Theorem numclwlk1lem2f 27225

Description: T is a function, mapping a closed walk having a fixed length and starting at a fixed vertex) with the last but 2 vertex is identical with the first (and therefore last) vertex to the pair of the shorter closed walk and its successor in the longer closed walk, which must be a neighbor of the first vertex. (Contributed by Alexander van der Vekens, 19-Sep-2018.) (Revised by AV, 29-May-2021.)

Hypotheses

Ref	Expression
extwwlkfab.v	|- V = (Vtx ` G )
extwwlkfab.f	|- F = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
extwwlkfab.c	|- C = ( v e. V , n e. ( ZZ>= ` 2 ) |-> { w e. ( n ClWWalksN G ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) = ( w ` 0 ) ) } )
numclwwlk.t	|- T = ( u e. ( X C N ) |-> <. ( u substr <. 0 , ( N - 2 ) >. ) , ( u ` ( N - 1 ) ) >. )

Assertion

Ref	Expression
numclwlk1lem2f	|- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> T : ( X C N ) --> ( ( X F ( N - 2 ) ) X. ( G NeighbVtx X ) ) )

Distinct variable groups:   n, G, u, v, w   n, N, u, v, w   n, V, v, w   n, X, u, v, w   w, F   u, C   u, F   u, V

Allowed substitution hints:   C( w, v, n)   T( w, v, u, n)   F( v, n)

Proof of Theorem numclwlk1lem2f

Step	Hyp	Ref	Expression
1		extwwlkfab.v	. . . . . . 7 |- V = (Vtx ` G )
2		extwwlkfab.f	. . . . . . 7 |- F = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
3		extwwlkfab.c	. . . . . . 7 |- C = ( v e. V , n e. ( ZZ>= ` 2 ) |-> { w e. ( n ClWWalksN G ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) = ( w ` 0 ) ) } )
4	1, 2, 3	extwwlkfab 27223	. . . . . 6 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( X C N ) = { w e. ( N ClWWalksN G ) | ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( w ` ( N - 2 ) ) = X ) } )
5	4	eleq2d 2687	. . . . 5 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( u e. ( X C N ) <-> u e. { w e. ( N ClWWalksN G ) | ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( w ` ( N - 2 ) ) = X ) } ) )
6		oveq1 6657	. . . . . . . 8 |- ( w = u -> ( w substr <. 0 , ( N - 2 ) >. ) = ( u substr <. 0 , ( N - 2 ) >. ) )
7	6	eleq1d 2686	. . . . . . 7 |- ( w = u -> ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) <-> ( u substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) ) )
8		fveq1 6190	. . . . . . . 8 |- ( w = u -> ( w ` ( N - 1 ) ) = ( u ` ( N - 1 ) ) )
9	8	eleq1d 2686	. . . . . . 7 |- ( w = u -> ( ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) <-> ( u ` ( N - 1 ) ) e. ( G NeighbVtx X ) ) )
10		fveq1 6190	. . . . . . . 8 |- ( w = u -> ( w ` ( N - 2 ) ) = ( u ` ( N - 2 ) ) )
11	10	eqeq1d 2624	. . . . . . 7 |- ( w = u -> ( ( w ` ( N - 2 ) ) = X <-> ( u ` ( N - 2 ) ) = X ) )
12	7, 9, 11	3anbi123d 1399	. . . . . 6 |- ( w = u -> ( ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( w ` ( N - 2 ) ) = X ) <-> ( ( u substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( u ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( u ` ( N - 2 ) ) = X ) ) )
13	12	elrab 3363	. . . . 5 |- ( u e. { w e. ( N ClWWalksN G ) | ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( w ` ( N - 2 ) ) = X ) } <-> ( u e. ( N ClWWalksN G ) /\ ( ( u substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( u ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( u ` ( N - 2 ) ) = X ) ) )
14	5, 13	syl6bb 276	. . . 4 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( u e. ( X C N ) <-> ( u e. ( N ClWWalksN G ) /\ ( ( u substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( u ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( u ` ( N - 2 ) ) = X ) ) ) )
15		simprr1 1109	. . . . . 6 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( u e. ( N ClWWalksN G ) /\ ( ( u substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( u ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( u ` ( N - 2 ) ) = X ) ) ) -> ( u substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) )
16		simprr2 1110	. . . . . 6 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( u e. ( N ClWWalksN G ) /\ ( ( u substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( u ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( u ` ( N - 2 ) ) = X ) ) ) -> ( u ` ( N - 1 ) ) e. ( G NeighbVtx X ) )
17	15, 16	opelxpd 5149	. . . . 5 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( u e. ( N ClWWalksN G ) /\ ( ( u substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( u ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( u ` ( N - 2 ) ) = X ) ) ) -> <. ( u substr <. 0 , ( N - 2 ) >. ) , ( u ` ( N - 1 ) ) >. e. ( ( X F ( N - 2 ) ) X. ( G NeighbVtx X ) ) )
18	17	ex 450	. . . 4 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( u e. ( N ClWWalksN G ) /\ ( ( u substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( u ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( u ` ( N - 2 ) ) = X ) ) -> <. ( u substr <. 0 , ( N - 2 ) >. ) , ( u ` ( N - 1 ) ) >. e. ( ( X F ( N - 2 ) ) X. ( G NeighbVtx X ) ) ) )
19	14, 18	sylbid 230	. . 3 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( u e. ( X C N ) -> <. ( u substr <. 0 , ( N - 2 ) >. ) , ( u ` ( N - 1 ) ) >. e. ( ( X F ( N - 2 ) ) X. ( G NeighbVtx X ) ) ) )
20	19	imp 445	. 2 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ u e. ( X C N ) ) -> <. ( u substr <. 0 , ( N - 2 ) >. ) , ( u ` ( N - 1 ) ) >. e. ( ( X F ( N - 2 ) ) X. ( G NeighbVtx X ) ) )
21		numclwwlk.t	. 2 |- T = ( u e. ( X C N ) |-> <. ( u substr <. 0 , ( N - 2 ) >. ) , ( u ` ( N - 1 ) ) >. )
22	20, 21	fmptd 6385	1 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> T : ( X C N ) --> ( ( X F ( N - 2 ) ) X. ( G NeighbVtx X ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   <.cop 4183   |-> cmpt 4729   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   - cmin 10266   NNcn 11020   2c2 11070   3c3 11071   ZZ>=cuz 11687   substr csubstr 13295  Vtxcvtx 25874   USGraph cusgr 26044   NeighbVtx cnbgr 26224   ClWWalksN cclwwlksn 26876

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-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-rmo 2920  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-2o 7561  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-cda 8990  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-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-substr 13303  df-edg 25940  df-upgr 25977  df-umgr 25978  df-usgr 26046  df-nbgr 26228  df-wwlks 26722  df-wwlksn 26723  df-clwwlks 26877  df-clwwlksn 26878

This theorem is referenced by:  numclwlk1lem2f1  27227  numclwlk1lem2fo  27228