( (/) concat  <" k "> ) )
118 s1cl 12293 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  e.  RR  ->  <" k ">  e. Word  RR )
119 ccatlid 12284 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( <" k ">  e. Word  RR  ->  ( (/) concat  <" k "> )  =  <" k "> )
120118, 119syl 16 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  e.  RR  ->  ( (/) concat  <" k "> )  =  <" k "> )
121117, 120sylan9eq 2495 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( e  =  (/)  /\  k  e.  RR )  ->  (
e concat  <" k "> )  =  <" k "> )
122121fveq2d 5695 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( e  =  (/)  /\  k  e.  RR )  ->  ( T `  ( e concat  <" k "> ) )  =  ( T `  <" k "> ) )
123122adantr 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e concat  <" k "> ) `  0 )  =/=  0 )  ->  ( T `  ( e concat  <" k "> ) )  =  ( T `  <" k "> ) )
124 simplr 754 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e concat  <" k "> ) `  0 )  =/=  0 )  ->  k  e.  RR )
12577, 78, 79, 80signstf0 26969 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  e.  RR  ->  ( T `  <" k "> )  =  <" (sgn `  k ) "> )
126124, 125syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e concat  <" k "> ) `  0 )  =/=  0 )  ->  ( T `  <" k "> )  =  <" (sgn `  k ) "> )
127123, 126eqtrd 2475 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e concat  <" k "> ) `  0 )  =/=  0 )  ->  ( T `  ( e concat  <" k "> ) )  =  <" (sgn `  k ) "> )
12866ad2antrr 725 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e concat  <" k "> ) `  0 )  =/=  0 )  ->  ( # `
 e )  =  0 )
129127, 128fveq12d 5697 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e concat  <" k "> ) `  0 )  =/=  0 )  ->  (
( T `  (
e concat  <" k "> ) ) `  ( # `  e ) )  =  ( <" (sgn `  k ) "> `  0 )
)
130 sgnclre 26922 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  RR  ->  (sgn `  k )  e.  RR )
131 s1fv 12298 . . . . . . . . . . . . . . . . . . 19  |-  ( (sgn
`  k )  e.  RR  ->  ( <" (sgn `  k ) "> `  0 )  =  (sgn `  k )
)
132124, 130, 1313syl 20 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e concat  <" k "> ) `  0 )  =/=  0 )  ->  ( <" (sgn `  k
) "> `  0
)  =  (sgn `  k ) )
133129, 132eqtrd 2475 . . . . . . . . . . . . . . . . 17  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e concat  <" k "> ) `  0 )  =/=  0 )  ->  (
( T `  (
e concat  <" k "> ) ) `  ( # `  e ) )  =  (sgn `  k ) )
134121fveq1d 5693 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( e  =  (/)  /\  k  e.  RR )  ->  (
( e concat  <" k "> ) `  0
)  =  ( <" k "> `  0 ) )
135 s1fv 12298 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  e.  RR  ->  ( <" k "> `  0 )  =  k )
136135adantl 466 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( e  =  (/)  /\  k  e.  RR )  ->  ( <" k "> `  0 )  =  k )
137134, 136eqtrd 2475 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  =  (/)  /\  k  e.  RR )  ->  (
( e concat  <" k "> ) `  0
)  =  k )
138137neeq1d 2621 . . . . . . . . . . . . . . . . . . 19  |-  ( ( e  =  (/)  /\  k  e.  RR )  ->  (
( ( e concat  <" k "> ) `  0 )  =/=  0  <->  k  =/=  0
) )
139138biimpa 484 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e concat  <" k "> ) `  0 )  =/=  0 )  ->  k  =/=  0 )
140 rexr 9429 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  e.  RR  ->  k  e.  RR* )
141 sgn0bi 26930 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  e.  RR*  ->  ( (sgn
`  k )  =  0  <->  k  =  0 ) )
142140, 141syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  e.  RR  ->  (
(sgn `  k )  =  0  <->  k  = 
0 ) )
143142necon3bid 2643 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  RR  ->  (
(sgn `  k )  =/=  0  <->  k  =/=  0
) )
144143biimpar 485 . . . . . . . . . . . . . . . . . 18  |-  ( ( k  e.  RR  /\  k  =/=  0 )  -> 
(sgn `  k )  =/=  0 )
145124, 139, 144syl2anc 661 . . . . . . . . . . . . . . . . 17  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e concat  <" k "> ) `  0 )  =/=  0 )  ->  (sgn `  k )  =/=  0
)
146133, 145eqnetrd 2626 . . . . . . . . . . . . . . . 16  |-  ( ( ( e  =  (/)  /\  k  e.  RR )  /\  ( ( e concat  <" k "> ) `  0 )  =/=  0 )  ->  (
( T `  (
e concat  <" k "> ) ) `  ( # `  e ) )  =/=  0 )
147113, 114, 116, 146syl21anc 1217 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  e  =  (/) )  ->  (
( T `  (
e concat  <" k "> ) ) `  ( # `  e ) )  =/=  0 )
148 simplll 757 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  (
( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
e  e. Word  RR )
149 simpr 461 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  (
( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  ->  -.  e  =  (/) )
150 elsn 3891 . . . . . . . . . . . . . . . . . . . 20  |-  ( e  e.  { (/) }  <->  e  =  (/) )
151149, 150sylnibr 305 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  (
( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  ->  -.  e  e.  { (/) } )
152148, 151eldifd 3339 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  (
( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
e  e.  (Word  RR  \  { (/) } ) )
153 simpllr 758 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  (
( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
k  e.  RR )
15477, 78, 79, 80signstfvn 26970 . . . . . . . . . . . . . . . . . 18  |-  ( ( e  e.  (Word  RR  \  { (/) } )  /\  k  e.  RR )  ->  ( ( T `  ( e concat  <" k "> ) ) `  ( # `  e ) )  =  ( ( ( T `  e
) `  ( ( # `
 e )  - 
1 ) )  .+^  (sgn `  k ) ) )
155152, 153, 154syl2anc 661 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  (
( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( ( T `  ( e concat  <" k "> ) ) `  ( # `  e ) )  =  ( ( ( T `  e
) `  ( ( # `
 e )  - 
1 ) )  .+^  (sgn `  k ) ) )
156155adantllr 718 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( ( T `  ( e concat  <" k "> ) ) `  ( # `  e ) )  =  ( ( ( T `  e
) `  ( ( # `
 e )  - 
1 ) )  .+^  (sgn `  k ) ) )
157149neneqad 2681 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  (
( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
e  =/=  (/) )
158148, 157, 91syl2anc 661 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  (
( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( # `  e )  e.  NN )
159158, 93syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  (
( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( ( # `  e
)  -  1 )  e.  ( 0..^ (
# `  e )
) )
16077, 78, 79, 80signstcl 26966 . . . . . . . . . . . . . . . . . . 19  |-  ( ( e  e. Word  RR  /\  ( ( # `  e
)  -  1 )  e.  ( 0..^ (
# `  e )
) )  ->  (
( T `  e
) `  ( ( # `
 e )  - 
1 ) )  e. 
{ -u 1 ,  0 ,  1 } )
161148, 159, 160syl2anc 661 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  (
( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( ( T `  e ) `  (
( # `  e )  -  1 ) )  e.  { -u 1 ,  0 ,  1 } )
162161adantllr 718 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( ( T `  e ) `  (
( # `  e )  -  1 ) )  e.  { -u 1 ,  0 ,  1 } )
163153rexrd 9433 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  (
( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
k  e.  RR* )
164 sgncl 26921 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  RR*  ->  (sgn `  k )  e.  { -u 1 ,  0 ,  1 } )
165163, 164syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  (
( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
(sgn `  k )  e.  { -u 1 ,  0 ,  1 } )
166165adantllr 718 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
(sgn `  k )  e.  { -u 1 ,  0 ,  1 } )
167159adantllr 718 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( ( # `  e
)  -  1 )  e.  ( 0..^ (
# `  e )
) )
168157adantllr 718 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
e  =/=  (/) )
169 simplll 757 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( e  e. Word  RR  /\  k  e.  RR ) )
170 simplrl 759 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )
171170simprd 463 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( ( e concat  <" k "> ) `  0 )  =/=  0 )
172169, 168, 171, 99syl21anc 1217 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( e `  0
)  =/=  0 )
173168, 172jca 532 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 ) )
174 simpllr 758 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( ( e  =/=  (/)  /\  ( e ` 
0 )  =/=  0
)  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )
175173, 174mpd 15 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  ->  A. n  e.  (
0..^ ( # `  e
) ) ( ( T `  e ) `
 n )  =/=  0 )
176 fveq2 5691 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  =  ( ( # `  e )  -  1 )  ->  ( ( T `  e ) `  n )  =  ( ( T `  e
) `  ( ( # `
 e )  - 
1 ) ) )
177176neeq1d 2621 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  =  ( ( # `  e )  -  1 )  ->  ( (
( T `  e
) `  n )  =/=  0  <->  ( ( T `
 e ) `  ( ( # `  e
)  -  1 ) )  =/=  0 ) )
178177rspcv 3069 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( # `  e
)  -  1 )  e.  ( 0..^ (
# `  e )
)  ->  ( A. n  e.  ( 0..^ ( # `  e
) ) ( ( T `  e ) `
 n )  =/=  0  ->  ( ( T `  e ) `  ( ( # `  e
)  -  1 ) )  =/=  0 ) )
179178imp 429 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( # `  e
)  -  1 )  e.  ( 0..^ (
# `  e )
)  /\  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 )  ->  (
( T `  e
) `  ( ( # `
 e )  - 
1 ) )  =/=  0 )
180167, 175, 179syl2anc 661 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( ( T `  e ) `  (
( # `  e )  -  1 ) )  =/=  0 )
18177, 78signswn0 26961 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( 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
)
182162, 166, 180, 181syl21anc 1217 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( ( ( T `
 e ) `  ( ( # `  e
)  -  1 ) )  .+^  (sgn `  k
) )  =/=  0
)
183156, 182eqnetrd 2626 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  /\  -.  e  =  (/) )  -> 
( ( T `  ( e concat  <" k "> ) ) `  ( # `  e ) )  =/=  0 )
184 exmidd 416 . . . . . . . . . . . . . . 15  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  ->  (
e  =  (/)  \/  -.  e  =  (/) ) )
185147, 183, 184mpjaodan 784 . . . . . . . . . . . . . 14  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ) )  ->  (
( T `  (
e concat  <" k "> ) ) `  ( # `  e ) )  =/=  0 )
186185anassrs 648 . . . . . . . . . . . . 13  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  ->  ( ( T `  ( e concat  <" k "> ) ) `  ( # `
 e ) )  =/=  0 )
187186adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e ` 
0 )  =/=  0
)  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  /\  m  =  ( # `  e
) )  ->  (
( T `  (
e concat  <" k "> ) ) `  ( # `  e ) )  =/=  0 )
188112, 187eqnetrd 2626 . . . . . . . . . . 11  |-  ( ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e ` 
0 )  =/=  0
)  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  /\  m  =  ( # `  e
) )  ->  (
( T `  (
e concat  <" k "> ) ) `  m )  =/=  0
)
189 lencl 12249 . . . . . . . . . . . . . 14  |-  ( e  e. Word  RR  ->  ( # `  e )  e.  NN0 )
190 nn0uz 10895 . . . . . . . . . . . . . 14  |-  NN0  =  ( ZZ>= `  0 )
191189, 190syl6eleq 2533 . . . . . . . . . . . . 13  |-  ( e  e. Word  RR  ->  ( # `  e )  e.  (
ZZ>= `  0 ) )
192191ad4antr 731 . . . . . . . . . . . 12  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  ->  ( # `  e
)  e.  ( ZZ>= ` 
0 ) )
193 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  ->  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) )
194 ccatlen 12275 . . . . . . . . . . . . . . . . . 18  |-  ( ( e  e. Word  RR  /\  <" k ">  e. Word  RR )  ->  ( # `
 ( e concat  <" k "> )
)  =  ( (
# `  e )  +  ( # `  <" k "> )
) )
195118, 194sylan2 474 . . . . . . . . . . . . . . . . 17  |-  ( ( e  e. Word  RR  /\  k  e.  RR )  ->  ( # `  (
e concat  <" k "> ) )  =  ( ( # `  e
)  +  ( # `  <" k "> ) ) )
196 s1len 12296 . . . . . . . . . . . . . . . . . 18  |-  ( # `  <" k "> )  =  1
197196oveq2i 6102 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  e )  +  ( # `  <" k "> )
)  =  ( (
# `  e )  +  1 )
198195, 197syl6eq 2491 . . . . . . . . . . . . . . . 16  |-  ( ( e  e. Word  RR  /\  k  e.  RR )  ->  ( # `  (
e concat  <" k "> ) )  =  ( ( # `  e
)  +  1 ) )
199198oveq2d 6107 . . . . . . . . . . . . . . 15  |-  ( ( e  e. Word  RR  /\  k  e.  RR )  ->  ( 0..^ ( # `  ( e concat  <" k "> ) ) )  =  ( 0..^ ( ( # `  e
)  +  1 ) ) )
200199eleq2d 2510 . . . . . . . . . . . . . 14  |-  ( ( e  e. Word  RR  /\  k  e.  RR )  ->  ( m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) )  <-> 
m  e.  ( 0..^ ( ( # `  e
)  +  1 ) ) ) )
201200biimpa 484 . . . . . . . . . . . . 13  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) )  ->  m  e.  ( 0..^ ( ( # `  e )  +  1 ) ) )
20284, 193, 201syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  ->  m  e.  ( 0..^ ( ( # `  e )  +  1 ) ) )
203 fzosplitsni 11634 . . . . . . . . . . . . 13  |-  ( (
# `  e )  e.  ( ZZ>= `  0 )  ->  ( m  e.  ( 0..^ ( ( # `  e )  +  1 ) )  <->  ( m  e.  ( 0..^ ( # `  e ) )  \/  m  =  ( # `  e ) ) ) )
204203biimpa 484 . . . . . . . . . . . 12  |-  ( ( ( # `  e
)  e.  ( ZZ>= ` 
0 )  /\  m  e.  ( 0..^ ( (
# `  e )  +  1 ) ) )  ->  ( m  e.  ( 0..^ ( # `  e ) )  \/  m  =  ( # `  e ) ) )
205192, 202, 204syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  ->  ( m  e.  ( 0..^ ( # `  e ) )  \/  m  =  ( # `  e ) ) )
206110, 188, 205mpjaodan 784 . . . . . . . . . 10  |-  ( ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  (
( e  =/=  (/)  /\  (
e `  0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  /\  m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) )  ->  ( ( T `  ( e concat  <" k "> ) ) `  m
)  =/=  0 )
207206ralrimiva 2799 . . . . . . . . 9  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. n  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  n )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  ->  A. m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) ( ( T `  ( e concat  <" k "> ) ) `  m )  =/=  0
)
20861, 207sylanbr 473 . . . . . . . 8  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. m  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  m )  =/=  0 ) )  /\  ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 ) )  ->  A. m  e.  ( 0..^ ( # `  ( e concat  <" k "> ) ) ) ( ( T `  ( e concat  <" k "> ) ) `  m )  =/=  0
)
209208ex 434 . . . . . . 7  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. m  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  m )  =/=  0 ) )  -> 
( ( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 )  ->  A. m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ( ( T `  ( e concat  <" k "> ) ) `  m )  =/=  0
) )
210209ex 434 . . . . . 6  |-  ( ( e  e. Word  RR  /\  k  e.  RR )  ->  ( ( ( e  =/=  (/)  /\  ( e `
 0 )  =/=  0 )  ->  A. m  e.  ( 0..^ ( # `  e ) ) ( ( T `  e
) `  m )  =/=  0 )  ->  (
( ( e concat  <" k "> )  =/=  (/)  /\  ( ( e concat  <" k "> ) `  0
)  =/=  0 )  ->  A. m  e.  ( 0..^ ( # `  (
e concat  <" k "> ) ) ) ( ( T `  ( e concat  <" k "> ) ) `  m )  =/=  0
) ) )
21120, 31, 42, 53, 56, 210wrdind 12371 . . . . 5  |-  ( F  e. Word  RR  ->  ( ( F  =/=  (/)  /\  ( F `  0 )  =/=  0 )  ->  A. m  e.  ( 0..^ ( # `  F ) ) ( ( T `  F
) `  m )  =/=  0 ) )
212211imp 429 . . . 4  |-  ( ( F  e. Word  RR  /\  ( F  =/=  (/)  /\  ( F `  0 )  =/=  0 ) )  ->  A. m  e.  (
0..^ ( # `  F
) ) ( ( T `  F ) `
 m )  =/=  0 )
213212adantrr 716 . . 3  |-  ( ( F  e. Word  RR  /\  ( ( F  =/=  (/)  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  A. m  e.  ( 0..^ ( # `  F
) ) ( ( T `  F ) `
 m )  =/=  0 )
214 fveq2 5691 . . . . . 6  |-  ( m  =  N  ->  (
( T `  F
) `  m )  =  ( ( T `
 F ) `  N ) )
215214neeq1d 2621 . . . . 5  |-  ( m  =  N  ->  (
( ( T `  F ) `  m
)  =/=  0  <->  (
( T `  F
) `  N )  =/=  0 ) )
216215rspcv 3069 . . . 4  |-  ( N  e.  ( 0..^ (
# `  F )
)  ->  ( A. m  e.  ( 0..^ ( # `  F
) ) ( ( T `  F ) `
 m )  =/=  0  ->  ( ( T `  F ) `  N )  =/=  0
) )
217216imp 429 . . 3  |-  ( ( N  e.  ( 0..^ ( # `  F
) )  /\  A. m  e.  ( 0..^ ( # `  F
) ) ( ( T `  F ) `
 m )  =/=  0 )  ->  (
( T `  F
) `  N )  =/=  0 )
2189, 213, 217syl2anc 661 . 2  |-  ( ( F  e. Word  RR  /\  ( ( F  =/=  (/)  /\  ( F ` 
0 )  =/=  0
)  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( ( T `
 F ) `  N )  =/=  0
)
2192, 7, 8, 218syl12anc 1216 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 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715    \ cdif 3325   (/)c0 3637   ifcif 3791   {csn 3877   {cpr 3879   {ctp 3881   <.cop 3883    e. cmpt 4350   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285   RR*cxr 9417    - cmin 9595   -ucneg 9596   NNcn 10322   NN0cn0 10579   ZZ>=cuz 10861   ...cfz 11437  ..^cfzo 11548   #chash 12103  Word cword 12221   concat cconcat 12223   <"cs1 12224  sgncsgn 12575   sum_csu 13163   ndxcnx 14171   Basecbs 14174   +g cplusg 14238    gsumg cgsu 14379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-seq 11807  df-hash 12104  df-word 12229  df-concat 12231  df-s1 12232  df-substr 12233  df-sgn 12576  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-plusg 14251  df-0g 14380  df-gsum 14381  df-mnd 15415  df-mulg 15548  df-cntz 15835
This theorem is referenced by:  signstfvcl  26974
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