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Theorem usgrnloop 25138
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 25096 . . 3  |-  ( F ( V Walks  E ) P  ->  ( ( # `
 F )  e. 
NN0  /\  ( V  e.  _V  /\  E  e. 
_V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
2 iswlk 25093 . . . . 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 1023 . . . 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 12663 . . . . . . . . 9  |-  ( F  e. Word  dom  E  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
5 ffvelrn 6035 . . . . . . . . . . . 12  |-  ( ( F : ( 0..^ ( # `  F
) ) --> dom  E  /\  k  e.  (
0..^ ( # `  F
) ) )  -> 
( F `  k
)  e.  dom  E
)
6 usgrafun 24922 . . . . . . . . . . . . 13  |-  ( V USGrph  E  ->  Fun  E )
7 fvelrn 6030 . . . . . . . . . . . . . . . 16  |-  ( ( Fun  E  /\  ( F `  k )  e.  dom  E )  -> 
( E `  ( F `  k )
)  e.  ran  E
)
8 eleq1 2501 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  ->  (
( E `  ( F `  k )
)  e.  ran  E  <->  { ( P `  k
) ,  ( P `
 ( k  +  1 ) ) }  e.  ran  E ) )
98anbi2d 708 . . . . . . . . . . . . . . . . . . 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 210 . . . . . . . . . . . . . . . . . 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 24954 . . . . . . . . . . . . . . . . . 18  |-  ( ( V USGrph  E  /\  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  e.  ran  E )  ->  ( P `  k )  =/=  ( P `  (
k  +  1 ) ) )
1210, 11syl6com 36 . . . . . . . . . . . . . . . . 17  |-  ( ( V USGrph  E  /\  ( E `  ( F `  k ) )  e. 
ran  E )  -> 
( ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  ->  ( P `  k )  =/=  ( P `  ( k  +  1 ) ) ) )
1312expcom 436 . . . . . . . . . . . . . . . 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 17 . . . . . . . . . . . . . . 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 435 . . . . . . . . . . . . . 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 81 . . . . . . . . . . . . 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 37 . . . . . . . . . . . 12  |-  ( V USGrph  E  ->  ( ( F `
 k )  e. 
dom  E  ->  ( ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  ->  ( P `  k )  =/=  ( P `  (
k  +  1 ) ) ) ) )
185, 17syl5com 31 . . . . . . . . . . 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 435 . . . . . . . . . 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 81 . . . . . . . . 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 17 . . . . . . . 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 433 . . . . . . 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 2840 . . . . . 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 441 . . . . 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 1024 . . . 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 231 . . 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 37 . 2  |-  ( F ( V Walks  E ) P  ->  ( V USGrph  E  ->  A. k  e.  ( 0..^ ( # `  F
) ) ( P `
 k )  =/=  ( P `  (
k  +  1 ) ) ) )
2827impcom 431 1  |-  ( ( V USGrph  E  /\  F ( V Walks  E ) P )  ->  A. k  e.  ( 0..^ ( # `  F ) ) ( P `  k )  =/=  ( P `  ( k  +  1 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   _Vcvv 3087   {cpr 4004   class class class wbr 4426   dom cdm 4854   ran crn 4855   Fun wfun 5595   -->wf 5597   ` cfv 5601  (class class class)co 6305   0cc0 9538   1c1 9539    + caddc 9541   NN0cn0 10869   ...cfz 11782  ..^cfzo 11913   #chash 12512  Word cword 12643   USGrph cusg 24903   Walks cwalk 25071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-fzo 11914  df-hash 12513  df-word 12651  df-usgra 24906  df-wlk 25081
This theorem is referenced by:  usgrcyclnl1  25213
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