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Theorem trls 24214
Description: The set of trails (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
Assertion
Ref Expression
trls  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V Trails  E )  =  { <. f ,  p >.  |  (
( f  e. Word  dom  E  /\  Fun  `' f )  /\  p : ( 0 ... ( # `
 f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) } )
Distinct variable groups:    f, E, k, p    f, V, p
Allowed substitution hints:    V( k)    X( f, k, p)    Y( f,
k, p)

Proof of Theorem trls
Dummy variables  e 
v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3122 . 2  |-  ( V  e.  X  ->  V  e.  _V )
2 elex 3122 . 2  |-  ( E  e.  Y  ->  E  e.  _V )
3 df-trail 24185 . . . 4  |- Trails  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Walks 
e ) p  /\  Fun  `' f ) } )
43a1i 11 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  -> Trails 
=  ( v  e. 
_V ,  e  e. 
_V  |->  { <. f ,  p >.  |  (
f ( v Walks  e
) p  /\  Fun  `' f ) } ) )
5 oveq12 6291 . . . . . . . 8  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v Walks  e )  =  ( V Walks  E
) )
65breqd 4458 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f ( v Walks 
e ) p  <->  f ( V Walks  E ) p ) )
76anbi1d 704 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( f ( v Walks  e ) p  /\  Fun  `' f )  <->  ( f ( V Walks  E ) p  /\  Fun  `' f ) ) )
87adantl 466 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( v  =  V  /\  e  =  E ) )  ->  (
( f ( v Walks 
e ) p  /\  Fun  `' f )  <->  ( f
( V Walks  E )
p  /\  Fun  `' f ) ) )
9 vex 3116 . . . . . . . . 9  |-  f  e. 
_V
10 vex 3116 . . . . . . . . 9  |-  p  e. 
_V
11 iswlk 24196 . . . . . . . . 9  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( f  e.  _V  /\  p  e.  _V )
)  ->  ( f
( V Walks  E )
p  <->  ( f  e. Word  dom  E  /\  p : ( 0 ... ( # `
 f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) ) )
129, 10, 11mpanr12 685 . . . . . . . 8  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( f ( V Walks 
E ) p  <->  ( f  e. Word  dom  E  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) ) )
1312adantr 465 . . . . . . 7  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( v  =  V  /\  e  =  E ) )  ->  (
f ( V Walks  E
) p  <->  ( f  e. Word  dom  E  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) ) )
1413anbi1d 704 . . . . . 6  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( v  =  V  /\  e  =  E ) )  ->  (
( f ( V Walks 
E ) p  /\  Fun  `' f )  <->  ( (
f  e. Word  dom  E  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } )  /\  Fun  `' f ) ) )
15 3anass 977 . . . . . . 7  |-  ( ( ( f  e. Word  dom  E  /\  Fun  `' f )  /\  p : ( 0 ... ( # `
 f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } )  <->  ( (
f  e. Word  dom  E  /\  Fun  `' f )  /\  ( p : ( 0 ... ( # `  f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) ) )
16 ancom 450 . . . . . . . 8  |-  ( ( f  e. Word  dom  E  /\  Fun  `' f )  <-> 
( Fun  `' f  /\  f  e. Word  dom  E
) )
1716anbi1i 695 . . . . . . 7  |-  ( ( ( f  e. Word  dom  E  /\  Fun  `' f )  /\  ( p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) )  <->  ( ( Fun  `' f  /\  f  e. Word  dom  E )  /\  ( p : ( 0 ... ( # `  f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) ) )
18 anass 649 . . . . . . . 8  |-  ( ( ( Fun  `' f  /\  f  e. Word  dom  E )  /\  ( p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) )  <->  ( Fun  `' f  /\  ( f  e. Word  dom  E  /\  ( p : ( 0 ... ( # `  f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) ) ) )
19 ancom 450 . . . . . . . 8  |-  ( ( Fun  `' f  /\  ( f  e. Word  dom  E  /\  ( p : ( 0 ... ( # `
 f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) ) )  <->  ( ( f  e. Word  dom  E  /\  ( p : ( 0 ... ( # `  f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) )  /\  Fun  `' f ) )
20 3anass 977 . . . . . . . . . 10  |-  ( ( f  e. Word  dom  E  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } )  <->  ( f  e. Word  dom  E  /\  (
p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) ) )
2120bicomi 202 . . . . . . . . 9  |-  ( ( f  e. Word  dom  E  /\  ( p : ( 0 ... ( # `  f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) )  <-> 
( f  e. Word  dom  E  /\  p : ( 0 ... ( # `  f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) )
2221anbi1i 695 . . . . . . . 8  |-  ( ( ( f  e. Word  dom  E  /\  ( p : ( 0 ... ( # `
 f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) )  /\  Fun  `' f )  <->  ( ( f  e. Word  dom  E  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } )  /\  Fun  `' f ) )
2318, 19, 223bitri 271 . . . . . . 7  |-  ( ( ( Fun  `' f  /\  f  e. Word  dom  E )  /\  ( p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) )  <->  ( (
f  e. Word  dom  E  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } )  /\  Fun  `' f ) )
2415, 17, 233bitri 271 . . . . . 6  |-  ( ( ( f  e. Word  dom  E  /\  Fun  `' f )  /\  p : ( 0 ... ( # `
 f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } )  <->  ( (
f  e. Word  dom  E  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } )  /\  Fun  `' f ) )
2514, 24syl6bbr 263 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( v  =  V  /\  e  =  E ) )  ->  (
( f ( V Walks 
E ) p  /\  Fun  `' f )  <->  ( (
f  e. Word  dom  E  /\  Fun  `' f )  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) ) )
268, 25bitrd 253 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( v  =  V  /\  e  =  E ) )  ->  (
( f ( v Walks 
e ) p  /\  Fun  `' f )  <->  ( (
f  e. Word  dom  E  /\  Fun  `' f )  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) ) )
2726opabbidv 4510 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( v  =  V  /\  e  =  E ) )  ->  { <. f ,  p >.  |  ( f ( v Walks  e
) p  /\  Fun  `' f ) }  =  { <. f ,  p >.  |  ( ( f  e. Word  dom  E  /\  Fun  `' f )  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) } )
28 simpl 457 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  V  e.  _V )
29 simpr 461 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  E  e.  _V )
30 anass 649 . . . . . 6  |-  ( ( ( f  e. Word  dom  E  /\  Fun  `' f )  /\  ( p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) )  <->  ( f  e. Word  dom  E  /\  ( Fun  `' f  /\  (
p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) ) ) )
3115, 30bitri 249 . . . . 5  |-  ( ( ( f  e. Word  dom  E  /\  Fun  `' f )  /\  p : ( 0 ... ( # `
 f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } )  <->  ( f  e. Word  dom  E  /\  ( Fun  `' f  /\  (
p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) ) ) )
3231opabbii 4511 . . . 4  |-  { <. f ,  p >.  |  ( ( f  e. Word  dom  E  /\  Fun  `' f )  /\  p : ( 0 ... ( # `
 f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) }  =  { <. f ,  p >.  |  (
f  e. Word  dom  E  /\  ( Fun  `' f  /\  ( p : ( 0 ... ( # `  f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) ) ) }
33 dmexg 6712 . . . . . . 7  |-  ( E  e.  _V  ->  dom  E  e.  _V )
3433adantl 466 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  dom  E  e.  _V )
35 wrdexg 12519 . . . . . 6  |-  ( dom 
E  e.  _V  -> Word  dom  E  e.  _V )
3634, 35syl 16 . . . . 5  |-  ( ( V  e.  _V  /\  E  e.  _V )  -> Word 
dom  E  e.  _V )
37 fzfi 12046 . . . . . . 7  |-  ( 0 ... ( # `  f
) )  e.  Fin
3828adantr 465 . . . . . . 7  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  f  e. Word  dom  E
)  ->  V  e.  _V )
39 mapex 7423 . . . . . . 7  |-  ( ( ( 0 ... ( # `
 f ) )  e.  Fin  /\  V  e.  _V )  ->  { p  |  p : ( 0 ... ( # `  f
) ) --> V }  e.  _V )
4037, 38, 39sylancr 663 . . . . . 6  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  f  e. Word  dom  E
)  ->  { p  |  p : ( 0 ... ( # `  f
) ) --> V }  e.  _V )
41 simpl 457 . . . . . . . . 9  |-  ( ( p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } )  ->  p : ( 0 ... ( # `  f
) ) --> V )
4241adantl 466 . . . . . . . 8  |-  ( ( Fun  `' f  /\  ( p : ( 0 ... ( # `  f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) )  ->  p : ( 0 ... ( # `  f ) ) --> V )
4342ss2abi 3572 . . . . . . 7  |-  { p  |  ( Fun  `' f  /\  ( p : ( 0 ... ( # `
 f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) ) }  C_  { p  |  p : ( 0 ... ( # `  f
) ) --> V }
4443a1i 11 . . . . . 6  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  f  e. Word  dom  E
)  ->  { p  |  ( Fun  `' f  /\  ( p : ( 0 ... ( # `
 f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) ) }  C_  { p  |  p : ( 0 ... ( # `  f
) ) --> V }
)
4540, 44ssexd 4594 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  f  e. Word  dom  E
)  ->  { p  |  ( Fun  `' f  /\  ( p : ( 0 ... ( # `
 f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) ) }  e.  _V )
4636, 45opabex3d 6759 . . . 4  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  { <. f ,  p >.  |  ( f  e. Word  dom  E  /\  ( Fun  `' f  /\  (
p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) ) ) }  e.  _V )
4732, 46syl5eqel 2559 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  { <. f ,  p >.  |  ( ( f  e. Word  dom  E  /\  Fun  `' f )  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) }  e.  _V )
484, 27, 28, 29, 47ovmpt2d 6412 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V Trails  E )  =  { <. f ,  p >.  |  (
( f  e. Word  dom  E  /\  Fun  `' f )  /\  p : ( 0 ... ( # `
 f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) } )
491, 2, 48syl2an 477 1  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V Trails  E )  =  { <. f ,  p >.  |  (
( f  e. Word  dom  E  /\  Fun  `' f )  /\  p : ( 0 ... ( # `
 f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2814   _Vcvv 3113    C_ wss 3476   {cpr 4029   class class class wbr 4447   {copab 4504   `'ccnv 4998   dom cdm 4999   Fun wfun 5580   -->wf 5582   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   Fincfn 7513   0cc0 9488   1c1 9489    + caddc 9491   ...cfz 11668  ..^cfzo 11788   #chash 12369  Word cword 12496   Walks cwalk 24174   Trails ctrail 24175
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-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  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-word 12504  df-wlk 24184  df-trail 24185
This theorem is referenced by:  istrl  24215
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