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Theorem 0wlkon 23414
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 457 . . . . 5  |-  ( ( P : ( 0 ... 0 ) --> V  /\  ( P ` 
0 )  =  N )  ->  P :
( 0 ... 0
) --> V )
21adantl 466 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  P : ( 0 ... 0 ) --> V )
3 simpl 457 . . . . . 6  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  N  e.  V )  ->  ( V  e.  X  /\  E  e.  Y )
)
43adantr 465 . . . . 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 11787 . . . . . 6  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  ( 0 ... 0 )  e.  Fin )
6 simplll 757 . . . . . 6  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  V  e.  X
)
7 fpmg 7230 . . . . . 6  |-  ( ( ( 0 ... 0
)  e.  Fin  /\  V  e.  X  /\  P : ( 0 ... 0 ) --> V )  ->  P  e.  ( V  ^pm  ( 0 ... 0 ) ) )
85, 6, 2, 7syl3anc 1218 . . . . 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 23412 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  ( V  ^pm  ( 0 ... 0 ) ) )  ->  ( (/) ( V Walks 
E ) P  <->  P :
( 0 ... 0
) --> V ) )
104, 8, 9syl2anc 661 . . . 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 756 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  ( P ` 
0 )  =  N )
13 hash0 12127 . . . . . 6  |-  ( # `  (/) )  =  0
14 id 22 . . . . . . . . . 10  |-  ( (
# `  (/) )  =  0  ->  ( # `  (/) )  =  0 )
1514eqcomd 2443 . . . . . . . . 9  |-  ( (
# `  (/) )  =  0  ->  0  =  ( # `  (/) ) )
1615fveq2d 5690 . . . . . . . 8  |-  ( (
# `  (/) )  =  0  ->  ( P `  0 )  =  ( P `  ( # `
 (/) ) ) )
1716eqeq1d 2446 . . . . . . 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 466 . . . 4  |-  ( ( P : ( 0 ... 0 ) --> V  /\  ( P ` 
0 )  =  N )  ->  ( P `  ( # `  (/) ) )  =  N )
2120adantl 466 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  /\  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )  ->  ( P `  ( # `  (/) ) )  =  N )
22 0ex 4417 . . . . 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 552 . . . . . 6  |-  ( N  e.  V  ->  ( N  e.  V  /\  N  e.  V )
)
2625adantl 466 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  N  e.  V )  ->  ( N  e.  V  /\  N  e.  V )
)
2726adantr 465 . . . 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 23398 . . . 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 1223 . . 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 1171 . 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 965    = wceq 1369    e. wcel 1756   _Vcvv 2967   (/)c0 3632   class class class wbr 4287   -->wf 5409   ` cfv 5413  (class class class)co 6086    ^pm cpm 7207   Fincfn 7302   0cc0 9274   ...cfz 11429   #chash 12095   Walks cwalk 23373   WalkOn cwlkon 23377
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-fzo 11541  df-hash 12096  df-word 12221  df-wlk 23383  df-wlkon 23389
This theorem is referenced by:  0trlon  23415  0pthon  23446
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