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Theorem 0pthon 24243
Description: A path of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
Assertion
Ref Expression
0pthon  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  N  e.  V )  ->  (
( P : ( 0 ... 0 ) --> V  /\  ( P `
 0 )  =  N )  ->  (/) ( N ( V PathOn  E ) N ) P ) )

Proof of Theorem 0pthon
StepHypRef Expression
1 0wlkon 24211 . . . . 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 simprl 755 . . . . 5  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  P : ( 0 ... 0 ) --> V )
4 simpll 753 . . . . . 6  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  ( V  e.  X  /\  E  e.  Y ) )
5 fzfid 12039 . . . . . . 7  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  ( 0 ... 0 )  e.  Fin )
6 simplll 757 . . . . . . 7  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  V  e.  X
)
7 fpmg 7434 . . . . . . 7  |-  ( ( ( 0 ... 0
)  e.  Fin  /\  V  e.  X  /\  P : ( 0 ... 0 ) --> V )  ->  P  e.  ( V  ^pm  ( 0 ... 0 ) ) )
85, 6, 3, 7syl3anc 1223 . . . . . 6  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  P  e.  ( V  ^pm  ( 0 ... 0 ) ) )
9 0pth 24234 . . . . . 6  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  ( V  ^pm  ( 0 ... 0 ) ) )  ->  ( (/) ( V Paths 
E ) P  <->  P :
( 0 ... 0
) --> V ) )
104, 8, 9syl2anc 661 . . . . 5  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  ( (/) ( V Paths 
E ) P  <->  P :
( 0 ... 0
) --> V ) )
113, 10mpbird 232 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  (/) ( V Paths  E
) P )
122, 11jca 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 Paths  E ) P ) )
13 0ex 4570 . . . . 5  |-  (/)  e.  _V
1413a1i 11 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  (/)  e.  _V )
15 simpr 461 . . . . . 6  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  N  e.  V )  ->  N  e.  V )
1615, 15jca 532 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  N  e.  V )  ->  ( N  e.  V  /\  N  e.  V )
)
1716adantr 465 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  ( N  e.  V  /\  N  e.  V ) )
18 ispthon 24240 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( (/)  e.  _V  /\  P  e.  ( V 
^pm  ( 0 ... 0 ) ) )  /\  ( N  e.  V  /\  N  e.  V ) )  -> 
( (/) ( N ( V PathOn  E ) N ) P  <->  ( (/) ( N ( V WalkOn  E ) N ) P  /\  (/) ( V Paths  E ) P ) ) )
194, 14, 8, 17, 18syl121anc 1228 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  ( (/) ( N ( V PathOn  E ) N ) P  <->  ( (/) ( N ( V WalkOn  E ) N ) P  /\  (/) ( V Paths  E ) P ) ) )
2012, 19mpbird 232 . 2  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  (/) ( N ( V PathOn  E ) N ) P )
2120ex 434 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  N  e.  V )  ->  (
( P : ( 0 ... 0 ) --> V  /\  ( P `
 0 )  =  N )  ->  (/) ( N ( V PathOn  E ) N ) P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3106   (/)c0 3778   class class class wbr 4440   -->wf 5575   ` cfv 5579  (class class class)co 6275    ^pm cpm 7411   Fincfn 7506   0cc0 9481   ...cfz 11661   Paths cpath 24162   WalkOn cwlkon 24164   PathOn cpthon 24166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-hash 12361  df-word 12495  df-wlk 24170  df-trail 24171  df-pth 24172  df-wlkon 24176  df-pthon 24178
This theorem is referenced by:  0pthon1  24244
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