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Theorem usgrnloop 24388
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 24346 . . 3  |-  ( F ( V Walks  E ) P  ->  ( ( # `
 F )  e. 
NN0  /\  ( V  e.  _V  /\  E  e. 
_V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
2 iswlk 24343 . . . . 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 1014 . . . 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 12534 . . . . . . . . 9  |-  ( F  e. Word  dom  E  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
5 ffvelrn 6030 . . . . . . . . . . . 12  |-  ( ( F : ( 0..^ ( # `  F
) ) --> dom  E  /\  k  e.  (
0..^ ( # `  F
) ) )  -> 
( F `  k
)  e.  dom  E
)
6 usgrafun 24172 . . . . . . . . . . . . 13  |-  ( V USGrph  E  ->  Fun  E )
7 fvelrn 6025 . . . . . . . . . . . . . . . 16  |-  ( ( Fun  E  /\  ( F `  k )  e.  dom  E )  -> 
( E `  ( F `  k )
)  e.  ran  E
)
8 eleq1 2539 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  ->  (
( E `  ( F `  k )
)  e.  ran  E  <->  { ( P `  k
) ,  ( P `
 ( k  +  1 ) ) }  e.  ran  E ) )
98anbi2d 703 . . . . . . . . . . . . . . . . . . 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 207 . . . . . . . . . . . . . . . . . 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 24204 . . . . . . . . . . . . . . . . . 18  |-  ( ( V USGrph  E  /\  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  e.  ran  E )  ->  ( P `  k )  =/=  ( P `  (
k  +  1 ) ) )
1210, 11syl6com 35 . . . . . . . . . . . . . . . . 17  |-  ( ( V USGrph  E  /\  ( E `  ( F `  k ) )  e. 
ran  E )  -> 
( ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  ->  ( P `  k )  =/=  ( P `  ( k  +  1 ) ) ) )
1312expcom 435 . . . . . . . . . . . . . . . 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 434 . . . . . . . . . . . . . 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 78 . . . . . . . . . . . . 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 36 . . . . . . . . . . . 12  |-  ( V USGrph  E  ->  ( ( F `
 k )  e. 
dom  E  ->  ( ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  ->  ( P `  k )  =/=  ( P `  (
k  +  1 ) ) ) ) )
185, 17syl5com 30 . . . . . . . . . . 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 434 . . . . . . . . . 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 78 . . . . . . . . 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 432 . . . . . . 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 2875 . . . . . 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 440 . . . . 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 1015 . . . 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 228 . . 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 36 . 2  |-  ( F ( V Walks  E ) P  ->  ( V USGrph  E  ->  A. k  e.  ( 0..^ ( # `  F
) ) ( P `
 k )  =/=  ( P `  (
k  +  1 ) ) ) )
2827impcom 430 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   _Vcvv 3118   {cpr 4035   class class class wbr 4453   dom cdm 5005   ran crn 5006   Fun wfun 5588   -->wf 5590   ` cfv 5594  (class class class)co 6295   0cc0 9504   1c1 9505    + caddc 9507   NN0cn0 10807   ...cfz 11684  ..^cfzo 11804   #chash 12385  Word cword 12515   USGrph cusg 24153   Walks cwalk 24321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-hash 12386  df-word 12523  df-usgra 24156  df-wlk 24331
This theorem is referenced by:  usgrcyclnl1  24463
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