Theorem numclwwlk3lem 24900

Description: Lemma for numclwwlk3 24901. (Contributed by Alexander van der Vekens, 6-Oct-2018.)

Hypotheses

Ref	Expression
numclwwlk.c	|- C = ( n e. NN0 |-> ( ( V ClWWalksN E ) ` n ) )
numclwwlk.f	|- F = ( v e. V , n e. NN0 |-> { w e. ( C ` n ) | ( w ` 0 ) = v } )
numclwwlk.g	|- G = ( v e. V , n e. ( ZZ>= ` 2 ) |-> { w e. ( C ` n ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) = ( w ` 0 ) ) } )
numclwwlk.q	|- Q = ( v e. V , n e. NN0 |-> { w e. ( ( V WWalksN E ) ` n ) | ( ( w ` 0 ) = v /\ ( lastS ` w ) =/= v ) } )
numclwwlk.h	|- H = ( v e. V , n e. NN0 |-> { w e. ( C ` n ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) =/= ( w ` 0 ) ) } )

Assertion

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

Distinct variable groups:   n, E   n, N   n, V   w, C   w, N   C, n, v, w   v, N   n, X, v, w   v, V   w, E   w, V   w, F   w, Q   w, G   v, E   v, H, w

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

Proof of Theorem numclwwlk3lem

Step	Hyp	Ref	Expression
1		simp3 998	. . . . 5 |- ( ( V USGrph E /\ V e. Fin /\ X e. V ) -> X e. V )
2		eluzge2nn0 11131	. . . . 5 |- ( N e. ( ZZ>= ` 2 ) -> N e. NN0 )
3		numclwwlk.c	. . . . . 6 |- C = ( n e. NN0 |-> ( ( V ClWWalksN E ) ` n ) )
4		numclwwlk.f	. . . . . 6 |- F = ( v e. V , n e. NN0 |-> { w e. ( C ` n ) | ( w ` 0 ) = v } )
5	3, 4	numclwwlkovf 24873	. . . . 5 |- ( ( X e. V /\ N e. NN0 ) -> ( X F N ) = { w e. ( C ` N ) | ( w ` 0 ) = X } )
6	1, 2, 5	syl2an 477	. . . 4 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( X F N ) = { w e. ( C ` N ) | ( w ` 0 ) = X } )
7	6	fveq2d 5875	. . 3 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( # ` ( X F N ) ) = ( # ` { w e. ( C ` N ) | ( w ` 0 ) = X } ) )
8		pm4.42 958	. . . . . . . 8 |- ( ( w ` 0 ) = X <-> ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) \/ ( ( w ` 0 ) = X /\ -. ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) ) )
9		nne 2668	. . . . . . . . . 10 |- ( -. ( w ` ( N - 2 ) ) =/= ( w ` 0 ) <-> ( w ` ( N - 2 ) ) = ( w ` 0 ) )
10	9	anbi2i 694	. . . . . . . . 9 |- ( ( ( w ` 0 ) = X /\ -. ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) <-> ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) )
11	10	orbi2i 519	. . . . . . . 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 ) ) ) )
12	8, 11	bitri 249	. . . . . . 7 |- ( ( w ` 0 ) = X <-> ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) \/ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) )
13	12	a1i 11	. . . . . 6 |- ( ( ( V USGrph E /\ V e. Fin /\ 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 ) ) ) ) )
14	13	rabbidv 3110	. . . . 5 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> { w e. ( C ` N ) | ( w ` 0 ) = X } = { w e. ( C ` N ) | ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) \/ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) } )
15		unrab 3774	. . . . 5 |- ( { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } u. { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) = { w e. ( C ` N ) | ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) \/ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) }
16	14, 15	syl6eqr 2526	. . . 4 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> { w e. ( C ` N ) | ( w ` 0 ) = X } = ( { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } u. { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) )
17	16	fveq2d 5875	. . 3 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( # ` { w e. ( C ` N ) | ( w ` 0 ) = X } ) = ( # ` ( { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } u. { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) ) )
18	3	numclwwlkfvc 24869	. . . . . . . 8 |- ( N e. NN0 -> ( C ` N ) = ( ( V ClWWalksN E ) ` N ) )
19	2, 18	syl 16	. . . . . . 7 |- ( N e. ( ZZ>= ` 2 ) -> ( C ` N ) = ( ( V ClWWalksN E ) ` N ) )
20	19	adantl 466	. . . . . 6 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( C ` N ) = ( ( V ClWWalksN E ) ` N ) )
21		simpl2 1000	. . . . . . 7 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> V e. Fin )
22		usgrav 24129	. . . . . . . . . 10 |- ( V USGrph E -> ( V e. _V /\ E e. _V ) )
23	22	simprd 463	. . . . . . . . 9 |- ( V USGrph E -> E e. _V )
24	23	3ad2ant1 1017	. . . . . . . 8 |- ( ( V USGrph E /\ V e. Fin /\ X e. V ) -> E e. _V )
25	24	adantr 465	. . . . . . 7 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> E e. _V )
26	2	adantl 466	. . . . . . 7 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> N e. NN0 )
27		clwwlknfi 24569	. . . . . . 7 |- ( ( V e. Fin /\ E e. _V /\ N e. NN0 ) -> ( ( V ClWWalksN E ) ` N ) e. Fin )
28	21, 25, 26, 27	syl3anc 1228	. . . . . 6 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( ( V ClWWalksN E ) ` N ) e. Fin )
29	20, 28	eqeltrd 2555	. . . . 5 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( C ` N ) e. Fin )
30		rabfi 7754	. . . . 5 |- ( ( C ` N ) e. Fin -> { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } e. Fin )
31	29, 30	syl 16	. . . 4 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } e. Fin )
32		rabfi 7754	. . . . 5 |- ( ( C ` N ) e. Fin -> { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } e. Fin )
33	29, 32	syl 16	. . . 4 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } e. Fin )
34		inrab 3775	. . . . 5 |- ( { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } i^i { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) = { w e. ( C ` N ) | ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) /\ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) }
35		df-ne 2664	. . . . . . . . . . . . 13 |- ( ( w ` ( N - 2 ) ) =/= ( w ` 0 ) <-> -. ( w ` ( N - 2 ) ) = ( w ` 0 ) )
36	35	biimpi 194	. . . . . . . . . . . 12 |- ( ( w ` ( N - 2 ) ) =/= ( w ` 0 ) -> -. ( w ` ( N - 2 ) ) = ( w ` 0 ) )
37	36	adantl 466	. . . . . . . . . . 11 |- ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) -> -. ( w ` ( N - 2 ) ) = ( w ` 0 ) )
38	37	intnand 914	. . . . . . . . . 10 |- ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) -> -. ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) )
39	38	imori 413	. . . . . . . . 9 |- ( -. ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) \/ -. ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) )
40		ianor 488	. . . . . . . . 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 ) ) ) )
41	39, 40	mpbir 209	. . . . . . . 8 |- -. ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) /\ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) )
42	41	a1i 11	. . . . . . 7 |- ( ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) /\ w e. ( C ` N ) ) -> -. ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) /\ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) )
43	42	ralrimiva 2881	. . . . . 6 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> A. w e. ( C ` N ) -. ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) /\ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) )
44		rabeq0 3812	. . . . . 6 |- ( { w e. ( C ` N ) | ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) /\ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) } = (/) <-> A. w e. ( C ` N ) -. ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) /\ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) )
45	43, 44	sylibr 212	. . . . 5 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> { w e. ( C ` N ) | ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) /\ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) } = (/) )
46	34, 45	syl5eq 2520	. . . 4 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } i^i { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) = (/) )
47		hashun 12428	. . . 4 |- ( ( { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } e. Fin /\ { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } e. Fin /\ ( { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } i^i { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) = (/) ) -> ( # ` ( { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } u. { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) ) = ( ( # ` { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } ) + ( # ` { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) ) )
48	31, 33, 46, 47	syl3anc 1228	. . 3 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( # ` ( { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } u. { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) ) = ( ( # ` { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } ) + ( # ` { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) ) )
49	7, 17, 48	3eqtrd 2512	. 2 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( # ` ( X F N ) ) = ( ( # ` { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } ) + ( # ` { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) ) )
50		numclwwlk.g	. . . . . 6 |- G = ( v e. V , n e. ( ZZ>= ` 2 ) |-> { w e. ( C ` n ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) = ( w ` 0 ) ) } )
51		numclwwlk.q	. . . . . 6 |- Q = ( v e. V , n e. NN0 |-> { w e. ( ( V WWalksN E ) ` n ) | ( ( w ` 0 ) = v /\ ( lastS ` w ) =/= v ) } )
52		numclwwlk.h	. . . . . 6 |- H = ( v e. V , n e. NN0 |-> { w e. ( C ` n ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) =/= ( w ` 0 ) ) } )
53	3, 4, 50, 51, 52	numclwwlkovh 24893	. . . . 5 |- ( ( X e. V /\ N e. NN0 ) -> ( X H N ) = { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } )
54	1, 2, 53	syl2an 477	. . . 4 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( X H N ) = { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } )
55	54	fveq2d 5875	. . 3 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( # ` ( X H N ) ) = ( # ` { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } ) )
56	3, 4, 50	numclwwlkovg 24879	. . . . 5 |- ( ( X e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( X G N ) = { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } )
57	1, 56	sylan 471	. . . 4 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( X G N ) = { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } )
58	57	fveq2d 5875	. . 3 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( # ` ( X G N ) ) = ( # ` { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) )
59	55, 58	oveq12d 6312	. 2 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( ( # ` ( X H N ) ) + ( # ` ( X G N ) ) ) = ( ( # ` { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } ) + ( # ` { w e. ( C ` N ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) ) )
60	49, 59	eqtr4d 2511	1 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( # ` ( X F N ) ) = ( ( # ` ( X H N ) ) + ( # ` ( X G N ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   A.wral 2817   {crab 2821   _Vcvv 3118   u. cun 3479   i^i cin 3480   (/)c0 3790   class class class wbr 4452   |-> cmpt 4510   ` cfv 5593  (class class class)co 6294   |-> cmpt2 6296   Fincfn 7526   0cc0 9502   + caddc 9505   - cmin 9815   2c2 10595   NN0cn0 10805   ZZ>=cuz 11092   #chash 12383   lastS clsw 12511   USGrph cusg 24121   WWalksN cwwlkn 24469   ClWWalksN cclwwlkn 24540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-1o 7140  df-2o 7141  df-oadd 7144  df-er 7321  df-map 7432  df-pm 7433  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-card 8330  df-cda 8558  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-nn 10547  df-2 10604  df-n0 10806  df-z 10875  df-uz 11093  df-fz 11683  df-fzo 11803  df-seq 12086  df-exp 12145  df-hash 12384  df-word 12518  df-usgra 24124  df-clwwlk 24542  df-clwwlkn 24543

This theorem is referenced by:  numclwwlk3  24901