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Theorem cyclispthon 24754
Description: A cycle is a path starting and ending at its first vertex. (Contributed by Alexander van der Vekens, 8-Nov-2017.)
Assertion
Ref Expression
cyclispthon  |-  ( F ( V Cycles  E ) P  ->  F (
( P `  0
) ( V PathOn  E
) ( P ` 
0 ) ) P )

Proof of Theorem cyclispthon
StepHypRef Expression
1 cycliswlk 24753 . . . . 5  |-  ( F ( V Cycles  E ) P  ->  F ( V Walks  E ) P )
2 wlkonwlk 24658 . . . . 5  |-  ( F ( V Walks  E ) P  ->  F (
( P `  0
) ( V WalkOn  E
) ( P `  ( # `  F ) ) ) P )
31, 2syl 16 . . . 4  |-  ( F ( V Cycles  E ) P  ->  F (
( P `  0
) ( V WalkOn  E
) ( P `  ( # `  F ) ) ) P )
4 wlkbprop 24644 . . . . . . . 8  |-  ( F ( V Walks  E ) P  ->  ( ( # `
 F )  e. 
NN0  /\  ( V  e.  _V  /\  E  e. 
_V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
51, 4syl 16 . . . . . . 7  |-  ( F ( V Cycles  E ) P  ->  ( ( # `
 F )  e. 
NN0  /\  ( V  e.  _V  /\  E  e. 
_V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
6 iscycl 24746 . . . . . . . . 9  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Cycles  E ) P 
<->  ( F ( V Paths 
E ) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) ) )
7 simpr 459 . . . . . . . . 9  |-  ( ( F ( V Paths  E
) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) )  ->  ( P `  0 )  =  ( P `  ( # `  F ) ) )
86, 7syl6bi 228 . . . . . . . 8  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Cycles  E ) P  ->  ( P ` 
0 )  =  ( P `  ( # `  F ) ) ) )
983adant1 1012 . . . . . . 7  |-  ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Cycles  E ) P  ->  ( P ` 
0 )  =  ( P `  ( # `  F ) ) ) )
105, 9mpcom 36 . . . . . 6  |-  ( F ( V Cycles  E ) P  ->  ( P `  0 )  =  ( P `  ( # `
 F ) ) )
1110oveq2d 6212 . . . . 5  |-  ( F ( V Cycles  E ) P  ->  ( ( P `  0 )
( V WalkOn  E )
( P `  0
) )  =  ( ( P `  0
) ( V WalkOn  E
) ( P `  ( # `  F ) ) ) )
1211breqd 4378 . . . 4  |-  ( F ( V Cycles  E ) P  ->  ( F
( ( P ` 
0 ) ( V WalkOn  E ) ( P `
 0 ) ) P  <->  F ( ( P `
 0 ) ( V WalkOn  E ) ( P `  ( # `  F ) ) ) P ) )
133, 12mpbird 232 . . 3  |-  ( F ( V Cycles  E ) P  ->  F (
( P `  0
) ( V WalkOn  E
) ( P ` 
0 ) ) P )
14 cyclispth 24750 . . 3  |-  ( F ( V Cycles  E ) P  ->  F ( V Paths  E ) P )
1513, 14jca 530 . 2  |-  ( F ( V Cycles  E ) P  ->  ( F
( ( P ` 
0 ) ( V WalkOn  E ) ( P `
 0 ) ) P  /\  F ( V Paths  E ) P ) )
16 simpl2 998 . . . . 5  |-  ( ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  /\  F ( V Walks  E ) P )  ->  ( V  e. 
_V  /\  E  e.  _V ) )
17 simpl3 999 . . . . 5  |-  ( ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  /\  F ( V Walks  E ) P )  ->  ( F  e. 
_V  /\  P  e.  _V ) )
18 2mwlk 24642 . . . . . . . . 9  |-  ( F ( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
19 0elfz 11695 . . . . . . . . . . . . 13  |-  ( (
# `  F )  e.  NN0  ->  0  e.  ( 0 ... ( # `
 F ) ) )
2019anim2i 567 . . . . . . . . . . . 12  |-  ( ( P : ( 0 ... ( # `  F
) ) --> V  /\  ( # `  F )  e.  NN0 )  -> 
( P : ( 0 ... ( # `  F ) ) --> V  /\  0  e.  ( 0 ... ( # `  F ) ) ) )
21 ffvelrn 5931 . . . . . . . . . . . 12  |-  ( ( P : ( 0 ... ( # `  F
) ) --> V  /\  0  e.  ( 0 ... ( # `  F
) ) )  -> 
( P `  0
)  e.  V )
22 id 22 . . . . . . . . . . . . 13  |-  ( ( P `  0 )  e.  V  ->  ( P `  0 )  e.  V )
2322, 22jca 530 . . . . . . . . . . . 12  |-  ( ( P `  0 )  e.  V  ->  (
( P `  0
)  e.  V  /\  ( P `  0 )  e.  V ) )
2420, 21, 233syl 20 . . . . . . . . . . 11  |-  ( ( P : ( 0 ... ( # `  F
) ) --> V  /\  ( # `  F )  e.  NN0 )  -> 
( ( P ` 
0 )  e.  V  /\  ( P `  0
)  e.  V ) )
2524ex 432 . . . . . . . . . 10  |-  ( P : ( 0 ... ( # `  F
) ) --> V  -> 
( ( # `  F
)  e.  NN0  ->  ( ( P `  0
)  e.  V  /\  ( P `  0 )  e.  V ) ) )
2625adantl 464 . . . . . . . . 9  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  ( ( # `  F )  e.  NN0  ->  ( ( P ` 
0 )  e.  V  /\  ( P `  0
)  e.  V ) ) )
2718, 26syl 16 . . . . . . . 8  |-  ( F ( V Walks  E ) P  ->  ( ( # `
 F )  e. 
NN0  ->  ( ( P `
 0 )  e.  V  /\  ( P `
 0 )  e.  V ) ) )
2827com12 31 . . . . . . 7  |-  ( (
# `  F )  e.  NN0  ->  ( F
( V Walks  E ) P  ->  ( ( P `
 0 )  e.  V  /\  ( P `
 0 )  e.  V ) ) )
29283ad2ant1 1015 . . . . . 6  |-  ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Walks  E ) P  ->  ( ( P `
 0 )  e.  V  /\  ( P `
 0 )  e.  V ) ) )
3029imp 427 . . . . 5  |-  ( ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  /\  F ( V Walks  E ) P )  ->  ( ( P `
 0 )  e.  V  /\  ( P `
 0 )  e.  V ) )
3116, 17, 303jca 1174 . . . 4  |-  ( ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  /\  F ( V Walks  E ) P )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  (
( P `  0
)  e.  V  /\  ( P `  0 )  e.  V ) ) )
324, 31mpancom 667 . . 3  |-  ( F ( V Walks  E ) P  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  (
( P `  0
)  e.  V  /\  ( P `  0 )  e.  V ) ) )
33 ispthon 24699 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( ( P ` 
0 )  e.  V  /\  ( P `  0
)  e.  V ) )  ->  ( F
( ( P ` 
0 ) ( V PathOn  E ) ( P `
 0 ) ) P  <->  ( F ( ( P `  0
) ( V WalkOn  E
) ( P ` 
0 ) ) P  /\  F ( V Paths 
E ) P ) ) )
341, 32, 333syl 20 . 2  |-  ( F ( V Cycles  E ) P  ->  ( F
( ( P ` 
0 ) ( V PathOn  E ) ( P `
 0 ) ) P  <->  ( F ( ( P `  0
) ( V WalkOn  E
) ( P ` 
0 ) ) P  /\  F ( V Paths 
E ) P ) ) )
3515, 34mpbird 232 1  |-  ( F ( V Cycles  E ) P  ->  F (
( P `  0
) ( V PathOn  E
) ( P ` 
0 ) ) P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826   _Vcvv 3034   class class class wbr 4367   dom cdm 4913   -->wf 5492   ` cfv 5496  (class class class)co 6196   0cc0 9403   NN0cn0 10712   ...cfz 11593   #chash 12307  Word cword 12438   Walks cwalk 24619   Paths cpath 24621   WalkOn cwlkon 24623   PathOn cpthon 24625   Cycles ccycl 24628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-map 7340  df-pm 7341  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-fzo 11718  df-hash 12308  df-word 12446  df-wlk 24629  df-trail 24630  df-pth 24631  df-cycl 24634  df-wlkon 24635  df-pthon 24637
This theorem is referenced by: (None)
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