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Theorem 0pthon 23476
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 23444 . . . . 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 11793 . . . . . . 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 7236 . . . . . . 7  |-  ( ( ( 0 ... 0
)  e.  Fin  /\  V  e.  X  /\  P : ( 0 ... 0 ) --> V )  ->  P  e.  ( V  ^pm  ( 0 ... 0 ) ) )
85, 6, 3, 7syl3anc 1218 . . . . . 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 23467 . . . . . 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 4420 . . . . 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 23473 . . . 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 1223 . . 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 1369    e. wcel 1756   _Vcvv 2970   (/)c0 3635   class class class wbr 4290   -->wf 5412   ` cfv 5416  (class class class)co 6089    ^pm cpm 7213   Fincfn 7308   0cc0 9280   ...cfz 11435   Paths cpath 23405   WalkOn cwlkon 23407   PathOn cpthon 23409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-map 7214  df-pm 7215  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-card 8107  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-n0 10578  df-z 10645  df-uz 10860  df-fz 11436  df-fzo 11547  df-hash 12102  df-word 12227  df-wlk 23413  df-trail 23414  df-pth 23415  df-wlkon 23419  df-pthon 23421
This theorem is referenced by:  0pthon1  23477
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