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Theorem 3cycld 40092
Description: Construction of a 3-cycle from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.) (Revised by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
Hypotheses
Ref Expression
31wlkd.p  |-  P  = 
<" A B C D ">
31wlkd.f  |-  F  = 
<" J K L ">
31wlkd.s  |-  ( ph  ->  ( ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
) )
31wlkd.n  |-  ( ph  ->  ( ( A  =/= 
B  /\  A  =/=  C )  /\  ( B  =/=  C  /\  B  =/=  D )  /\  C  =/=  D ) )
31wlkd.e  |-  ( ph  ->  ( { A ,  B }  C_  ( I `
 J )  /\  { B ,  C }  C_  ( I `  K
)  /\  { C ,  D }  C_  (
I `  L )
) )
31wlkd.v  |-  V  =  (Vtx `  G )
31wlkd.i  |-  I  =  (iEdg `  G )
3trld.n  |-  ( ph  ->  ( J  =/=  K  /\  J  =/=  L  /\  K  =/=  L
) )
3cycld.e  |-  ( ph  ->  A  =  D )
Assertion
Ref Expression
3cycld  |-  ( ph  ->  F (CycleS `  G
) P )

Proof of Theorem 3cycld
StepHypRef Expression
1 31wlkd.p . . 3  |-  P  = 
<" A B C D ">
2 31wlkd.f . . 3  |-  F  = 
<" J K L ">
3 31wlkd.s . . 3  |-  ( ph  ->  ( ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
) )
4 31wlkd.n . . 3  |-  ( ph  ->  ( ( A  =/= 
B  /\  A  =/=  C )  /\  ( B  =/=  C  /\  B  =/=  D )  /\  C  =/=  D ) )
5 31wlkd.e . . 3  |-  ( ph  ->  ( { A ,  B }  C_  ( I `
 J )  /\  { B ,  C }  C_  ( I `  K
)  /\  { C ,  D }  C_  (
I `  L )
) )
6 31wlkd.v . . 3  |-  V  =  (Vtx `  G )
7 31wlkd.i . . 3  |-  I  =  (iEdg `  G )
8 3trld.n . . 3  |-  ( ph  ->  ( J  =/=  K  /\  J  =/=  L  /\  K  =/=  L
) )
91, 2, 3, 4, 5, 6, 7, 83pthd 40088 . 2  |-  ( ph  ->  F (PathS `  G
) P )
10 simpr 468 . . 3  |-  ( (
ph  /\  F (PathS `  G ) P )  ->  F (PathS `  G ) P )
11 3cycld.e . . . . 5  |-  ( ph  ->  A  =  D )
121fveq1i 5880 . . . . . . . 8  |-  ( P `
 0 )  =  ( <" A B C D "> `  0 )
13 s4fv0 13049 . . . . . . . 8  |-  ( A  e.  V  ->  ( <" A B C D "> `  0
)  =  A )
1412, 13syl5eq 2517 . . . . . . 7  |-  ( A  e.  V  ->  ( P `  0 )  =  A )
1514ad3antrrr 744 . . . . . 6  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  /\  A  =  D )  ->  ( P `  0 )  =  A )
16 simpr 468 . . . . . 6  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  /\  A  =  D )  ->  A  =  D )
172fveq2i 5882 . . . . . . . . . . . 12  |-  ( # `  F )  =  (
# `  <" J K L "> )
18 s3len 13048 . . . . . . . . . . . 12  |-  ( # `  <" J K L "> )  =  3
1917, 18eqtri 2493 . . . . . . . . . . 11  |-  ( # `  F )  =  3
201, 19fveq12i 5884 . . . . . . . . . 10  |-  ( P `
 ( # `  F
) )  =  (
<" A B C D "> `  3
)
21 s4fv3 13052 . . . . . . . . . 10  |-  ( D  e.  V  ->  ( <" A B C D "> `  3
)  =  D )
2220, 21syl5req 2518 . . . . . . . . 9  |-  ( D  e.  V  ->  D  =  ( P `  ( # `  F ) ) )
2322adantl 473 . . . . . . . 8  |-  ( ( C  e.  V  /\  D  e.  V )  ->  D  =  ( P `
 ( # `  F
) ) )
2423adantl 473 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( C  e.  V  /\  D  e.  V ) )  ->  D  =  ( P `  ( # `  F
) ) )
2524adantr 472 . . . . . 6  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  /\  A  =  D )  ->  D  =  ( P `  ( # `  F ) ) )
2615, 16, 253eqtrd 2509 . . . . 5  |-  ( ( ( ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  /\  A  =  D )  ->  ( P `  0 )  =  ( P `  ( # `  F ) ) )
273, 11, 26syl2anc 673 . . . 4  |-  ( ph  ->  ( P `  0
)  =  ( P `
 ( # `  F
) ) )
2827adantr 472 . . 3  |-  ( (
ph  /\  F (PathS `  G ) P )  ->  ( P ` 
0 )  =  ( P `  ( # `  F ) ) )
29 pthis1wlk 39921 . . . . . 6  |-  ( F (PathS `  G ) P  ->  F (1Walks `  G ) P )
30 wlkv 39817 . . . . . 6  |-  ( F (1Walks `  G ) P  ->  ( G  e. 
_V  /\  F  e.  _V  /\  P  e.  _V ) )
3129, 30syl 17 . . . . 5  |-  ( F (PathS `  G ) P  ->  ( G  e. 
_V  /\  F  e.  _V  /\  P  e.  _V ) )
3231adantl 473 . . . 4  |-  ( (
ph  /\  F (PathS `  G ) P )  ->  ( G  e. 
_V  /\  F  e.  _V  /\  P  e.  _V ) )
33 isCycl 39974 . . . 4  |-  ( ( G  e.  _V  /\  F  e.  _V  /\  P  e.  _V )  ->  ( F (CycleS `  G ) P 
<->  ( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) ) )
3432, 33syl 17 . . 3  |-  ( (
ph  /\  F (PathS `  G ) P )  ->  ( F (CycleS `  G ) P  <->  ( F
(PathS `  G ) P  /\  ( P ` 
0 )  =  ( P `  ( # `  F ) ) ) ) )
3510, 28, 34mpbir2and 936 . 2  |-  ( (
ph  /\  F (PathS `  G ) P )  ->  F (CycleS `  G ) P )
369, 35mpdan 681 1  |-  ( ph  ->  F (CycleS `  G
) P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   _Vcvv 3031    C_ wss 3390   {cpr 3961   class class class wbr 4395   ` cfv 5589   0cc0 9557   3c3 10682   #chash 12553   <"cs3 12997   <"cs4 12998  Vtxcvtx 39251  iEdgciedg 39252  1Walksc1wlks 39800  PathScpths 39907  CycleSccycls 39968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-ifp 984  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-concat 12713  df-s1 12714  df-s2 13003  df-s3 13004  df-s4 13005  df-1wlks 39804  df-trls 39889  df-pths 39911  df-cycls 39970
This theorem is referenced by:  3cyclpd  40093
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