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Theorem signstfvc 28714
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 ++  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 6204 . . . . . . . 8  |-  ( g  =  (/)  ->  ( F ++  g )  =  ( F ++  (/) ) )
21fveq2d 5778 . . . . . . 7  |-  ( g  =  (/)  ->  ( T `
 ( F ++  g
) )  =  ( T `  ( F ++  (/) ) ) )
32fveq1d 5776 . . . . . 6  |-  ( g  =  (/)  ->  ( ( T `  ( F ++  g ) ) `  N )  =  ( ( T `  ( F ++  (/) ) ) `  N ) )
43eqeq1d 2384 . . . . 5  |-  ( g  =  (/)  ->  ( ( ( T `  ( F ++  g ) ) `  N )  =  ( ( T `  F
) `  N )  <->  ( ( T `  ( F ++  (/) ) ) `  N )  =  ( ( T `  F
) `  N )
) )
54imbi2d 314 . . . 4  |-  ( g  =  (/)  ->  ( ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( ( T `  ( F ++  g )
) `  N )  =  ( ( T `
 F ) `  N ) )  <->  ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 ( F ++  (/) ) ) `
 N )  =  ( ( T `  F ) `  N
) ) ) )
6 oveq2 6204 . . . . . . . 8  |-  ( g  =  e  ->  ( F ++  g )  =  ( F ++  e ) )
76fveq2d 5778 . . . . . . 7  |-  ( g  =  e  ->  ( T `  ( F ++  g ) )  =  ( T `  ( F ++  e ) ) )
87fveq1d 5776 . . . . . 6  |-  ( g  =  e  ->  (
( T `  ( F ++  g ) ) `  N )  =  ( ( T `  ( F ++  e ) ) `  N ) )
98eqeq1d 2384 . . . . 5  |-  ( g  =  e  ->  (
( ( T `  ( F ++  g )
) `  N )  =  ( ( T `
 F ) `  N )  <->  ( ( T `  ( F ++  e ) ) `  N )  =  ( ( T `  F
) `  N )
) )
109imbi2d 314 . . . 4  |-  ( g  =  e  ->  (
( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( ( T `  ( F ++  g )
) `  N )  =  ( ( T `
 F ) `  N ) )  <->  ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 ( F ++  e
) ) `  N
)  =  ( ( T `  F ) `
 N ) ) ) )
11 oveq2 6204 . . . . . . . 8  |-  ( g  =  ( e ++  <" k "> )  ->  ( F ++  g )  =  ( F ++  (
e ++  <" k "> ) ) )
1211fveq2d 5778 . . . . . . 7  |-  ( g  =  ( e ++  <" k "> )  ->  ( T `  ( F ++  g ) )  =  ( T `  ( F ++  ( e ++  <" k "> ) ) ) )
1312fveq1d 5776 . . . . . 6  |-  ( g  =  ( e ++  <" k "> )  ->  ( ( T `  ( F ++  g )
) `  N )  =  ( ( T `
 ( F ++  (
e ++  <" k "> ) ) ) `
 N ) )
1413eqeq1d 2384 . . . . 5  |-  ( g  =  ( e ++  <" k "> )  ->  ( ( ( T `
 ( F ++  g
) ) `  N
)  =  ( ( T `  F ) `
 N )  <->  ( ( T `  ( F ++  ( e ++  <" k "> ) ) ) `
 N )  =  ( ( T `  F ) `  N
) ) )
1514imbi2d 314 . . . 4  |-  ( g  =  ( e ++  <" k "> )  ->  ( ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 ( F ++  g
) ) `  N
)  =  ( ( T `  F ) `
 N ) )  <-> 
( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( ( T `  ( F ++  ( e ++  <" k "> ) ) ) `  N )  =  ( ( T `  F
) `  N )
) ) )
16 oveq2 6204 . . . . . . . 8  |-  ( g  =  G  ->  ( F ++  g )  =  ( F ++  G ) )
1716fveq2d 5778 . . . . . . 7  |-  ( g  =  G  ->  ( T `  ( F ++  g ) )  =  ( T `  ( F ++  G ) ) )
1817fveq1d 5776 . . . . . 6  |-  ( g  =  G  ->  (
( T `  ( F ++  g ) ) `  N )  =  ( ( T `  ( F ++  G ) ) `  N ) )
1918eqeq1d 2384 . . . . 5  |-  ( g  =  G  ->  (
( ( T `  ( F ++  g )
) `  N )  =  ( ( T `
 F ) `  N )  <->  ( ( T `  ( F ++  G ) ) `  N )  =  ( ( T `  F
) `  N )
) )
2019imbi2d 314 . . . 4  |-  ( g  =  G  ->  (
( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( ( T `  ( F ++  g )
) `  N )  =  ( ( T `
 F ) `  N ) )  <->  ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 ( F ++  G
) ) `  N
)  =  ( ( T `  F ) `
 N ) ) ) )
21 ccatrid 12513 . . . . . . 7  |-  ( F  e. Word  RR  ->  ( F ++  (/) )  =  F
)
2221fveq2d 5778 . . . . . 6  |-  ( F  e. Word  RR  ->  ( T `
 ( F ++  (/) ) )  =  ( T `  F ) )
2322fveq1d 5776 . . . . 5  |-  ( F  e. Word  RR  ->  ( ( T `  ( F ++  (/) ) ) `  N
)  =  ( ( T `  F ) `
 N ) )
2423adantr 463 . . . 4  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( ( T `  ( F ++  (/) ) ) `
 N )  =  ( ( T `  F ) `  N
) )
25 simprl 754 . . . . . . . . . . . 12  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  F  e. Word  RR )
26 simpll 751 . . . . . . . . . . . 12  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  e  e. Word  RR )
27 simplr 753 . . . . . . . . . . . . 13  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  k  e.  RR )
2827s1cld 12524 . . . . . . . . . . . 12  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  <" k ">  e. Word  RR )
29 ccatass 12514 . . . . . . . . . . . 12  |-  ( ( F  e. Word  RR  /\  e  e. Word  RR  /\  <" k ">  e. Word  RR )  ->  ( ( F ++  e ) ++  <" k "> )  =  ( F ++  ( e ++  <" k "> )
) )
3025, 26, 28, 29syl3anc 1226 . . . . . . . . . . 11  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( ( F ++  e ) ++  <" k "> )  =  ( F ++  ( e ++  <" k "> )
) )
3130fveq2d 5778 . . . . . . . . . 10  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( T `  ( ( F ++  e
) ++  <" k "> ) )  =  ( T `  ( F ++  ( e ++  <" k "> ) ) ) )
3231fveq1d 5776 . . . . . . . . 9  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( ( T `
 ( ( F ++  e ) ++  <" k "> ) ) `  N )  =  ( ( T `  ( F ++  ( e ++  <" k "> ) ) ) `
 N ) )
33 ccatcl 12502 . . . . . . . . . . 11  |-  ( ( F  e. Word  RR  /\  e  e. Word  RR )  -> 
( F ++  e )  e. Word  RR )
3425, 26, 33syl2anc 659 . . . . . . . . . 10  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( F ++  e
)  e. Word  RR )
35 lencl 12469 . . . . . . . . . . . . . . 15  |-  ( F  e. Word  RR  ->  ( # `  F )  e.  NN0 )
3625, 35syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( # `  F
)  e.  NN0 )
3736nn0zd 10882 . . . . . . . . . . . . 13  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( # `  F
)  e.  ZZ )
38 lencl 12469 . . . . . . . . . . . . . . 15  |-  ( ( F ++  e )  e. Word  RR  ->  ( # `  ( F ++  e ) )  e. 
NN0 )
3934, 38syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( # `  ( F ++  e ) )  e. 
NN0 )
4039nn0zd 10882 . . . . . . . . . . . . 13  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( # `  ( F ++  e ) )  e.  ZZ )
4136nn0red 10770 . . . . . . . . . . . . . . 15  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( # `  F
)  e.  RR )
42 lencl 12469 . . . . . . . . . . . . . . . 16  |-  ( e  e. Word  RR  ->  ( # `  e )  e.  NN0 )
4326, 42syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( # `  e
)  e.  NN0 )
44 nn0addge1 10759 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  F
)  e.  RR  /\  ( # `  e )  e.  NN0 )  -> 
( # `  F )  <_  ( ( # `  F )  +  (
# `  e )
) )
4541, 43, 44syl2anc 659 . . . . . . . . . . . . . 14  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( # `  F
)  <_  ( ( # `
 F )  +  ( # `  e
) ) )
46 ccatlen 12503 . . . . . . . . . . . . . . 15  |-  ( ( F  e. Word  RR  /\  e  e. Word  RR )  -> 
( # `  ( F ++  e ) )  =  ( ( # `  F
)  +  ( # `  e ) ) )
4725, 26, 46syl2anc 659 . . . . . . . . . . . . . 14  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( # `  ( F ++  e ) )  =  ( ( # `  F
)  +  ( # `  e ) ) )
4845, 47breqtrrd 4393 . . . . . . . . . . . . 13  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( # `  F
)  <_  ( # `  ( F ++  e ) ) )
49 eluz2 11007 . . . . . . . . . . . . 13  |-  ( (
# `  ( F ++  e ) )  e.  ( ZZ>= `  ( # `  F
) )  <->  ( ( # `
 F )  e.  ZZ  /\  ( # `  ( F ++  e ) )  e.  ZZ  /\  ( # `  F )  <_  ( # `  ( F ++  e ) ) ) )
5037, 40, 48, 49syl3anbrc 1178 . . . . . . . . . . . 12  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( # `  ( F ++  e ) )  e.  ( ZZ>= `  ( # `  F
) ) )
51 fzoss2 11748 . . . . . . . . . . . 12  |-  ( (
# `  ( F ++  e ) )  e.  ( ZZ>= `  ( # `  F
) )  ->  (
0..^ ( # `  F
) )  C_  (
0..^ ( # `  ( F ++  e ) ) ) )
5250, 51syl 16 . . . . . . . . . . 11  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( 0..^ (
# `  F )
)  C_  ( 0..^ ( # `  ( F ++  e ) ) ) )
53 simprr 755 . . . . . . . . . . 11  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  N  e.  ( 0..^ ( # `  F
) ) )
5452, 53sseldd 3418 . . . . . . . . . 10  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  N  e.  ( 0..^ ( # `  ( F ++  e ) ) ) )
55 signsv.p . . . . . . . . . . 11  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
56 signsv.w . . . . . . . . . . 11  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
57 signsv.t . . . . . . . . . . 11  |-  T  =  ( f  e. Word  RR  |->  ( n  e.  (
0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) ) )
58 signsv.v . . . . . . . . . . 11  |-  V  =  ( f  e. Word  RR  |->  sum_ j  e.  ( 1..^ ( # `  f
) ) if ( ( ( T `  f ) `  j
)  =/=  ( ( T `  f ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
5955, 56, 57, 58signstfvp 28711 . . . . . . . . . 10  |-  ( ( ( F ++  e )  e. Word  RR  /\  k  e.  RR  /\  N  e.  ( 0..^ ( # `  ( F ++  e ) ) ) )  -> 
( ( T `  ( ( F ++  e
) ++  <" k "> ) ) `  N )  =  ( ( T `  ( F ++  e ) ) `  N ) )
6034, 27, 54, 59syl3anc 1226 . . . . . . . . 9  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( ( T `
 ( ( F ++  e ) ++  <" k "> ) ) `  N )  =  ( ( T `  ( F ++  e ) ) `  N ) )
6132, 60eqtr3d 2425 . . . . . . . 8  |-  ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  ->  ( ( T `
 ( F ++  (
e ++  <" k "> ) ) ) `
 N )  =  ( ( T `  ( F ++  e )
) `  N )
)
6261adantr 463 . . . . . . 7  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  /\  ( ( T `
 ( F ++  e
) ) `  N
)  =  ( ( T `  F ) `
 N ) )  ->  ( ( T `
 ( F ++  (
e ++  <" k "> ) ) ) `
 N )  =  ( ( T `  ( F ++  e )
) `  N )
)
63 simpr 459 . . . . . . 7  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  /\  ( ( T `
 ( F ++  e
) ) `  N
)  =  ( ( T `  F ) `
 N ) )  ->  ( ( T `
 ( F ++  e
) ) `  N
)  =  ( ( T `  F ) `
 N ) )
6462, 63eqtrd 2423 . . . . . 6  |-  ( ( ( ( e  e. Word  RR  /\  k  e.  RR )  /\  ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) ) )  /\  ( ( T `
 ( F ++  e
) ) `  N
)  =  ( ( T `  F ) `
 N ) )  ->  ( ( T `
 ( F ++  (
e ++  <" k "> ) ) ) `
 N )  =  ( ( T `  F ) `  N
) )
6564exp31 602 . . . . 5  |-  ( ( e  e. Word  RR  /\  k  e.  RR )  ->  ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( ( ( T `
 ( F ++  e
) ) `  N
)  =  ( ( T `  F ) `
 N )  -> 
( ( T `  ( F ++  ( e ++  <" k "> ) ) ) `  N )  =  ( ( T `  F
) `  N )
) ) )
6665a2d 26 . . . 4  |-  ( ( e  e. Word  RR  /\  k  e.  RR )  ->  ( ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 ( F ++  e
) ) `  N
)  =  ( ( T `  F ) `
 N ) )  ->  ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 ( F ++  (
e ++  <" k "> ) ) ) `
 N )  =  ( ( T `  F ) `  N
) ) ) )
675, 10, 15, 20, 24, 66wrdind 12613 . . 3  |-  ( G  e. Word  RR  ->  ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( ( T `  ( F ++  G )
) `  N )  =  ( ( T `
 F ) `  N ) ) )
68673impib 1192 . 2  |-  ( ( G  e. Word  RR  /\  F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 ( F ++  G
) ) `  N
)  =  ( ( T `  F ) `
 N ) )
69683com12 1198 1  |-  ( ( F  e. Word  RR  /\  G  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 ( F ++  G
) ) `  N
)  =  ( ( T `  F ) `
 N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577    C_ wss 3389   (/)c0 3711   ifcif 3857   {cpr 3946   {ctp 3948   <.cop 3950   class class class wbr 4367    |-> cmpt 4425   ` cfv 5496  (class class class)co 6196    |-> cmpt2 6198   RRcr 9402   0cc0 9403   1c1 9404    + caddc 9406    <_ cle 9540    - cmin 9718   -ucneg 9719   NN0cn0 10712   ZZcz 10781   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-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-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-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  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-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-hash 12308  df-word 12446  df-lsw 12447  df-concat 12448  df-s1 12449  df-substr 12450
This theorem is referenced by:  signstres  28715
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