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Theorem 0trlon 24373
Description: A trail of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
Assertion
Ref Expression
0trlon  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  N  e.  V )  ->  (
( P : ( 0 ... 0 ) --> V  /\  ( P `
 0 )  =  N )  ->  (/) ( N ( V TrailOn  E ) N ) P ) )

Proof of Theorem 0trlon
StepHypRef Expression
1 0wlkon 24372 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  N  e.  V )  ->  (
( P : ( 0 ... 0 ) --> V  /\  ( P `
 0 )  =  N )  ->  (/) ( N ( V WalkOn  E ) N ) P ) )
21imp 429 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  (/) ( N ( V WalkOn  E ) N ) P )
3 simpl 457 . . . . . 6  |-  ( ( P : ( 0 ... 0 ) --> V  /\  ( P ` 
0 )  =  N )  ->  P :
( 0 ... 0
) --> V )
43adantl 466 . . . . 5  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  P : ( 0 ... 0 ) --> V )
5 simpl 457 . . . . . . 7  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  N  e.  V )  ->  ( V  e.  X  /\  E  e.  Y )
)
65adantr 465 . . . . . 6  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  ( V  e.  X  /\  E  e.  Y ) )
7 fzfid 12063 . . . . . . 7  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  ( 0 ... 0 )  e.  Fin )
8 simplll 757 . . . . . . 7  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  V  e.  X
)
9 fpmg 7456 . . . . . . 7  |-  ( ( ( 0 ... 0
)  e.  Fin  /\  V  e.  X  /\  P : ( 0 ... 0 ) --> V )  ->  P  e.  ( V  ^pm  ( 0 ... 0 ) ) )
107, 8, 4, 9syl3anc 1228 . . . . . 6  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  P  e.  ( V  ^pm  ( 0 ... 0 ) ) )
11 0trl 24371 . . . . . 6  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  ( V  ^pm  ( 0 ... 0 ) ) )  ->  ( (/) ( V Trails  E ) P  <->  P :
( 0 ... 0
) --> V ) )
126, 10, 11syl2anc 661 . . . . 5  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  ( (/) ( V Trails  E ) P  <->  P :
( 0 ... 0
) --> V ) )
134, 12mpbird 232 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  (/) ( V Trails  E
) P )
142, 13jca 532 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  ( (/) ( N ( V WalkOn  E ) N ) P  /\  (/) ( V Trails  E ) P ) )
15 0ex 4583 . . . . 5  |-  (/)  e.  _V
1610, 15jctil 537 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  ( (/)  e.  _V  /\  P  e.  ( V 
^pm  ( 0 ... 0 ) ) ) )
17 id 22 . . . . . . 7  |-  ( N  e.  V  ->  N  e.  V )
1817, 17jca 532 . . . . . 6  |-  ( N  e.  V  ->  ( N  e.  V  /\  N  e.  V )
)
1918adantl 466 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  N  e.  V )  ->  ( N  e.  V  /\  N  e.  V )
)
2019adantr 465 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  ( N  e.  V  /\  N  e.  V ) )
21 istrlon 24366 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( (/)  e.  _V  /\  P  e.  ( V 
^pm  ( 0 ... 0 ) ) )  /\  ( N  e.  V  /\  N  e.  V ) )  -> 
( (/) ( N ( V TrailOn  E ) N ) P  <->  ( (/) ( N ( V WalkOn  E ) N ) P  /\  (/) ( V Trails  E ) P ) ) )
226, 16, 20, 21syl3anc 1228 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  ( (/) ( N ( V TrailOn  E ) N ) P  <->  ( (/) ( N ( V WalkOn  E ) N ) P  /\  (/) ( V Trails  E ) P ) ) )
2314, 22mpbird 232 . 2  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  (/) ( N ( V TrailOn  E ) N ) P )
2423ex 434 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  N  e.  V )  ->  (
( P : ( 0 ... 0 ) --> V  /\  ( P `
 0 )  =  N )  ->  (/) ( N ( V TrailOn  E ) N ) P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118   (/)c0 3790   class class class wbr 4453   -->wf 5590   ` cfv 5594  (class class class)co 6295    ^pm cpm 7433   Fincfn 7528   0cc0 9504   ...cfz 11684   Trails ctrail 24322   WalkOn cwlkon 24325   TrailOn ctrlon 24326
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-hash 12386  df-word 12523  df-wlk 24331  df-trail 24332  df-wlkon 24337  df-trlon 24338
This theorem is referenced by: (None)
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