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Theorem trls 23582
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 3081 . 2  |-  ( V  e.  X  ->  V  e.  _V )
2 elex 3081 . 2  |-  ( E  e.  Y  ->  E  e.  _V )
3 df-trail 23563 . . . 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 6204 . . . . . . . 8  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v Walks  e )  =  ( V Walks  E
) )
65breqd 4406 . . . . . . 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 3075 . . . . . . . . 9  |-  f  e. 
_V
10 vex 3075 . . . . . . . . 9  |-  p  e. 
_V
11 iswlk 23573 . . . . . . . . 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 969 . . . . . . 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 969 . . . . . . . . . 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 4458 . . 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 4459 . . . 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 6614 . . . . . . 7  |-  ( E  e.  _V  ->  dom  E  e.  _V )
3433adantl 466 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  dom  E  e.  _V )
35 wrdexg 12357 . . . . . 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 11906 . . . . . . 7  |-  ( 0 ... ( # `  f
) )  e.  Fin
3828adantr 465 . . . . . . 7  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  f  e. Word  dom  E
)  ->  V  e.  _V )
39 mapex 7325 . . . . . . 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 3527 . . . . . . 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 4542 . . . . 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 6660 . . . 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 2544 . . 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 6323 . 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 965    = wceq 1370    e. wcel 1758   {cab 2437   A.wral 2796   _Vcvv 3072    C_ wss 3431   {cpr 3982   class class class wbr 4395   {copab 4452   `'ccnv 4942   dom cdm 4943   Fun wfun 5515   -->wf 5517   ` cfv 5521  (class class class)co 6195    |-> cmpt2 6197   Fincfn 7415   0cc0 9388   1c1 9389    + caddc 9391   ...cfz 11549  ..^cfzo 11660   #chash 12215  Word cword 12334   Walks cwalk 23552   Trails ctrail 23553
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-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-map 7321  df-pm 7322  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  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-n0 10686  df-z 10753  df-uz 10968  df-fz 11550  df-fzo 11661  df-word 12342  df-wlk 23562  df-trail 23563
This theorem is referenced by:  istrl  23583
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