Theorem numclwwlk3lem 27241

Description: Lemma for numclwwlk3 27243. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 1-Jun-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 ) ) } )
numclwwlk.c	|- C = ( v e. V , n e. ( ZZ>= ` 2 ) |-> { w e. ( n ClWWalksN G ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) = ( w ` 0 ) ) } )

Assertion

Ref	Expression
numclwwlk3lem	|- ( ( ( G e. FinUSGraph /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( # ` ( X F N ) ) = ( ( # ` ( X H N ) ) + ( # ` ( X C N ) ) ) )

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

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

Proof of Theorem numclwwlk3lem

Step	Hyp	Ref	Expression
1		simpr 477	. . . . 5 |- ( ( G e. FinUSGraph /\ X e. V ) -> X e. V )
2		eluz2nn 11726	. . . . 5 |- ( N e. ( ZZ>= ` 2 ) -> N e. NN )
3		numclwwlk.f	. . . . . 6 |- F = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
4	3	numclwwlkovf 27213	. . . . 5 |- ( ( X e. V /\ N e. NN ) -> ( X F N ) = { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } )
5	1, 2, 4	syl2an 494	. . . 4 |- ( ( ( G e. FinUSGraph /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( X F N ) = { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } )
6	5	fveq2d 6195	. . 3 |- ( ( ( G e. FinUSGraph /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( # ` ( X F N ) ) = ( # ` { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } ) )
7		pm4.42 1004	. . . . . . . 8 |- ( ( w ` 0 ) = X <-> ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) \/ ( ( w ` 0 ) = X /\ -. ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) ) )
8		nne 2798	. . . . . . . . . 10 |- ( -. ( w ` ( N - 2 ) ) =/= ( w ` 0 ) <-> ( w ` ( N - 2 ) ) = ( w ` 0 ) )
9	8	anbi2i 730	. . . . . . . . 9 |- ( ( ( w ` 0 ) = X /\ -. ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) <-> ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) )
10	9	orbi2i 541	. . . . . . . 8 |- ( ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) \/ ( ( w ` 0 ) = X /\ -. ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) ) <-> ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) \/ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) )
11	7, 10	bitri 264	. . . . . . 7 |- ( ( w ` 0 ) = X <-> ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) \/ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) )
12	11	a1i 11	. . . . . 6 |- ( ( ( G e. FinUSGraph /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( ( w ` 0 ) = X <-> ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) \/ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) ) )
13	12	rabbidv 3189	. . . . 5 |- ( ( ( G e. FinUSGraph /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } = { w e. ( N ClWWalksN G ) | ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) \/ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) } )
14		unrab 3898	. . . . 5 |- ( { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } u. { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) = { w e. ( N ClWWalksN G ) | ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) \/ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) }
15	13, 14	syl6eqr 2674	. . . 4 |- ( ( ( G e. FinUSGraph /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } = ( { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } u. { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) )
16	15	fveq2d 6195	. . 3 |- ( ( ( G e. FinUSGraph /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( # ` { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } ) = ( # ` ( { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } u. { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) ) )
17		numclwwlk.v	. . . . . . 7 |- V = (Vtx ` G )
18	17	fusgrvtxfi 26211	. . . . . . . 8 |- ( G e. FinUSGraph -> V e. Fin )
19	18	ad2antrr 762	. . . . . . 7 |- ( ( ( G e. FinUSGraph /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> V e. Fin )
20	17, 19	syl5eqelr 2706	. . . . . 6 |- ( ( ( G e. FinUSGraph /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> (Vtx ` G ) e. Fin )
21		clwwlksnfi 26913	. . . . . 6 |- ( (Vtx ` G ) e. Fin -> ( N ClWWalksN G ) e. Fin )
22	20, 21	syl 17	. . . . 5 |- ( ( ( G e. FinUSGraph /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( N ClWWalksN G ) e. Fin )
23		rabfi 8185	. . . . 5 |- ( ( N ClWWalksN G ) e. Fin -> { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } e. Fin )
24	22, 23	syl 17	. . . 4 |- ( ( ( G e. FinUSGraph /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } e. Fin )
25		rabfi 8185	. . . . 5 |- ( ( N ClWWalksN G ) e. Fin -> { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } e. Fin )
26	22, 25	syl 17	. . . 4 |- ( ( ( G e. FinUSGraph /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } e. Fin )
27		inrab 3899	. . . . 5 |- ( { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } i^i { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) = { w e. ( N ClWWalksN G ) | ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) /\ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) }
28		neneq 2800	. . . . . . . . . . . 12 |- ( ( w ` ( N - 2 ) ) =/= ( w ` 0 ) -> -. ( w ` ( N - 2 ) ) = ( w ` 0 ) )
29	28	adantl 482	. . . . . . . . . . 11 |- ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) -> -. ( w ` ( N - 2 ) ) = ( w ` 0 ) )
30	29	intnand 962	. . . . . . . . . 10 |- ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) -> -. ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) )
31	30	imori 429	. . . . . . . . 9 |- ( -. ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) \/ -. ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) )
32		ianor 509	. . . . . . . . 9 |- ( -. ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) /\ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) <-> ( -. ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) \/ -. ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) )
33	31, 32	mpbir 221	. . . . . . . 8 |- -. ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) /\ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) )
34	33	a1i 11	. . . . . . 7 |- ( ( ( ( G e. FinUSGraph /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) /\ w e. ( N ClWWalksN G ) ) -> -. ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) /\ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) )
35	34	ralrimiva 2966	. . . . . 6 |- ( ( ( G e. FinUSGraph /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> A. w e. ( N ClWWalksN G ) -. ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) /\ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) )
36		rabeq0 3957	. . . . . 6 |- ( { w e. ( N ClWWalksN G ) | ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) /\ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) } = (/) <-> A. w e. ( N ClWWalksN G ) -. ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) /\ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) )
37	35, 36	sylibr 224	. . . . 5 |- ( ( ( G e. FinUSGraph /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> { w e. ( N ClWWalksN G ) | ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) /\ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) } = (/) )
38	27, 37	syl5eq 2668	. . . 4 |- ( ( ( G e. FinUSGraph /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } i^i { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) = (/) )
39		hashun 13171	. . . 4 |- ( ( { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } e. Fin /\ { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } e. Fin /\ ( { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } i^i { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) = (/) ) -> ( # ` ( { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } u. { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) ) = ( ( # ` { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } ) + ( # ` { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) ) )
40	24, 26, 38, 39	syl3anc 1326	. . 3 |- ( ( ( G e. FinUSGraph /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( # ` ( { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } u. { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) ) = ( ( # ` { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } ) + ( # ` { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) ) )
41	6, 16, 40	3eqtrd 2660	. 2 |- ( ( ( G e. FinUSGraph /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( # ` ( X F N ) ) = ( ( # ` { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } ) + ( # ` { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) ) )
42		numclwwlk.q	. . . . . 6 |- Q = ( v e. V , n e. NN |-> { w e. ( n WWalksN G ) | ( ( w ` 0 ) = v /\ ( lastS ` w ) =/= v ) } )
43		numclwwlk.h	. . . . . 6 |- H = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) =/= ( w ` 0 ) ) } )
44	17, 42, 3, 43	numclwwlkovh 27234	. . . . 5 |- ( ( X e. V /\ N e. NN ) -> ( X H N ) = { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } )
45	1, 2, 44	syl2an 494	. . . 4 |- ( ( ( G e. FinUSGraph /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( X H N ) = { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } )
46	45	fveq2d 6195	. . 3 |- ( ( ( G e. FinUSGraph /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( # ` ( X H N ) ) = ( # ` { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } ) )
47		numclwwlk.c	. . . . . 6 |- C = ( v e. V , n e. ( ZZ>= ` 2 ) |-> { w e. ( n ClWWalksN G ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) = ( w ` 0 ) ) } )
48	47	numclwwlkovg 27220	. . . . 5 |- ( ( X e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( X C N ) = { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } )
49	48	adantll 750	. . . 4 |- ( ( ( G e. FinUSGraph /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( X C N ) = { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } )
50	49	fveq2d 6195	. . 3 |- ( ( ( G e. FinUSGraph /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( # ` ( X C N ) ) = ( # ` { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) )
51	46, 50	oveq12d 6668	. 2 |- ( ( ( G e. FinUSGraph /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( ( # ` ( X H N ) ) + ( # ` ( X C N ) ) ) = ( ( # ` { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } ) + ( # ` { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) ) )
52	41, 51	eqtr4d 2659	1 |- ( ( ( G e. FinUSGraph /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( # ` ( X F N ) ) = ( ( # ` ( X H N ) ) + ( # ` ( X C N ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   u. cun 3572   i^i cin 3573   (/)c0 3915   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955   0cc0 9936   + caddc 9939   - cmin 10266   NNcn 11020   2c2 11070   ZZ>=cuz 11687   #chash 13117   lastS clsw 13292  Vtxcvtx 25874   FinUSGraph cfusgr 26208   WWalksN cwwlksn 26718   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-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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-fusgr 26209  df-clwwlks 26877  df-clwwlksn 26878

This theorem is referenced by:  numclwwlk3OLD  27242  numclwwlk3  27243