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Theorem wlkonwlk 21535
Description: A walk is a walk between its endpoints. (Contributed by Alexander van der Vekens, 2-Nov-2017.)
Assertion
Ref Expression
wlkonwlk  |-  ( F ( V Walks  E ) P  ->  F (
( P `  0
) ( V WalkOn  E
) ( P `  ( # `  F ) ) ) P )

Proof of Theorem wlkonwlk
StepHypRef Expression
1 id 20 . 2  |-  ( F ( V Walks  E ) P  ->  F ( V Walks  E ) P )
2 eqidd 2437 . 2  |-  ( F ( V Walks  E ) P  ->  ( P `  0 )  =  ( P `  0
) )
3 eqidd 2437 . 2  |-  ( F ( V Walks  E ) P  ->  ( P `  ( # `  F
) )  =  ( P `  ( # `  F ) ) )
4 wlkbprop 21534 . . . 4  |-  ( F ( V Walks  E ) P  ->  ( ( # `
 F )  e. 
NN0  /\  ( V  e.  _V  /\  E  e. 
_V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
54simp2d 970 . . 3  |-  ( F ( V Walks  E ) P  ->  ( V  e.  _V  /\  E  e. 
_V ) )
64simp3d 971 . . 3  |-  ( F ( V Walks  E ) P  ->  ( F  e.  _V  /\  P  e. 
_V ) )
7 2mwlk 21528 . . . 4  |-  ( F ( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
8 simpr 448 . . . . . 6  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  P : ( 0 ... ( # `  F ) ) --> V )
9 lencl 11735 . . . . . . . . 9  |-  ( F  e. Word  dom  E  ->  (
# `  F )  e.  NN0 )
10 0nn0 10236 . . . . . . . . . . 11  |-  0  e.  NN0
1110a1i 11 . . . . . . . . . 10  |-  ( (
# `  F )  e.  NN0  ->  0  e.  NN0 )
12 id 20 . . . . . . . . . 10  |-  ( (
# `  F )  e.  NN0  ->  ( # `  F
)  e.  NN0 )
13 nn0ge0 10247 . . . . . . . . . 10  |-  ( (
# `  F )  e.  NN0  ->  0  <_  (
# `  F )
)
1411, 12, 133jca 1134 . . . . . . . . 9  |-  ( (
# `  F )  e.  NN0  ->  ( 0  e.  NN0  /\  ( # `
 F )  e. 
NN0  /\  0  <_  (
# `  F )
) )
159, 14syl 16 . . . . . . . 8  |-  ( F  e. Word  dom  E  ->  ( 0  e.  NN0  /\  ( # `  F )  e.  NN0  /\  0  <_  ( # `  F
) ) )
1615adantr 452 . . . . . . 7  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  ( 0  e. 
NN0  /\  ( # `  F
)  e.  NN0  /\  0  <_  ( # `  F
) ) )
17 elfz2nn0 11082 . . . . . . 7  |-  ( 0  e.  ( 0 ... ( # `  F
) )  <->  ( 0  e.  NN0  /\  ( # `
 F )  e. 
NN0  /\  0  <_  (
# `  F )
) )
1816, 17sylibr 204 . . . . . 6  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  0  e.  ( 0 ... ( # `  F ) ) )
198, 18ffvelrnd 5871 . . . . 5  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  ( P ` 
0 )  e.  V
)
20 nn0re 10230 . . . . . . . . . . 11  |-  ( (
# `  F )  e.  NN0  ->  ( # `  F
)  e.  RR )
2120leidd 9593 . . . . . . . . . 10  |-  ( (
# `  F )  e.  NN0  ->  ( # `  F
)  <_  ( # `  F
) )
2212, 12, 213jca 1134 . . . . . . . . 9  |-  ( (
# `  F )  e.  NN0  ->  ( ( # `
 F )  e. 
NN0  /\  ( # `  F
)  e.  NN0  /\  ( # `  F )  <_  ( # `  F
) ) )
239, 22syl 16 . . . . . . . 8  |-  ( F  e. Word  dom  E  ->  ( ( # `  F
)  e.  NN0  /\  ( # `  F )  e.  NN0  /\  ( # `
 F )  <_ 
( # `  F ) ) )
2423adantr 452 . . . . . . 7  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  ( ( # `  F )  e.  NN0  /\  ( # `  F
)  e.  NN0  /\  ( # `  F )  <_  ( # `  F
) ) )
25 elfz2nn0 11082 . . . . . . 7  |-  ( (
# `  F )  e.  ( 0 ... ( # `
 F ) )  <-> 
( ( # `  F
)  e.  NN0  /\  ( # `  F )  e.  NN0  /\  ( # `
 F )  <_ 
( # `  F ) ) )
2624, 25sylibr 204 . . . . . 6  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  ( # `  F
)  e.  ( 0 ... ( # `  F
) ) )
278, 26ffvelrnd 5871 . . . . 5  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  ( P `  ( # `  F ) )  e.  V )
2819, 27jca 519 . . . 4  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  ( ( P `
 0 )  e.  V  /\  ( P `
 ( # `  F
) )  e.  V
) )
297, 28syl 16 . . 3  |-  ( F ( V Walks  E ) P  ->  ( ( P `  0 )  e.  V  /\  ( P `  ( # `  F
) )  e.  V
) )
30 iswlkon 21531 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( ( P ` 
0 )  e.  V  /\  ( P `  ( # `
 F ) )  e.  V ) )  ->  ( F ( ( P `  0
) ( V WalkOn  E
) ( P `  ( # `  F ) ) ) P  <->  ( F
( V Walks  E ) P  /\  ( P ` 
0 )  =  ( P `  0 )  /\  ( P `  ( # `  F ) )  =  ( P `
 ( # `  F
) ) ) ) )
315, 6, 29, 30syl3anc 1184 . 2  |-  ( F ( V Walks  E ) P  ->  ( F
( ( P ` 
0 ) ( V WalkOn  E ) ( P `
 ( # `  F
) ) ) P  <-> 
( F ( V Walks 
E ) P  /\  ( P `  0 )  =  ( P ` 
0 )  /\  ( P `  ( # `  F
) )  =  ( P `  ( # `  F ) ) ) ) )
321, 2, 3, 31mpbir3and 1137 1  |-  ( F ( V Walks  E ) P  ->  F (
( P `  0
) ( V WalkOn  E
) ( P `  ( # `  F ) ) ) P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2956   class class class wbr 4212   dom cdm 4878   -->wf 5450   ` cfv 5454  (class class class)co 6081   0cc0 8990    <_ cle 9121   NN0cn0 10221   ...cfz 11043   #chash 11618  Word cword 11717   Walks cwalk 21506   WalkOn cwlkon 21510
This theorem is referenced by:  cyclispthon  21620
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-fzo 11136  df-hash 11619  df-word 11723  df-wlk 21516  df-wlkon 21522
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