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Theorem 0wlkon 23269
Description: A walk of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
Assertion
Ref Expression
0wlkon  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  N  e.  V )  ->  (
( P : ( 0 ... 0 ) --> V  /\  ( P `
 0 )  =  N )  ->  (/) ( N ( V WalkOn  E ) N ) P ) )

Proof of Theorem 0wlkon
StepHypRef Expression
1 simpl 454 . . . . 5  |-  ( ( P : ( 0 ... 0 ) --> V  /\  ( P ` 
0 )  =  N )  ->  P :
( 0 ... 0
) --> V )
21adantl 463 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  P : ( 0 ... 0 ) --> V )
3 simpl 454 . . . . . 6  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  N  e.  V )  ->  ( V  e.  X  /\  E  e.  Y )
)
43adantr 462 . . . . 5  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  ( V  e.  X  /\  E  e.  Y ) )
5 fzfid 11779 . . . . . 6  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  ( 0 ... 0 )  e.  Fin )
6 simplll 750 . . . . . 6  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  V  e.  X
)
7 fpmg 7226 . . . . . 6  |-  ( ( ( 0 ... 0
)  e.  Fin  /\  V  e.  X  /\  P : ( 0 ... 0 ) --> V )  ->  P  e.  ( V  ^pm  ( 0 ... 0 ) ) )
85, 6, 2, 7syl3anc 1211 . . . . 5  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  P  e.  ( V  ^pm  ( 0 ... 0 ) ) )
9 0wlk 23267 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  ( V  ^pm  ( 0 ... 0 ) ) )  ->  ( (/) ( V Walks 
E ) P  <->  P :
( 0 ... 0
) --> V ) )
104, 8, 9syl2anc 654 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  ( (/) ( V Walks 
E ) P  <->  P :
( 0 ... 0
) --> V ) )
112, 10mpbird 232 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  (/) ( V Walks  E
) P )
12 simprr 749 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  ( P ` 
0 )  =  N )
13 hash0 12119 . . . . . 6  |-  ( # `  (/) )  =  0
14 id 22 . . . . . . . . . 10  |-  ( (
# `  (/) )  =  0  ->  ( # `  (/) )  =  0 )
1514eqcomd 2438 . . . . . . . . 9  |-  ( (
# `  (/) )  =  0  ->  0  =  ( # `  (/) ) )
1615fveq2d 5683 . . . . . . . 8  |-  ( (
# `  (/) )  =  0  ->  ( P `  0 )  =  ( P `  ( # `
 (/) ) ) )
1716eqeq1d 2441 . . . . . . 7  |-  ( (
# `  (/) )  =  0  ->  ( ( P `  0 )  =  N  <->  ( P `  ( # `  (/) ) )  =  N ) )
1817biimpd 207 . . . . . 6  |-  ( (
# `  (/) )  =  0  ->  ( ( P `  0 )  =  N  ->  ( P `
 ( # `  (/) ) )  =  N ) )
1913, 18ax-mp 5 . . . . 5  |-  ( ( P `  0 )  =  N  ->  ( P `  ( # `  (/) ) )  =  N )
2019adantl 463 . . . 4  |-  ( ( P : ( 0 ... 0 ) --> V  /\  ( P ` 
0 )  =  N )  ->  ( P `  ( # `  (/) ) )  =  N )
2120adantl 463 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  ( P `  ( # `  (/) ) )  =  N )
22 0ex 4410 . . . . 5  |-  (/)  e.  _V
2322a1i 11 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  (/)  e.  _V )
24 id 22 . . . . . . 7  |-  ( N  e.  V  ->  N  e.  V )
2524ancri 547 . . . . . 6  |-  ( N  e.  V  ->  ( N  e.  V  /\  N  e.  V )
)
2625adantl 463 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  N  e.  V )  ->  ( N  e.  V  /\  N  e.  V )
)
2726adantr 462 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  ( N  e.  V  /\  N  e.  V ) )
28 iswlkon 23253 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( (/)  e.  _V  /\  P  e.  ( V 
^pm  ( 0 ... 0 ) ) )  /\  ( N  e.  V  /\  N  e.  V ) )  -> 
( (/) ( N ( V WalkOn  E ) N ) P  <->  ( (/) ( V Walks 
E ) P  /\  ( P `  0 )  =  N  /\  ( P `  ( # `  (/) ) )  =  N ) ) )
294, 23, 8, 27, 28syl121anc 1216 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  ( (/) ( N ( V WalkOn  E ) N ) P  <->  ( (/) ( V Walks 
E ) P  /\  ( P `  0 )  =  N  /\  ( P `  ( # `  (/) ) )  =  N ) ) )
3011, 12, 21, 29mpbir3and 1164 . 2  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  (/) ( N ( V WalkOn  E ) N ) P )
3130ex 434 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  N  e.  V )  ->  (
( P : ( 0 ... 0 ) --> V  /\  ( P `
 0 )  =  N )  ->  (/) ( N ( V WalkOn  E ) N ) P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 958    = wceq 1362    e. wcel 1755   _Vcvv 2962   (/)c0 3625   class class class wbr 4280   -->wf 5402   ` cfv 5406  (class class class)co 6080    ^pm cpm 7203   Fincfn 7298   0cc0 9270   ...cfz 11424   #chash 12087   Walks cwalk 23228   WalkOn cwlkon 23232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-card 8097  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-n0 10568  df-z 10635  df-uz 10850  df-fz 11425  df-fzo 11533  df-hash 12088  df-word 12213  df-wlk 23238  df-wlkon 23244
This theorem is referenced by:  0trlon  23270  0pthon  23301
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