Theorem numclwwlk2lem3 27239

Description: In a friendship graph, the size of the set of walks of length N starting with a fixed vertex X and ending not at this vertex equals the size of the set of all closed walks of length ( N + 2 ) starting at this vertex X and not having this vertex as last but 2 vertex. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 31-May-2021.)

Hypotheses

Ref	Expression
numclwwlk.v	|- V = (Vtx ` G )
numclwwlk.q	|- Q = ( v e. V , n e. NN |-> { w e. ( n WWalksN G ) | ( ( w ` 0 ) = v /\ ( lastS ` w ) =/= v ) } )
numclwwlk.f	|- F = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
numclwwlk.h	|- H = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) =/= ( w ` 0 ) ) } )

Assertion

Ref	Expression
numclwwlk2lem3	|- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> ( # ` ( X Q N ) ) = ( # ` ( X H ( N + 2 ) ) ) )

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

Allowed substitution hints:   Q( w, v, n)   F( w, v, n)   H( w, v, n)

Proof of Theorem numclwwlk2lem3

Dummy variable h is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovexd 6680	. . 3 |- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> ( X H ( N + 2 ) ) e. _V )
2		numclwwlk.v	. . . 4 |- V = (Vtx ` G )
3		numclwwlk.q	. . . 4 |- Q = ( v e. V , n e. NN |-> { w e. ( n WWalksN G ) | ( ( w ` 0 ) = v /\ ( lastS ` w ) =/= v ) } )
4		numclwwlk.f	. . . 4 |- F = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
5		numclwwlk.h	. . . 4 |- H = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) =/= ( w ` 0 ) ) } )
6		eqid 2622	. . . 4 |- ( h e. ( X H ( N + 2 ) ) |-> ( h substr <. 0 , ( N + 1 ) >. ) ) = ( h e. ( X H ( N + 2 ) ) |-> ( h substr <. 0 , ( N + 1 ) >. ) )
7	2, 3, 4, 5, 6	numclwlk2lem2f1o 27238	. . 3 |- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> ( h e. ( X H ( N + 2 ) ) |-> ( h substr <. 0 , ( N + 1 ) >. ) ) : ( X H ( N + 2 ) ) -1-1-onto-> ( X Q N ) )
8	1, 7	hasheqf1od 13144	. 2 |- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> ( # ` ( X H ( N + 2 ) ) ) = ( # ` ( X Q N ) ) )
9	8	eqcomd 2628	1 |- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> ( # ` ( X Q N ) ) = ( # ` ( X H ( N + 2 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   <.cop 4183   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   NNcn 11020   2c2 11070   #chash 13117   lastS clsw 13292   substr csubstr 13295  Vtxcvtx 25874   WWalksN cwwlksn 26718   ClWWalksN cclwwlksn 26876   FriendGraph cfrgr 27120

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-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-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-wwlks 26722  df-wwlksn 26723  df-clwwlks 26877  df-clwwlksn 26878  df-frgr 27121

This theorem is referenced by:  numclwwlk2  27240