Theorem numclwwlk3lem 30706

Description: Lemma for numclwwlk3 30707. (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 990	. . . . 5 |- ( ( V USGrph E /\ V e. Fin /\ X e. V ) -> X e. V )
2		eluzge2nn0 30197	. . . . 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 30679	. . . . 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 5700	. . 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		pm4.42 951	. . . . . . . 8 |- ( ( w ` 0 ) = X <-> ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) \/ ( ( w ` 0 ) = X /\ -. ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) ) )
9		nne 2617	. . . . . . . . . 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 2969	. . . . 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 3626	. . . . 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 2493	. . . 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 5700	. . 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 30675	. . . . . . . 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 992	. . . . . . 7 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> V e. Fin )
22		usgrav 23275	. . . . . . . . . 10 |- ( V USGrph E -> ( V e. _V /\ E e. _V ) )
23	22	simprd 463	. . . . . . . . 9 |- ( V USGrph E -> E e. _V )
24	23	3ad2ant1 1009	. . . . . . . 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 30482	. . . . . . 7 |- ( ( V e. Fin /\ E e. _V /\ N e. NN0 ) -> ( ( V ClWWalksN E ) ` N ) e. Fin )
28	21, 25, 26, 27	syl3anc 1218	. . . . . 6 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( ( V ClWWalksN E ) ` N ) e. Fin )
29	20, 28	eqeltrd 2517	. . . . 5 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( C ` N ) e. Fin )
30		rabfi 7542	. . . . 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 7542	. . . . 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 3627	. . . . 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 2613	. . . . . . . . . . . . 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 907	. . . . . . . . . 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 2804	. . . . . 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 3664	. . . . . 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 2487	. . . 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 12150	. . . 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 1218	. . 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 2479	. 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 30699	. . . . 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 5700	. . 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 30685	. . . . 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 5700	. . 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 6114	. 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 2478	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 965   = wceq 1369   e. wcel 1756   =/= wne 2611   A.wral 2720   {crab 2724   _Vcvv 2977   u. cun 3331   i^i cin 3332   (/)c0 3642   class class class wbr 4297   e. cmpt 4355   ` cfv 5423  (class class class)co 6096   e. cmpt2 6098   Fincfn 7315   0cc0 9287   + caddc 9290   - cmin 9600   2c2 10376   NN0cn0 10584   ZZ>=cuz 10866   #chash 12108   lastS clsw 12227   USGrph cusg 23269   WWalksN cwwlkn 30317   ClWWalksN cclwwlkn 30419

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-fzo 11554  df-seq 11812  df-exp 11871  df-hash 12109  df-word 12234  df-usgra 23271  df-clwwlk 30421  df-clwwlkn 30422

This theorem is referenced by:  numclwwlk3  30707