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Theorem signstfvc 28171
Description: Zero-skipping sign in a word compared to a shorter word. (Contributed by Thierry Arnoux, 11-Oct-2018.)
Hypotheses
Ref Expression
signsv.p  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
signsv.w  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
signsv.t  |-  T  =  ( f  e. Word  RR  |->  ( n  e.  (
0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) ) )
signsv.v  |-  V  =  ( f  e. Word  RR  |->  sum_ j  e.  ( 1..^ ( # `  f
) ) if ( ( ( T `  f ) `  j
)  =/=  ( ( T `  f ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
Assertion
Ref Expression
signstfvc  |-  ( ( F  e. Word  RR  /\  G  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 ( F concat  G
) ) `  N
)  =  ( ( T `  F ) `
 N ) )
Distinct variable groups:    a, b,  .+^    f, i, n, F    f, W, i, n    i, N, n
Allowed substitution hints:    .+^ ( f, i,
j, n)    T( f,
i, j, n, a, b)    F( j, a, b)    G( f, i, j, n, a, b)    N( f, j, a, b)    V( f, i, j, n, a, b)    W( j, a, b)

Proof of Theorem signstfvc
Dummy variables  e 
g  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6290 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( F concat 
g )  =  ( F concat  (/) ) )
21fveq2d 5868 . . . . . . . 8  |-  ( g  =  (/)  ->  ( T `
 ( F concat  g
) )  =  ( T `  ( F concat  (/) ) ) )
32fveq1d 5866 . . . . . . 7  |-  ( g  =  (/)  ->  ( ( T `  ( F concat 
g ) ) `  N )  =  ( ( T `  ( F concat 
(/) ) ) `  N ) )
43eqeq1d 2469 . . . . . 6  |-  ( g  =  (/)  ->  ( ( ( T `  ( F concat  g ) ) `  N )  =  ( ( T `  F
) `  N )  <->  ( ( T `  ( F concat 
(/) ) ) `  N )  =  ( ( T `  F
) `  N )
) )
54imbi2d 316 . . . . 5  |-  ( g  =  (/)  ->  ( ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( ( T `  ( F concat  g )
) `  N )  =  ( ( T `
 F ) `  N ) )  <->  ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 ( F concat  (/) ) ) `
 N )  =  ( ( T `  F ) `  N
) ) ) )
6 oveq2 6290 . . . . . . . . 9  |-  ( g  =  e  ->  ( F concat  g )  =  ( F concat  e ) )
76fveq2d 5868 . . . . . . . 8  |-  ( g  =  e  ->  ( T `  ( F concat  g ) )  =  ( T `  ( F concat 
e ) ) )
87fveq1d 5866 . . . . . . 7  |-  ( g  =  e  ->  (
( T `  ( F concat  g ) ) `  N )  =  ( ( T `  ( F concat  e ) ) `  N ) )
98eqeq1d 2469 . . . . . 6  |-  ( g  =  e  ->  (
( ( T `  ( F concat  g )
) `  N )  =  ( ( T `
 F ) `  N )  <->  ( ( T `  ( F concat  e ) ) `  N
)  =  ( ( T `  F ) `
 N ) ) )
109imbi2d 316 . . . . 5  |-  ( g  =  e  ->  (
( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( ( T `  ( F concat  g )
) `  N )  =  ( ( T `
 F ) `  N ) )  <->  ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 ( F concat  e
) ) `  N
)  =  ( ( T `  F ) `
 N ) ) ) )
11 oveq2 6290 . . . . . . . . 9  |-  ( g  =  ( e concat  <" k "> )  ->  ( F concat  g )  =  ( F concat  (
e concat  <" k "> ) ) )
1211fveq2d 5868 . . . . . . . 8  |-  ( g  =  ( e concat  <" k "> )  ->  ( T `  ( F concat  g ) )  =  ( T `  ( F concat  ( e concat  <" k "> ) ) ) )
1312fveq1d 5866 . . . . . . 7  |-  ( g  =  ( e concat  <" k "> )  ->  ( ( T `  ( F concat  g )
) `  N )  =  ( ( T `
 ( F concat  (
e concat  <" k "> ) ) ) `
 N ) )
1413eqeq1d 2469 . . . . . 6  |-  ( g  =  ( e concat  <" k "> )  ->  ( ( ( T `
 ( F concat  g
) ) `  N
)  =  ( ( T `  F ) `
 N )  <->  ( ( T `  ( F concat  ( e concat  <" k "> ) ) ) `
 N )  =  ( ( T `  F ) `  N
) ) )
1514imbi2d 316 . . . . 5  |-  ( g  =  ( e concat  <" k "> )  ->  ( ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 ( F concat  g
) ) `  N
)  =  ( ( T `  F ) `
 N ) )  <-> 
( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( ( T `  ( F concat  ( e concat  <" k "> ) ) ) `  N )  =  ( ( T `  F
) `  N )
) ) )
16 oveq2 6290 . . . . . . . . 9  |-  ( g  =  G  ->  ( F concat  g )  =  ( F concat  G ) )
1716fveq2d 5868 . . . . . . . 8  |-  ( g  =  G  ->  ( T `  ( F concat  g ) )  =  ( T `  ( F concat  G ) ) )
1817fveq1d 5866 . . . . . . 7  |-  ( g  =  G  ->  (
( T `  ( F concat  g ) ) `  N )  =  ( ( T `  ( F concat  G ) ) `  N ) )
1918eqeq1d 2469 . . . . . 6  |-  ( g  =  G  ->  (
( ( T `  ( F concat  g )
) `  N )  =  ( ( T `
 F ) `  N )  <->  ( ( T `  ( F concat  G ) ) `  N
)  =  ( ( T `  F ) `
 N ) ) )
2019imbi2d 316 . . . . 5  |-  ( g  =  G  ->  (
( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( ( T `  ( F concat  g )
) `  N )  =  ( ( T `
 F ) `  N ) )  <->  ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 ( F concat  G
) ) `  N
)  =  ( ( T `  F ) `
 N ) ) ) )
21 ccatrid 12565 . . . . . . . 8  |-  ( F  e. Word  RR  ->  ( F concat  (/) )  =  F )
2221fveq2d 5868 . . . . . . 7  |-  ( F  e. Word  RR  ->  ( T `
 ( F concat  (/) ) )  =  ( T `  F ) )
2322fveq1d 5866 . . . . . 6  |-  ( F  e. Word  RR  ->  ( ( T `  ( F concat  (/) ) ) `  N
)  =  ( ( T `  F ) `
 N ) )
2423adantr 465 . . . . 5  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( ( T `  ( F concat  (/) ) ) `
 N )  =  ( ( T `  F ) `  N
) )
25 simprl 755 . . . . . . . . . . . . . 14  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  F  e. Word  RR )
26 simpll 753 . . . . . . . . . . . . . 14  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  e  e. Word  RR )
27 simplr 754 . . . . . . . . . . . . . . 15  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  k  e.  RR )
2827s1cld 12574 . . . . . . . . . . . . . 14  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  <" k ">  e. Word  RR )
29 ccatass 12566 . . . . . . . . . . . . . 14  |-  ( ( F  e. Word  RR  /\  e  e. Word  RR  /\  <" k ">  e. Word  RR )  ->  ( ( F concat  e ) concat  <" k "> )  =  ( F concat  ( e concat  <" k "> )
) )
3025, 26, 28, 29syl3anc 1228 . . . . . . . . . . . . 13  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( ( F concat 
e ) concat  <" k "> )  =  ( F concat  ( e concat  <" k "> )
) )
3130fveq2d 5868 . . . . . . . . . . . 12  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( T `  ( ( F concat  e
) concat  <" k "> ) )  =  ( T `  ( F concat  ( e concat  <" k "> ) ) ) )
3231fveq1d 5866 . . . . . . . . . . 11  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( ( T `
 ( ( F concat 
e ) concat  <" k "> ) ) `  N )  =  ( ( T `  ( F concat  ( e concat  <" k "> ) ) ) `
 N ) )
33 ccatcl 12554 . . . . . . . . . . . . 13  |-  ( ( F  e. Word  RR  /\  e  e. Word  RR )  -> 
( F concat  e )  e. Word  RR )
3425, 26, 33syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( F concat  e
)  e. Word  RR )
35 lencl 12524 . . . . . . . . . . . . . . . . . 18  |-  ( F  e. Word  RR  ->  ( # `  F )  e.  NN0 )
3625, 35syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( # `  F
)  e.  NN0 )
3736nn0zd 10960 . . . . . . . . . . . . . . . 16  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( # `  F
)  e.  ZZ )
38 lencl 12524 . . . . . . . . . . . . . . . . . 18  |-  ( ( F concat  e )  e. Word  RR  ->  ( # `  ( F concat  e ) )  e. 
NN0 )
3934, 38syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( # `  ( F concat  e ) )  e. 
NN0 )
4039nn0zd 10960 . . . . . . . . . . . . . . . 16  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( # `  ( F concat  e ) )  e.  ZZ )
4136nn0red 10849 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( # `  F
)  e.  RR )
42 lencl 12524 . . . . . . . . . . . . . . . . . . 19  |-  ( e  e. Word  RR  ->  ( # `  e )  e.  NN0 )
4326, 42syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( # `  e
)  e.  NN0 )
44 nn0addge1 10838 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( # `  F
)  e.  RR  /\  ( # `  e )  e.  NN0 )  -> 
( # `  F )  <_  ( ( # `  F )  +  (
# `  e )
) )
4541, 43, 44syl2anc 661 . . . . . . . . . . . . . . . . 17  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( # `  F
)  <_  ( ( # `
 F )  +  ( # `  e
) ) )
46 ccatlen 12555 . . . . . . . . . . . . . . . . . 18  |-  ( ( F  e. Word  RR  /\  e  e. Word  RR )  -> 
( # `  ( F concat 
e ) )  =  ( ( # `  F
)  +  ( # `  e ) ) )
4725, 26, 46syl2anc 661 . . . . . . . . . . . . . . . . 17  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( # `  ( F concat  e ) )  =  ( ( # `  F
)  +  ( # `  e ) ) )
4845, 47breqtrrd 4473 . . . . . . . . . . . . . . . 16  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( # `  F
)  <_  ( # `  ( F concat  e ) ) )
4937, 40, 483jca 1176 . . . . . . . . . . . . . . 15  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( ( # `  F )  e.  ZZ  /\  ( # `  ( F concat  e ) )  e.  ZZ  /\  ( # `  F )  <_  ( # `
 ( F concat  e
) ) ) )
50 eluz2 11084 . . . . . . . . . . . . . . 15  |-  ( (
# `  ( F concat  e ) )  e.  (
ZZ>= `  ( # `  F
) )  <->  ( ( # `
 F )  e.  ZZ  /\  ( # `  ( F concat  e ) )  e.  ZZ  /\  ( # `  F )  <_  ( # `  ( F concat  e ) ) ) )
5149, 50sylibr 212 . . . . . . . . . . . . . 14  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( # `  ( F concat  e ) )  e.  ( ZZ>= `  ( # `  F
) ) )
52 fzoss2 11817 . . . . . . . . . . . . . 14  |-  ( (
# `  ( F concat  e ) )  e.  (
ZZ>= `  ( # `  F
) )  ->  (
0..^ ( # `  F
) )  C_  (
0..^ ( # `  ( F concat  e ) ) ) )
5351, 52syl 16 . . . . . . . . . . . . 13  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( 0..^ (
# `  F )
)  C_  ( 0..^ ( # `  ( F concat  e ) ) ) )
54 simprr 756 . . . . . . . . . . . . 13  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  N  e.  ( 0..^ ( # `  F
) ) )
5553, 54sseldd 3505 . . . . . . . . . . . 12  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  N  e.  ( 0..^ ( # `  ( F concat  e ) ) ) )
56 signsv.p . . . . . . . . . . . . 13  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
57 signsv.w . . . . . . . . . . . . 13  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
58 signsv.t . . . . . . . . . . . . 13  |-  T  =  ( f  e. Word  RR  |->  ( n  e.  (
0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) ) )
59 signsv.v . . . . . . . . . . . . 13  |-  V  =  ( f  e. Word  RR  |->  sum_ j  e.  ( 1..^ ( # `  f
) ) if ( ( ( T `  f ) `  j
)  =/=  ( ( T `  f ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
6056, 57, 58, 59signstfvp 28168 . . . . . . . . . . . 12  |-  ( ( ( F concat  e )  e. Word  RR  /\  k  e.  RR  /\  N  e.  ( 0..^ ( # `  ( F concat  e ) ) ) )  -> 
( ( T `  ( ( F concat  e
) concat  <" k "> ) ) `  N )  =  ( ( T `  ( F concat  e ) ) `  N ) )
6134, 27, 55, 60syl3anc 1228 . . . . . . . . . . 11  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( ( T `
 ( ( F concat 
e ) concat  <" k "> ) ) `  N )  =  ( ( T `  ( F concat  e ) ) `  N ) )
6232, 61eqtr3d 2510 . . . . . . . . . 10  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( ( T `
 ( F concat  (
e concat  <" k "> ) ) ) `
 N )  =  ( ( T `  ( F concat  e )
) `  N )
)
6362adantr 465 . . . . . . . . 9  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  /\  ( ( T `
 ( F concat  e
) ) `  N
)  =  ( ( T `  F ) `
 N ) )  ->  ( ( T `
 ( F concat  (
e concat  <" k "> ) ) ) `
 N )  =  ( ( T `  ( F concat  e )
) `  N )
)
64 simpr 461 . . . . . . . . 9  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  /\  ( ( T `
 ( F concat  e
) ) `  N
)  =  ( ( T `  F ) `
 N ) )  ->  ( ( T `
 ( F concat  e
) ) `  N
)  =  ( ( T `  F ) `
 N ) )
6563, 64eqtrd 2508 . . . . . . . 8  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  /\  ( ( T `
 ( F concat  e
) ) `  N
)  =  ( ( T `  F ) `
 N ) )  ->  ( ( T `
 ( F concat  (
e concat  <" k "> ) ) ) `
 N )  =  ( ( T `  F ) `  N
) )
6665ex 434 . . . . . . 7  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( ( ( T `  ( F concat 
e ) ) `  N )  =  ( ( T `  F
) `  N )  ->  ( ( T `  ( F concat  ( e concat  <" k "> ) ) ) `  N )  =  ( ( T `  F
) `  N )
) )
6766ex 434 . . . . . 6  |-  ( ( e  e. Word  RR  /\  k  e.  RR )  ->  ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( ( ( T `
 ( F concat  e
) ) `  N
)  =  ( ( T `  F ) `
 N )  -> 
( ( T `  ( F concat  ( e concat  <" k "> ) ) ) `  N )  =  ( ( T `  F
) `  N )
) ) )
6867a2d 26 . . . . 5  |-  ( ( e  e. Word  RR  /\  k  e.  RR )  ->  ( ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 ( F concat  e
) ) `  N
)  =  ( ( T `  F ) `
 N ) )  ->  ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 ( F concat  (
e concat  <" k "> ) ) ) `
 N )  =  ( ( T `  F ) `  N
) ) ) )
695, 10, 15, 20, 24, 68wrdind 12661 . . . 4  |-  ( G  e. Word  RR  ->  ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( ( T `  ( F concat  G ) ) `
 N )  =  ( ( T `  F ) `  N
) ) )
7069imp 429 . . 3  |-  ( ( G  e. Word  RR  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( ( T `
 ( F concat  G
) ) `  N
)  =  ( ( T `  F ) `
 N ) )
71703impb 1192 . 2  |-  ( ( G  e. Word  RR  /\  F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 ( F concat  G
) ) `  N
)  =  ( ( T `  F ) `
 N ) )
72713com12 1200 1  |-  ( ( F  e. Word  RR  /\  G  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 ( F concat  G
) ) `  N
)  =  ( ( T `  F ) `
 N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662    C_ wss 3476   (/)c0 3785   ifcif 3939   {cpr 4029   {ctp 4031   <.cop 4033   class class class wbr 4447    |-> cmpt 4505   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   RRcr 9487   0cc0 9488   1c1 9489    + caddc 9491    <_ cle 9625    - cmin 9801   -ucneg 9802   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11078   ...cfz 11668  ..^cfzo 11788   #chash 12369  Word cword 12496   concat cconcat 12498   <"cs1 12499  sgncsgn 12878   sum_csu 13467   ndxcnx 14483   Basecbs 14486   +g cplusg 14551    gsumg cgsu 14692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-hash 12370  df-word 12504  df-concat 12506  df-s1 12507  df-substr 12508
This theorem is referenced by:  signstres  28172
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