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Theorem signstres 27115
Description: Restriction of a zero skipping sign to a subword. (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
signstres  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( ( T `  F )  |`  (
0..^ N ) )  =  ( T `  ( F  |`  ( 0..^ N ) ) ) )
Distinct variable groups:    a, b,  .+^    f, i, n, F    f, W, i, n    f, N, i, n
Allowed substitution hints:    .+^ ( f, i,
j, n)    T( f,
i, j, n, a, b)    F( j, a, b)    N( j, a, b)    V( f, i, j, n, a, b)    W( j, a, b)

Proof of Theorem signstres
Dummy variables  g  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvres 5808 . . . . . 6  |-  ( m  e.  ( 0..^ N )  ->  ( (
( T `  F
)  |`  ( 0..^ N ) ) `  m
)  =  ( ( T `  F ) `
 m ) )
21ad3antlr 730 . . . . 5  |-  ( ( ( ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `
 F ) ) )  /\  m  e.  ( 0..^ N ) )  /\  g  e. Word  RR )  /\  F  =  ( ( F  |`  ( 0..^ N ) ) concat 
g ) )  -> 
( ( ( T `
 F )  |`  ( 0..^ N ) ) `
 m )  =  ( ( T `  F ) `  m
) )
3 simpr 461 . . . . . . . 8  |-  ( ( ( ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `
 F ) ) )  /\  m  e.  ( 0..^ N ) )  /\  g  e. Word  RR )  /\  F  =  ( ( F  |`  ( 0..^ N ) ) concat 
g ) )  ->  F  =  ( ( F  |`  ( 0..^ N ) ) concat  g ) )
43fveq2d 5798 . . . . . . 7  |-  ( ( ( ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `
 F ) ) )  /\  m  e.  ( 0..^ N ) )  /\  g  e. Word  RR )  /\  F  =  ( ( F  |`  ( 0..^ N ) ) concat 
g ) )  -> 
( T `  F
)  =  ( T `
 ( ( F  |`  ( 0..^ N ) ) concat  g ) ) )
54fveq1d 5796 . . . . . 6  |-  ( ( ( ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `
 F ) ) )  /\  m  e.  ( 0..^ N ) )  /\  g  e. Word  RR )  /\  F  =  ( ( F  |`  ( 0..^ N ) ) concat 
g ) )  -> 
( ( T `  F ) `  m
)  =  ( ( T `  ( ( F  |`  ( 0..^ N ) ) concat  g
) ) `  m
) )
6 wrdres 27077 . . . . . . . 8  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( F  |`  (
0..^ N ) )  e. Word  RR )
76ad3antrrr 729 . . . . . . 7  |-  ( ( ( ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `
 F ) ) )  /\  m  e.  ( 0..^ N ) )  /\  g  e. Word  RR )  /\  F  =  ( ( F  |`  ( 0..^ N ) ) concat 
g ) )  -> 
( F  |`  (
0..^ N ) )  e. Word  RR )
8 simplr 754 . . . . . . 7  |-  ( ( ( ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `
 F ) ) )  /\  m  e.  ( 0..^ N ) )  /\  g  e. Word  RR )  /\  F  =  ( ( F  |`  ( 0..^ N ) ) concat 
g ) )  -> 
g  e. Word  RR )
9 wrdf 12353 . . . . . . . . . . . . . . . 16  |-  ( F  e. Word  RR  ->  F :
( 0..^ ( # `  F ) ) --> RR )
10 ffn 5662 . . . . . . . . . . . . . . . 16  |-  ( F : ( 0..^ (
# `  F )
) --> RR  ->  F  Fn  ( 0..^ ( # `  F ) ) )
119, 10syl 16 . . . . . . . . . . . . . . 15  |-  ( F  e. Word  RR  ->  F  Fn  ( 0..^ ( # `  F
) ) )
1211adantr 465 . . . . . . . . . . . . . 14  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  ->  F  Fn  ( 0..^ ( # `  F
) ) )
13 elfzuz3 11562 . . . . . . . . . . . . . . . 16  |-  ( N  e.  ( 0 ... ( # `  F
) )  ->  ( # `
 F )  e.  ( ZZ>= `  N )
)
14 fzoss2 11689 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  e.  ( ZZ>= `  N )  ->  ( 0..^ N ) 
C_  ( 0..^ (
# `  F )
) )
1513, 14syl 16 . . . . . . . . . . . . . . 15  |-  ( N  e.  ( 0 ... ( # `  F
) )  ->  (
0..^ N )  C_  ( 0..^ ( # `  F
) ) )
1615adantl 466 . . . . . . . . . . . . . 14  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( 0..^ N ) 
C_  ( 0..^ (
# `  F )
) )
17 fnssres 5627 . . . . . . . . . . . . . 14  |-  ( ( F  Fn  ( 0..^ ( # `  F
) )  /\  (
0..^ N )  C_  ( 0..^ ( # `  F
) ) )  -> 
( F  |`  (
0..^ N ) )  Fn  ( 0..^ N ) )
1812, 16, 17syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( F  |`  (
0..^ N ) )  Fn  ( 0..^ N ) )
19 hashfn 12251 . . . . . . . . . . . . 13  |-  ( ( F  |`  ( 0..^ N ) )  Fn  ( 0..^ N )  ->  ( # `  ( F  |`  ( 0..^ N ) ) )  =  ( # `  (
0..^ N ) ) )
2018, 19syl 16 . . . . . . . . . . . 12  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( # `  ( F  |`  ( 0..^ N ) ) )  =  (
# `  ( 0..^ N ) ) )
21 elfznn0 11593 . . . . . . . . . . . . . 14  |-  ( N  e.  ( 0 ... ( # `  F
) )  ->  N  e.  NN0 )
22 hashfzo0 12304 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  ( # `  ( 0..^ N ) )  =  N )
2321, 22syl 16 . . . . . . . . . . . . 13  |-  ( N  e.  ( 0 ... ( # `  F
) )  ->  ( # `
 ( 0..^ N ) )  =  N )
2423adantl 466 . . . . . . . . . . . 12  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( # `  ( 0..^ N ) )  =  N )
2520, 24eqtrd 2493 . . . . . . . . . . 11  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( # `  ( F  |`  ( 0..^ N ) ) )  =  N )
2625oveq2d 6211 . . . . . . . . . 10  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( 0..^ ( # `  ( F  |`  (
0..^ N ) ) ) )  =  ( 0..^ N ) )
2726eleq2d 2522 . . . . . . . . 9  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( m  e.  ( 0..^ ( # `  ( F  |`  ( 0..^ N ) ) ) )  <-> 
m  e.  ( 0..^ N ) ) )
2827biimpar 485 . . . . . . . 8  |-  ( ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  /\  m  e.  ( 0..^ N ) )  ->  m  e.  ( 0..^ ( # `  ( F  |`  ( 0..^ N ) ) ) ) )
2928ad2antrr 725 . . . . . . 7  |-  ( ( ( ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `
 F ) ) )  /\  m  e.  ( 0..^ N ) )  /\  g  e. Word  RR )  /\  F  =  ( ( F  |`  ( 0..^ N ) ) concat 
g ) )  ->  m  e.  ( 0..^ ( # `  ( F  |`  ( 0..^ N ) ) ) ) )
30 signsv.p . . . . . . . 8  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
31 signsv.w . . . . . . . 8  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
32 signsv.t . . . . . . . 8  |-  T  =  ( f  e. Word  RR  |->  ( n  e.  (
0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) ) )
33 signsv.v . . . . . . . 8  |-  V  =  ( f  e. Word  RR  |->  sum_ j  e.  ( 1..^ ( # `  f
) ) if ( ( ( T `  f ) `  j
)  =/=  ( ( T `  f ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
3430, 31, 32, 33signstfvc 27114 . . . . . . 7  |-  ( ( ( F  |`  (
0..^ N ) )  e. Word  RR  /\  g  e. Word  RR  /\  m  e.  ( 0..^ ( # `  ( F  |`  (
0..^ N ) ) ) ) )  -> 
( ( T `  ( ( F  |`  ( 0..^ N ) ) concat 
g ) ) `  m )  =  ( ( T `  ( F  |`  ( 0..^ N ) ) ) `  m ) )
357, 8, 29, 34syl3anc 1219 . . . . . 6  |-  ( ( ( ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `
 F ) ) )  /\  m  e.  ( 0..^ N ) )  /\  g  e. Word  RR )  /\  F  =  ( ( F  |`  ( 0..^ N ) ) concat 
g ) )  -> 
( ( T `  ( ( F  |`  ( 0..^ N ) ) concat 
g ) ) `  m )  =  ( ( T `  ( F  |`  ( 0..^ N ) ) ) `  m ) )
365, 35eqtrd 2493 . . . . 5  |-  ( ( ( ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `
 F ) ) )  /\  m  e.  ( 0..^ N ) )  /\  g  e. Word  RR )  /\  F  =  ( ( F  |`  ( 0..^ N ) ) concat 
g ) )  -> 
( ( T `  F ) `  m
)  =  ( ( T `  ( F  |`  ( 0..^ N ) ) ) `  m
) )
372, 36eqtrd 2493 . . . 4  |-  ( ( ( ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `
 F ) ) )  /\  m  e.  ( 0..^ N ) )  /\  g  e. Word  RR )  /\  F  =  ( ( F  |`  ( 0..^ N ) ) concat 
g ) )  -> 
( ( ( T `
 F )  |`  ( 0..^ N ) ) `
 m )  =  ( ( T `  ( F  |`  ( 0..^ N ) ) ) `
 m ) )
38 wrdsplex 27078 . . . . 5  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  ->  E. g  e. Word  RR F  =  ( ( F  |`  ( 0..^ N ) ) concat  g ) )
3938adantr 465 . . . 4  |-  ( ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  /\  m  e.  ( 0..^ N ) )  ->  E. g  e. Word  RR F  =  ( ( F  |`  ( 0..^ N ) ) concat  g ) )
4037, 39r19.29a 2962 . . 3  |-  ( ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  /\  m  e.  ( 0..^ N ) )  -> 
( ( ( T `
 F )  |`  ( 0..^ N ) ) `
 m )  =  ( ( T `  ( F  |`  ( 0..^ N ) ) ) `
 m ) )
4140ralrimiva 2827 . 2  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  ->  A. m  e.  (
0..^ N ) ( ( ( T `  F )  |`  (
0..^ N ) ) `
 m )  =  ( ( T `  ( F  |`  ( 0..^ N ) ) ) `
 m ) )
4230, 31, 32, 33signstf 27106 . . . . . . . 8  |-  ( F  e. Word  RR  ->  ( T `
 F )  e. Word  RR )
43 wrdf 12353 . . . . . . . 8  |-  ( ( T `  F )  e. Word  RR  ->  ( T `
 F ) : ( 0..^ ( # `  ( T `  F
) ) ) --> RR )
44 ffn 5662 . . . . . . . 8  |-  ( ( T `  F ) : ( 0..^ (
# `  ( T `  F ) ) ) --> RR  ->  ( T `  F )  Fn  (
0..^ ( # `  ( T `  F )
) ) )
4542, 43, 443syl 20 . . . . . . 7  |-  ( F  e. Word  RR  ->  ( T `
 F )  Fn  ( 0..^ ( # `  ( T `  F
) ) ) )
4630, 31, 32, 33signstlen 27107 . . . . . . . . 9  |-  ( F  e. Word  RR  ->  ( # `  ( T `  F
) )  =  (
# `  F )
)
4746oveq2d 6211 . . . . . . . 8  |-  ( F  e. Word  RR  ->  ( 0..^ ( # `  ( T `  F )
) )  =  ( 0..^ ( # `  F
) ) )
4847fneq2d 5605 . . . . . . 7  |-  ( F  e. Word  RR  ->  ( ( T `  F )  Fn  ( 0..^ (
# `  ( T `  F ) ) )  <-> 
( T `  F
)  Fn  ( 0..^ ( # `  F
) ) ) )
4945, 48mpbid 210 . . . . . 6  |-  ( F  e. Word  RR  ->  ( T `
 F )  Fn  ( 0..^ ( # `  F ) ) )
50 fnresin 26093 . . . . . 6  |-  ( ( T `  F )  Fn  ( 0..^ (
# `  F )
)  ->  ( ( T `  F )  |`  ( 0..^ N ) )  Fn  ( ( 0..^ ( # `  F
) )  i^i  (
0..^ N ) ) )
5149, 50syl 16 . . . . 5  |-  ( F  e. Word  RR  ->  ( ( T `  F )  |`  ( 0..^ N ) )  Fn  ( ( 0..^ ( # `  F
) )  i^i  (
0..^ N ) ) )
5251adantr 465 . . . 4  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( ( T `  F )  |`  (
0..^ N ) )  Fn  ( ( 0..^ ( # `  F
) )  i^i  (
0..^ N ) ) )
53 incom 3646 . . . . . . 7  |-  ( ( 0..^ N )  i^i  ( 0..^ ( # `  F ) ) )  =  ( ( 0..^ ( # `  F
) )  i^i  (
0..^ N ) )
54 df-ss 3445 . . . . . . . 8  |-  ( ( 0..^ N )  C_  ( 0..^ ( # `  F
) )  <->  ( (
0..^ N )  i^i  ( 0..^ ( # `  F ) ) )  =  ( 0..^ N ) )
5554biimpi 194 . . . . . . 7  |-  ( ( 0..^ N )  C_  ( 0..^ ( # `  F
) )  ->  (
( 0..^ N )  i^i  ( 0..^ (
# `  F )
) )  =  ( 0..^ N ) )
5653, 55syl5eqr 2507 . . . . . 6  |-  ( ( 0..^ N )  C_  ( 0..^ ( # `  F
) )  ->  (
( 0..^ ( # `  F ) )  i^i  ( 0..^ N ) )  =  ( 0..^ N ) )
5756fneq2d 5605 . . . . 5  |-  ( ( 0..^ N )  C_  ( 0..^ ( # `  F
) )  ->  (
( ( T `  F )  |`  (
0..^ N ) )  Fn  ( ( 0..^ ( # `  F
) )  i^i  (
0..^ N ) )  <-> 
( ( T `  F )  |`  (
0..^ N ) )  Fn  ( 0..^ N ) ) )
5816, 57syl 16 . . . 4  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( ( ( T `
 F )  |`  ( 0..^ N ) )  Fn  ( ( 0..^ ( # `  F
) )  i^i  (
0..^ N ) )  <-> 
( ( T `  F )  |`  (
0..^ N ) )  Fn  ( 0..^ N ) ) )
5952, 58mpbid 210 . . 3  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( ( T `  F )  |`  (
0..^ N ) )  Fn  ( 0..^ N ) )
6030, 31, 32, 33signstf 27106 . . . . 5  |-  ( ( F  |`  ( 0..^ N ) )  e. Word  RR  ->  ( T `  ( F  |`  ( 0..^ N ) ) )  e. Word  RR )
61 wrdf 12353 . . . . 5  |-  ( ( T `  ( F  |`  ( 0..^ N ) ) )  e. Word  RR  ->  ( T `  ( F  |`  ( 0..^ N ) ) ) : ( 0..^ ( # `  ( T `  ( F  |`  ( 0..^ N ) ) ) ) ) --> RR )
62 ffn 5662 . . . . 5  |-  ( ( T `  ( F  |`  ( 0..^ N ) ) ) : ( 0..^ ( # `  ( T `  ( F  |`  ( 0..^ N ) ) ) ) ) --> RR  ->  ( T `  ( F  |`  (
0..^ N ) ) )  Fn  ( 0..^ ( # `  ( T `  ( F  |`  ( 0..^ N ) ) ) ) ) )
636, 60, 61, 624syl 21 . . . 4  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( T `  ( F  |`  ( 0..^ N ) ) )  Fn  ( 0..^ ( # `  ( T `  ( F  |`  ( 0..^ N ) ) ) ) ) )
6430, 31, 32, 33signstlen 27107 . . . . . . . 8  |-  ( ( F  |`  ( 0..^ N ) )  e. Word  RR  ->  ( # `  ( T `  ( F  |`  ( 0..^ N ) ) ) )  =  ( # `  ( F  |`  ( 0..^ N ) ) ) )
656, 64syl 16 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( # `  ( T `
 ( F  |`  ( 0..^ N ) ) ) )  =  (
# `  ( F  |`  ( 0..^ N ) ) ) )
6665, 20, 243eqtrd 2497 . . . . . 6  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( # `  ( T `
 ( F  |`  ( 0..^ N ) ) ) )  =  N )
6766oveq2d 6211 . . . . 5  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( 0..^ ( # `  ( T `  ( F  |`  ( 0..^ N ) ) ) ) )  =  ( 0..^ N ) )
6867fneq2d 5605 . . . 4  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( ( T `  ( F  |`  ( 0..^ N ) ) )  Fn  ( 0..^ (
# `  ( T `  ( F  |`  (
0..^ N ) ) ) ) )  <->  ( T `  ( F  |`  (
0..^ N ) ) )  Fn  ( 0..^ N ) ) )
6963, 68mpbid 210 . . 3  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( T `  ( F  |`  ( 0..^ N ) ) )  Fn  ( 0..^ N ) )
70 eqfnfv 5901 . . 3  |-  ( ( ( ( T `  F )  |`  (
0..^ N ) )  Fn  ( 0..^ N )  /\  ( T `
 ( F  |`  ( 0..^ N ) ) )  Fn  ( 0..^ N ) )  -> 
( ( ( T `
 F )  |`  ( 0..^ N ) )  =  ( T `  ( F  |`  ( 0..^ N ) ) )  <->  A. m  e.  (
0..^ N ) ( ( ( T `  F )  |`  (
0..^ N ) ) `
 m )  =  ( ( T `  ( F  |`  ( 0..^ N ) ) ) `
 m ) ) )
7159, 69, 70syl2anc 661 . 2  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( ( ( T `
 F )  |`  ( 0..^ N ) )  =  ( T `  ( F  |`  ( 0..^ N ) ) )  <->  A. m  e.  (
0..^ N ) ( ( ( T `  F )  |`  (
0..^ N ) ) `
 m )  =  ( ( T `  ( F  |`  ( 0..^ N ) ) ) `
 m ) ) )
7241, 71mpbird 232 1  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0 ... ( # `  F
) ) )  -> 
( ( T `  F )  |`  (
0..^ N ) )  =  ( T `  ( F  |`  ( 0..^ N ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2645   A.wral 2796   E.wrex 2797    i^i cin 3430    C_ wss 3431   ifcif 3894   {cpr 3982   {ctp 3984   <.cop 3986    |-> cmpt 4453    |` cres 4945    Fn wfn 5516   -->wf 5517   ` cfv 5521  (class class class)co 6195    |-> cmpt2 6197   RRcr 9387   0cc0 9388   1c1 9389    - cmin 9701   -ucneg 9702   NN0cn0 10685   ZZ>=cuz 10967   ...cfz 11549  ..^cfzo 11660   #chash 12215  Word cword 12334   concat cconcat 12336  sgncsgn 12688   sum_csu 13276   ndxcnx 14284   Basecbs 14287   +g cplusg 14352    gsumg cgsu 14493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-card 8215  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-n0 10686  df-z 10753  df-uz 10968  df-fz 11550  df-fzo 11661  df-seq 11919  df-hash 12216  df-word 12342  df-concat 12344  df-s1 12345  df-substr 12346  df-sgn 12689  df-struct 14289  df-ndx 14290  df-slot 14291  df-base 14292  df-plusg 14365  df-0g 14494  df-gsum 14495  df-mnd 15529
This theorem is referenced by: (None)
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