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Theorem wlkonwlk1l 26559
Description: A walk is a walk from its first vertex to its last vertex. (Contributed by AV, 7-Feb-2021.) (Revised by AV, 22-Mar-2021.)
Hypothesis
Ref Expression
wlkonwlk1l.w |- ( ph -> F (Walks ` G ) P )
Assertion
Ref Expression
wlkonwlk1l |- ( ph -> F ( ( P ` 0 ) (WalksOn ` G ) ( lastS ` P ) ) P )
Proof of Theorem wlkonwlk1l
StepHypRef Expression
1 wlkonwlk1l.w . 2 |- ( ph -> F (Walks ` G ) P )
2 eqidd 2623 . 2 |- ( ph -> ( P ` 0 ) = ( P ` 0 ) )
3 wlklenvm1 26517 . . . . 5 |- ( F (Walks ` G ) P -> ( # ` F ) = ( ( # ` P ) - 1 ) )
43fveq2d 6195 . . . 4 |- ( F (Walks ` G ) P -> ( P ` ( # ` F ) ) = ( P ` ( ( # ` P ) - 1 ) ) )
5 eqid 2622 . . . . . 6 |- (Vtx ` G ) = (Vtx ` G )
65wlkpwrd 26513 . . . . 5 |- ( F (Walks ` G ) P -> P e. Word (Vtx ` G ) )
7 lsw 13351 . . . . 5 |- ( P e. Word (Vtx ` G ) -> ( lastS ` P ) = ( P ` ( ( # ` P ) - 1 ) ) )
86, 7syl 17 . . . 4 |- ( F (Walks ` G ) P -> ( lastS ` P ) = ( P ` ( ( # ` P ) - 1 ) ) )
94, 8eqtr4d 2659 . . 3 |- ( F (Walks ` G ) P -> ( P ` ( # ` F ) ) = ( lastS ` P ) )
101, 9syl 17 . 2 |- ( ph -> ( P ` ( # ` F ) ) = ( lastS ` P ) )
11 wlkcl 26511 . . . . . . . 8 |- ( F (Walks ` G ) P -> ( # ` F ) e. NN0 )
12 nn0p1nn 11332 . . . . . . . 8 |- ( ( # ` F ) e. NN0 -> ( ( # ` F ) + 1 ) e. NN )
1311, 12syl 17 . . . . . . 7 |- ( F (Walks ` G ) P -> ( ( # ` F ) + 1 ) e. NN )
14 wlklenvp1 26514 . . . . . . 7 |- ( F (Walks ` G ) P -> ( # ` P ) = ( ( # ` F ) + 1 ) )
1513, 6, 14jca32 558 . . . . . 6 |- ( F (Walks ` G ) P -> ( ( ( # ` F ) + 1 ) e. NN /\ ( P e. Word (Vtx ` G ) /\ ( # ` P ) = ( ( # ` F ) + 1 ) ) ) )
16 fstwrdne0 13345 . . . . . . 7 |- ( ( ( ( # ` F ) + 1 ) e. NN /\ ( P e. Word (Vtx ` G ) /\ ( # ` P ) = ( ( # ` F ) + 1 ) ) ) -> ( P ` 0 ) e. (Vtx ` G ) )
17 lswlgt0cl 13356 . . . . . . 7 |- ( ( ( ( # ` F ) + 1 ) e. NN /\ ( P e. Word (Vtx ` G ) /\ ( # ` P ) = ( ( # ` F ) + 1 ) ) ) -> ( lastS ` P ) e. (Vtx ` G ) )
1816, 17jca 554 . . . . . 6 |- ( ( ( ( # ` F ) + 1 ) e. NN /\ ( P e. Word (Vtx ` G ) /\ ( # ` P ) = ( ( # ` F ) + 1 ) ) ) -> ( ( P ` 0 ) e. (Vtx ` G ) /\ ( lastS ` P ) e. (Vtx ` G ) ) )
1915, 18syl 17 . . . . 5 |- ( F (Walks ` G ) P -> ( ( P ` 0 ) e. (Vtx ` G ) /\ ( lastS ` P ) e. (Vtx ` G ) ) )
20 eqid 2622 . . . . . . 7 |- (iEdg ` G ) = (iEdg ` G )
2120wlkf 26510 . . . . . 6 |- ( F (Walks ` G ) P -> F e. Word dom (iEdg ` G ) )
22 wrdv 13320 . . . . . 6 |- ( F e. Word dom (iEdg ` G ) -> F e. Word _V )
2321, 22syl 17 . . . . 5 |- ( F (Walks ` G ) P -> F e. Word _V )
2419, 23, 6jca32 558 . . . 4 |- ( F (Walks ` G ) P -> ( ( ( P ` 0 ) e. (Vtx ` G ) /\ ( lastS ` P ) e. (Vtx ` G ) ) /\ ( F e. Word _V /\ P e. Word (Vtx ` G ) ) ) )
251, 24syl 17 . . 3 |- ( ph -> ( ( ( P ` 0 ) e. (Vtx ` G ) /\ ( lastS ` P ) e. (Vtx ` G ) ) /\ ( F e. Word _V /\ P e. Word (Vtx ` G ) ) ) )
265iswlkon 26553 . . 3 |- ( ( ( ( P ` 0 ) e. (Vtx ` G ) /\ ( lastS ` P ) e. (Vtx ` G ) ) /\ ( F e. Word _V /\ P e. Word (Vtx ` G ) ) ) -> ( F ( ( P ` 0 ) (WalksOn ` G ) ( lastS ` P ) ) P <-> ( F (Walks ` G ) P /\ ( P ` 0 ) = ( P ` 0 ) /\ ( P ` ( # ` F ) ) = ( lastS ` P ) ) ) )
2725, 26syl 17 . 2 |- ( ph -> ( F ( ( P ` 0 ) (WalksOn ` G ) ( lastS ` P ) ) P <-> ( F (Walks ` G ) P /\ ( P ` 0 ) = ( P ` 0 ) /\ ( P ` ( # ` F ) ) = ( lastS ` P ) ) ) )
281, 2, 10, 27mpbir3and 1245 1 |- ( ph -> F ( ( P ` 0 ) (WalksOn ` G ) ( lastS ` P ) ) P )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   dom cdm 5114   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   NNcn 11020   NN0cn0 11292   #chash 13117  Word cword 13291   lastS clsw 13292  Vtxcvtx 25874  iEdgciedg 25875  Walkscwlks 26492  WalksOncwlkson 26493
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-oadd 7564  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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-wlks 26495  df-wlkson 26496
This theorem is referenced by:  3wlkond  27031
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