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Theorem lfgrwlkprop 26584
Description: Two adjacent vertices in a walk are different in a loop-free graph. (Contributed by AV, 28-Jan-2021.)
Hypothesis
Ref Expression
lfgrwlkprop.i |- I = (iEdg ` G )
Assertion
Ref Expression
lfgrwlkprop |- ( ( F (Walks ` G ) P /\ I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) -> A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) )
Distinct variable groups:   k, F, x   k, G   k, I, x   P, k   k, V, x
Allowed substitution hints:   P( x)   G( x)
Proof of Theorem lfgrwlkprop
StepHypRef Expression
1 wlkv 26508 . . . . 5 |- ( F (Walks ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) )
2 eqid 2622 . . . . . 6 |- (Vtx ` G ) = (Vtx ` G )
3 lfgrwlkprop.i . . . . . 6 |- I = (iEdg ` G )
42, 3iswlk 26506 . . . . 5 |- ( ( G e. _V /\ F e. _V /\ P e. _V ) -> ( F (Walks ` G ) P <-> ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) ) )
51, 4syl 17 . . . 4 |- ( F (Walks ` G ) P -> ( F (Walks ` G ) P <-> ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) ) )
6 ifptru 1023 . . . . . . . . . . . 12 |- ( ( P ` k ) = ( P ` ( k + 1 ) ) -> (if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) <-> ( I ` ( F ` k ) ) = { ( P ` k ) } ) )
76adantr 481 . . . . . . . . . . 11 |- ( ( ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) /\ k e. ( 0..^ ( # ` F ) ) ) ) -> (if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) <-> ( I ` ( F ` k ) ) = { ( P ` k ) } ) )
8 simplr 792 . . . . . . . . . . . . . 14 |- ( ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) /\ k e. ( 0..^ ( # ` F ) ) ) -> I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } )
9 wrdsymbcl 13318 . . . . . . . . . . . . . . 15 |- ( ( F e. Word dom I /\ k e. ( 0..^ ( # ` F ) ) ) -> ( F ` k ) e. dom I )
109ad4ant14 1293 . . . . . . . . . . . . . 14 |- ( ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) /\ k e. ( 0..^ ( # ` F ) ) ) -> ( F ` k ) e. dom I )
118, 10ffvelrnd 6360 . . . . . . . . . . . . 13 |- ( ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) /\ k e. ( 0..^ ( # ` F ) ) ) -> ( I ` ( F ` k ) ) e. { x e. ~P V | 2 <_ ( # ` x ) } )
12 fveq2 6191 . . . . . . . . . . . . . . . 16 |- ( x = ( I ` ( F ` k ) ) -> ( # ` x ) = ( # ` ( I ` ( F ` k ) ) ) )
1312breq2d 4665 . . . . . . . . . . . . . . 15 |- ( x = ( I ` ( F ` k ) ) -> ( 2 <_ ( # ` x ) <-> 2 <_ ( # ` ( I ` ( F ` k ) ) ) ) )
1413elrab 3363 . . . . . . . . . . . . . 14 |- ( ( I ` ( F ` k ) ) e. { x e. ~P V | 2 <_ ( # ` x ) } <-> ( ( I ` ( F ` k ) ) e. ~P V /\ 2 <_ ( # ` ( I ` ( F ` k ) ) ) ) )
15 fveq2 6191 . . . . . . . . . . . . . . . . . . 19 |- ( ( I ` ( F ` k ) ) = { ( P ` k ) } -> ( # ` ( I ` ( F ` k ) ) ) = ( # ` { ( P ` k ) } ) )
1615breq2d 4665 . . . . . . . . . . . . . . . . . 18 |- ( ( I ` ( F ` k ) ) = { ( P ` k ) } -> ( 2 <_ ( # ` ( I ` ( F ` k ) ) ) <-> 2 <_ ( # ` { ( P ` k ) } ) ) )
17 fvex 6201 . . . . . . . . . . . . . . . . . . . . 21 |- ( P ` k ) e. _V
18 hashsng 13159 . . . . . . . . . . . . . . . . . . . . 21 |- ( ( P ` k ) e. _V -> ( # ` { ( P ` k ) } ) = 1 )
1917, 18ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 |- ( # ` { ( P ` k ) } ) = 1
2019breq2i 4661 . . . . . . . . . . . . . . . . . . 19 |- ( 2 <_ ( # ` { ( P ` k ) } ) <-> 2 <_ 1 )
21 1lt2 11194 . . . . . . . . . . . . . . . . . . . 20 |- 1 < 2
22 1re 10039 . . . . . . . . . . . . . . . . . . . . . 22 |- 1 e. RR
23 2re 11090 . . . . . . . . . . . . . . . . . . . . . 22 |- 2 e. RR
2422, 23ltnlei 10158 . . . . . . . . . . . . . . . . . . . . 21 |- ( 1 < 2 <-> -. 2 <_ 1 )
25 pm2.21 120 . . . . . . . . . . . . . . . . . . . . 21 |- ( -. 2 <_ 1 -> ( 2 <_ 1 -> ( P ` k ) =/= ( P ` ( k + 1 ) ) ) )
2624, 25sylbi 207 . . . . . . . . . . . . . . . . . . . 20 |- ( 1 < 2 -> ( 2 <_ 1 -> ( P ` k ) =/= ( P ` ( k + 1 ) ) ) )
2721, 26ax-mp 5 . . . . . . . . . . . . . . . . . . 19 |- ( 2 <_ 1 -> ( P ` k ) =/= ( P ` ( k + 1 ) ) )
2820, 27sylbi 207 . . . . . . . . . . . . . . . . . 18 |- ( 2 <_ ( # ` { ( P ` k ) } ) -> ( P ` k ) =/= ( P ` ( k + 1 ) ) )
2916, 28syl6bi 243 . . . . . . . . . . . . . . . . 17 |- ( ( I ` ( F ` k ) ) = { ( P ` k ) } -> ( 2 <_ ( # ` ( I ` ( F ` k ) ) ) -> ( P ` k ) =/= ( P ` ( k + 1 ) ) ) )
3029com12 32 . . . . . . . . . . . . . . . 16 |- ( 2 <_ ( # ` ( I ` ( F ` k ) ) ) -> ( ( I ` ( F ` k ) ) = { ( P ` k ) } -> ( P ` k ) =/= ( P ` ( k + 1 ) ) ) )
3130adantl 482 . . . . . . . . . . . . . . 15 |- ( ( ( I ` ( F ` k ) ) e. ~P V /\ 2 <_ ( # ` ( I ` ( F ` k ) ) ) ) -> ( ( I ` ( F ` k ) ) = { ( P ` k ) } -> ( P ` k ) =/= ( P ` ( k + 1 ) ) ) )
3231a1i 11 . . . . . . . . . . . . . 14 |- ( ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) /\ k e. ( 0..^ ( # ` F ) ) ) -> ( ( ( I ` ( F ` k ) ) e. ~P V /\ 2 <_ ( # ` ( I ` ( F ` k ) ) ) ) -> ( ( I ` ( F ` k ) ) = { ( P ` k ) } -> ( P ` k ) =/= ( P ` ( k + 1 ) ) ) ) )
3314, 32syl5bi 232 . . . . . . . . . . . . 13 |- ( ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) /\ k e. ( 0..^ ( # ` F ) ) ) -> ( ( I ` ( F ` k ) ) e. { x e. ~P V | 2 <_ ( # ` x ) } -> ( ( I ` ( F ` k ) ) = { ( P ` k ) } -> ( P ` k ) =/= ( P ` ( k + 1 ) ) ) ) )
3411, 33mpd 15 . . . . . . . . . . . 12 |- ( ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) /\ k e. ( 0..^ ( # ` F ) ) ) -> ( ( I ` ( F ` k ) ) = { ( P ` k ) } -> ( P ` k ) =/= ( P ` ( k + 1 ) ) ) )
3534adantl 482 . . . . . . . . . . 11 |- ( ( ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) /\ k e. ( 0..^ ( # ` F ) ) ) ) -> ( ( I ` ( F ` k ) ) = { ( P ` k ) } -> ( P ` k ) =/= ( P ` ( k + 1 ) ) ) )
367, 35sylbid 230 . . . . . . . . . 10 |- ( ( ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) /\ k e. ( 0..^ ( # ` F ) ) ) ) -> (if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> ( P ` k ) =/= ( P ` ( k + 1 ) ) ) )
3736ex 450 . . . . . . . . 9 |- ( ( P ` k ) = ( P ` ( k + 1 ) ) -> ( ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) /\ k e. ( 0..^ ( # ` F ) ) ) -> (if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> ( P ` k ) =/= ( P ` ( k + 1 ) ) ) ) )
38 neqne 2802 . . . . . . . . . 10 |- ( -. ( P ` k ) = ( P ` ( k + 1 ) ) -> ( P ` k ) =/= ( P ` ( k + 1 ) ) )
39382a1d 26 . . . . . . . . 9 |- ( -. ( P ` k ) = ( P ` ( k + 1 ) ) -> ( ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) /\ k e. ( 0..^ ( # ` F ) ) ) -> (if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> ( P ` k ) =/= ( P ` ( k + 1 ) ) ) ) )
4037, 39pm2.61i 176 . . . . . . . 8 |- ( ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) /\ k e. ( 0..^ ( # ` F ) ) ) -> (if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> ( P ` k ) =/= ( P ` ( k + 1 ) ) ) )
4140ralimdva 2962 . . . . . . 7 |- ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) -> ( A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) ) )
4241ex 450 . . . . . 6 |- ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) -> ( I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } -> ( A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) ) ) )
4342com23 86 . . . . 5 |- ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) -> ( A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> ( I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } -> A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) ) ) )
44433impia 1261 . . . 4 |- ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) -> ( I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } -> A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) ) )
455, 44syl6bi 243 . . 3 |- ( F (Walks ` G ) P -> ( F (Walks ` G ) P -> ( I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } -> A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) ) ) )
4645pm2.43i 52 . 2 |- ( F (Walks ` G ) P -> ( I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } -> A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) ) )
4746imp 445 1 |- ( ( F (Walks ` G ) P /\ I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) -> A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384  if-wif 1012   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   {csn 4177   {cpr 4179   class class class wbr 4653   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   2c2 11070   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291  Vtxcvtx 25874  iEdgciedg 25875  Walkscwlks 26492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-wlks 26495
This theorem is referenced by:  lfgriswlk  26585
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