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Theorem usgrnloop 21516
Description: In an undirected simple graph, each walk has no loops! (Contributed by Alexander van der Vekens, 7-Nov-2017.)
Assertion
Ref Expression
usgrnloop  |-  ( ( V USGrph  E  /\  F ( V Walks  E ) P )  ->  A. k  e.  ( 0..^ ( # `  F ) ) ( P `  k )  =/=  ( P `  ( k  +  1 ) ) )
Distinct variable groups:    k, E    k, F    P, k    k, V

Proof of Theorem usgrnloop
StepHypRef Expression
1 wlkbprop 21487 . . 3  |-  ( F ( V Walks  E ) P  ->  ( ( # `
 F )  e. 
NN0  /\  ( V  e.  _V  /\  E  e. 
_V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
2 iswlk 21480 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Walks  E ) P 
<->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) } ) ) )
323adant1 975 . . . 4  |-  ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Walks  E ) P 
<->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) } ) ) )
4 wrdf 11688 . . . . . . . . 9  |-  ( F  e. Word  dom  E  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
5 ffvelrn 5827 . . . . . . . . . . . 12  |-  ( ( F : ( 0..^ ( # `  F
) ) --> dom  E  /\  k  e.  (
0..^ ( # `  F
) ) )  -> 
( F `  k
)  e.  dom  E
)
6 usgrafun 21331 . . . . . . . . . . . . 13  |-  ( V USGrph  E  ->  Fun  E )
7 fvelrn 5825 . . . . . . . . . . . . . . . 16  |-  ( ( Fun  E  /\  ( F `  k )  e.  dom  E )  -> 
( E `  ( F `  k )
)  e.  ran  E
)
8 eleq1 2464 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  ->  (
( E `  ( F `  k )
)  e.  ran  E  <->  { ( P `  k
) ,  ( P `
 ( k  +  1 ) ) }  e.  ran  E ) )
98anbi2d 685 . . . . . . . . . . . . . . . . . . 19  |-  ( ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  ->  (
( V USGrph  E  /\  ( E `  ( F `
 k ) )  e.  ran  E )  <-> 
( V USGrph  E  /\  { ( P `  k
) ,  ( P `
 ( k  +  1 ) ) }  e.  ran  E ) ) )
109biimpd 199 . . . . . . . . . . . . . . . . . 18  |-  ( ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  ->  (
( V USGrph  E  /\  ( E `  ( F `
 k ) )  e.  ran  E )  ->  ( V USGrph  E  /\  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  e.  ran  E
) ) )
11 usgraedgrn 21354 . . . . . . . . . . . . . . . . . 18  |-  ( ( V USGrph  E  /\  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  e.  ran  E )  ->  ( P `  k )  =/=  ( P `  (
k  +  1 ) ) )
1210, 11syl6com 33 . . . . . . . . . . . . . . . . 17  |-  ( ( V USGrph  E  /\  ( E `  ( F `  k ) )  e. 
ran  E )  -> 
( ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  ->  ( P `  k )  =/=  ( P `  ( k  +  1 ) ) ) )
1312expcom 425 . . . . . . . . . . . . . . . 16  |-  ( ( E `  ( F `
 k ) )  e.  ran  E  -> 
( V USGrph  E  ->  ( ( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  ->  ( P `  k )  =/=  ( P `  ( k  +  1 ) ) ) ) )
147, 13syl 16 . . . . . . . . . . . . . . 15  |-  ( ( Fun  E  /\  ( F `  k )  e.  dom  E )  -> 
( V USGrph  E  ->  ( ( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  ->  ( P `  k )  =/=  ( P `  ( k  +  1 ) ) ) ) )
1514ex 424 . . . . . . . . . . . . . 14  |-  ( Fun 
E  ->  ( ( F `  k )  e.  dom  E  ->  ( V USGrph  E  ->  ( ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  ->  ( P `  k )  =/=  ( P `  ( k  +  1 ) ) ) ) ) )
1615com23 74 . . . . . . . . . . . . 13  |-  ( Fun 
E  ->  ( V USGrph  E  ->  ( ( F `
 k )  e. 
dom  E  ->  ( ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  ->  ( P `  k )  =/=  ( P `  (
k  +  1 ) ) ) ) ) )
176, 16mpcom 34 . . . . . . . . . . . 12  |-  ( V USGrph  E  ->  ( ( F `
 k )  e. 
dom  E  ->  ( ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  ->  ( P `  k )  =/=  ( P `  (
k  +  1 ) ) ) ) )
185, 17syl5com 28 . . . . . . . . . . 11  |-  ( ( F : ( 0..^ ( # `  F
) ) --> dom  E  /\  k  e.  (
0..^ ( # `  F
) ) )  -> 
( V USGrph  E  ->  ( ( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  ->  ( P `  k )  =/=  ( P `  ( k  +  1 ) ) ) ) )
1918ex 424 . . . . . . . . . 10  |-  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  ( k  e.  ( 0..^ ( # `  F
) )  ->  ( V USGrph  E  ->  ( ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  ->  ( P `  k )  =/=  ( P `  ( k  +  1 ) ) ) ) ) )
2019com23 74 . . . . . . . . 9  |-  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  ( V USGrph  E  ->  ( k  e.  ( 0..^ (
# `  F )
)  ->  ( ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  ->  ( P `  k )  =/=  ( P `  ( k  +  1 ) ) ) ) ) )
214, 20syl 16 . . . . . . . 8  |-  ( F  e. Word  dom  E  ->  ( V USGrph  E  ->  ( k  e.  ( 0..^ (
# `  F )
)  ->  ( ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  ->  ( P `  k )  =/=  ( P `  ( k  +  1 ) ) ) ) ) )
2221imp31 422 . . . . . . 7  |-  ( ( ( F  e. Word  dom  E  /\  V USGrph  E )  /\  k  e.  (
0..^ ( # `  F
) ) )  -> 
( ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  ->  ( P `  k )  =/=  ( P `  ( k  +  1 ) ) ) )
2322ralimdva 2744 . . . . . 6  |-  ( ( F  e. Word  dom  E  /\  V USGrph  E )  -> 
( A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  ->  A. k  e.  ( 0..^ ( # `  F ) ) ( P `  k )  =/=  ( P `  ( k  +  1 ) ) ) )
2423impancom 428 . . . . 5  |-  ( ( F  e. Word  dom  E  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )  ->  ( V USGrph  E  ->  A. k  e.  ( 0..^ ( # `  F ) ) ( P `  k )  =/=  ( P `  ( k  +  1 ) ) ) )
25243adant2 976 . . . 4  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )  ->  ( V USGrph  E  ->  A. k  e.  ( 0..^ ( # `  F ) ) ( P `  k )  =/=  ( P `  ( k  +  1 ) ) ) )
263, 25syl6bi 220 . . 3  |-  ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Walks  E ) P  ->  ( V USGrph  E  ->  A. k  e.  ( 0..^ ( # `  F
) ) ( P `
 k )  =/=  ( P `  (
k  +  1 ) ) ) ) )
271, 26mpcom 34 . 2  |-  ( F ( V Walks  E ) P  ->  ( V USGrph  E  ->  A. k  e.  ( 0..^ ( # `  F
) ) ( P `
 k )  =/=  ( P `  (
k  +  1 ) ) ) )
2827impcom 420 1  |-  ( ( V USGrph  E  /\  F ( V Walks  E ) P )  ->  A. k  e.  ( 0..^ ( # `  F ) ) ( P `  k )  =/=  ( P `  ( k  +  1 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   _Vcvv 2916   {cpr 3775   class class class wbr 4172   dom cdm 4837   ran crn 4838   Fun wfun 5407   -->wf 5409   ` cfv 5413  (class class class)co 6040   0cc0 8946   1c1 8947    + caddc 8949   NN0cn0 10177   ...cfz 10999  ..^cfzo 11090   #chash 11573  Word cword 11672   USGrph cusg 21318   Walks cwalk 21459
This theorem is referenced by:  usgrcyclnl1  21580
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-hash 11574  df-word 11678  df-usgra 21320  df-wlk 21469
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