Theorem numclwwlk3lem 24973

Description: Lemma for numclwwlk3 24974. (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 997	. . . . 5 |- ( ( V USGrph E /\ V e. Fin /\ X e. V ) -> X e. V )
2		eluzge2nn0 11124	. . . . 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 24946	. . . . 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 5856	. . 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 2642	. . . . . . . . . 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 3085	. . . . 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 3751	. . . . 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 2500	. . . 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 5856	. . 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 24942	. . . . . . . 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 999	. . . . . . 7 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> V e. Fin )
22		usgrav 24203	. . . . . . . . . 10 |- ( V USGrph E -> ( V e. _V /\ E e. _V ) )
23	22	simprd 463	. . . . . . . . 9 |- ( V USGrph E -> E e. _V )
24	23	3ad2ant1 1016	. . . . . . . 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 24643	. . . . . . 7 |- ( ( V e. Fin /\ E e. _V /\ N e. NN0 ) -> ( ( V ClWWalksN E ) ` N ) e. Fin )
28	21, 25, 26, 27	syl3anc 1227	. . . . . 6 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( ( V ClWWalksN E ) ` N ) e. Fin )
29	20, 28	eqeltrd 2529	. . . . 5 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( C ` N ) e. Fin )
30		rabfi 7742	. . . . 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 7742	. . . . 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 3752	. . . . 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 2638	. . . . . . . . . . . . 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 2855	. . . . . 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 3789	. . . . . 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 2494	. . . 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 12424	. . . 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 1227	. . 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 2486	. 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 24966	. . . . 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 5856	. . 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 24952	. . . . 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 5856	. . 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 6295	. 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 2485	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 972   = wceq 1381   e. wcel 1802   =/= wne 2636   A.wral 2791   {crab 2795   _Vcvv 3093   u. cun 3456   i^i cin 3457   (/)c0 3767   class class class wbr 4433   |-> cmpt 4491   ` cfv 5574  (class class class)co 6277   |-> cmpt2 6279   Fincfn 7514   0cc0 9490   + caddc 9493   - cmin 9805   2c2 10586   NN0cn0 10796   ZZ>=cuz 11085   #chash 12379   lastS clsw 12509   USGrph cusg 24195   WWalksN cwwlkn 24543   ClWWalksN cclwwlkn 24614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-2o 7129  df-oadd 7132  df-er 7309  df-map 7420  df-pm 7421  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-card 8318  df-cda 8546  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11086  df-fz 11677  df-fzo 11799  df-seq 12082  df-exp 12141  df-hash 12380  df-word 12516  df-usgra 24198  df-clwwlk 24616  df-clwwlkn 24617

This theorem is referenced by:  numclwwlk3  24974