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Theorem cyclispthon 23672
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 23671 . . . . 5  |-  ( F ( V Cycles  E ) P  ->  F ( V Walks  E ) P )
2 wlkonwlk 23587 . . . . 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 23586 . . . . . . . 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 23664 . . . . . . . . 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 461 . . . . . . . . 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 1006 . . . . . . 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 6217 . . . . 5  |-  ( F ( V Cycles  E ) P  ->  ( ( P `  0 )
( V WalkOn  E )
( P `  0
) )  =  ( ( P `  0
) ( V WalkOn  E
) ( P `  ( # `  F ) ) ) )
1211breqd 4412 . . . 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 23668 . . 3  |-  ( F ( V Cycles  E ) P  ->  F ( V Paths  E ) P )
1513, 14jca 532 . 2  |-  ( F ( V Cycles  E ) P  ->  ( F
( ( P ` 
0 ) ( V WalkOn  E ) ( P `
 0 ) ) P  /\  F ( V Paths  E ) P ) )
16 simpl2 992 . . . . 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 993 . . . . 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 23580 . . . . . . . . 9  |-  ( F ( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
19 0nn0 10706 . . . . . . . . . . . . . . 15  |-  0  e.  NN0
2019a1i 11 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  e.  NN0  ->  0  e.  NN0 )
21 id 22 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  e.  NN0  ->  ( # `  F
)  e.  NN0 )
22 nn0ge0 10717 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  e.  NN0  ->  0  <_  (
# `  F )
)
23 elfz2nn0 11598 . . . . . . . . . . . . . 14  |-  ( 0  e.  ( 0 ... ( # `  F
) )  <->  ( 0  e.  NN0  /\  ( # `
 F )  e. 
NN0  /\  0  <_  (
# `  F )
) )
2420, 21, 22, 23syl3anbrc 1172 . . . . . . . . . . . . 13  |-  ( (
# `  F )  e.  NN0  ->  0  e.  ( 0 ... ( # `
 F ) ) )
2524anim2i 569 . . . . . . . . . . . 12  |-  ( ( P : ( 0 ... ( # `  F
) ) --> V  /\  ( # `  F )  e.  NN0 )  -> 
( P : ( 0 ... ( # `  F ) ) --> V  /\  0  e.  ( 0 ... ( # `  F ) ) ) )
26 ffvelrn 5951 . . . . . . . . . . . 12  |-  ( ( P : ( 0 ... ( # `  F
) ) --> V  /\  0  e.  ( 0 ... ( # `  F
) ) )  -> 
( P `  0
)  e.  V )
27 id 22 . . . . . . . . . . . . 13  |-  ( ( P `  0 )  e.  V  ->  ( P `  0 )  e.  V )
2827, 27jca 532 . . . . . . . . . . . 12  |-  ( ( P `  0 )  e.  V  ->  (
( P `  0
)  e.  V  /\  ( P `  0 )  e.  V ) )
2925, 26, 283syl 20 . . . . . . . . . . 11  |-  ( ( P : ( 0 ... ( # `  F
) ) --> V  /\  ( # `  F )  e.  NN0 )  -> 
( ( P ` 
0 )  e.  V  /\  ( P `  0
)  e.  V ) )
3029ex 434 . . . . . . . . . 10  |-  ( P : ( 0 ... ( # `  F
) ) --> V  -> 
( ( # `  F
)  e.  NN0  ->  ( ( P `  0
)  e.  V  /\  ( P `  0 )  e.  V ) ) )
3130adantl 466 . . . . . . . . 9  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  ( ( # `  F )  e.  NN0  ->  ( ( P ` 
0 )  e.  V  /\  ( P `  0
)  e.  V ) ) )
3218, 31syl 16 . . . . . . . 8  |-  ( F ( V Walks  E ) P  ->  ( ( # `
 F )  e. 
NN0  ->  ( ( P `
 0 )  e.  V  /\  ( P `
 0 )  e.  V ) ) )
3332com12 31 . . . . . . 7  |-  ( (
# `  F )  e.  NN0  ->  ( F
( V Walks  E ) P  ->  ( ( P `
 0 )  e.  V  /\  ( P `
 0 )  e.  V ) ) )
34333ad2ant1 1009 . . . . . 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 ) ) )
3534imp 429 . . . . 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 ) )
3616, 17, 353jca 1168 . . . 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 ) ) )
374, 36mpancom 669 . . 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 ) ) )
38 ispthon 23628 . . 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 ) ) )
391, 37, 383syl 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 ) ) )
4015, 39mpbird 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3078   class class class wbr 4401   dom cdm 4949   -->wf 5523   ` cfv 5527  (class class class)co 6201   0cc0 9394    <_ cle 9531   NN0cn0 10691   ...cfz 11555   #chash 12221  Word cword 12340   Walks cwalk 23558   Paths cpath 23560   WalkOn cwlkon 23562   PathOn cpthon 23564   Cycles ccycl 23567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-map 7327  df-pm 7328  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667  df-hash 12222  df-word 12348  df-wlk 23568  df-trail 23569  df-pth 23570  df-cycl 23573  df-wlkon 23574  df-pthon 23576
This theorem is referenced by: (None)
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