Theorem numclwwlk3lem 25310

Description: Lemma for numclwwlk3 25311. (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 996	. . . . 5 |- ( ( V USGrph E /\ V e. Fin /\ X e. V ) -> X e. V )
2		eluzge2nn0 11121	. . . . 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 25283	. . . . 5 |- ( ( X e. V /\ N e. NN0 ) -> ( X F N ) = { w e. ( C ` N ) | ( w ` 0 ) = X } )
6	1, 2, 5	syl2an 475	. . . 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 5852	. . 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 2655	. . . . . . . . . 10 |- ( -. ( w ` ( N - 2 ) ) =/= ( w ` 0 ) <-> ( w ` ( N - 2 ) ) = ( w ` 0 ) )
10	9	anbi2i 692	. . . . . . . . 9 |- ( ( ( w ` 0 ) = X /\ -. ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) <-> ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) )
11	10	orbi2i 517	. . . . . . . 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 3098	. . . . 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 3766	. . . . 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 2513	. . . 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 5852	. . 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 25279	. . . . . . . 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 464	. . . . . 6 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( C ` N ) = ( ( V ClWWalksN E ) ` N ) )
21		simpl2 998	. . . . . . 7 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> V e. Fin )
22		usgrav 24540	. . . . . . . . . 10 |- ( V USGrph E -> ( V e. _V /\ E e. _V ) )
23	22	simprd 461	. . . . . . . . 9 |- ( V USGrph E -> E e. _V )
24	23	3ad2ant1 1015	. . . . . . . 8 |- ( ( V USGrph E /\ V e. Fin /\ X e. V ) -> E e. _V )
25	24	adantr 463	. . . . . . 7 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> E e. _V )
26	2	adantl 464	. . . . . . 7 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> N e. NN0 )
27		clwwlknfi 24980	. . . . . . 7 |- ( ( V e. Fin /\ E e. _V /\ N e. NN0 ) -> ( ( V ClWWalksN E ) ` N ) e. Fin )
28	21, 25, 26, 27	syl3anc 1226	. . . . . 6 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( ( V ClWWalksN E ) ` N ) e. Fin )
29	20, 28	eqeltrd 2542	. . . . 5 |- ( ( ( V USGrph E /\ V e. Fin /\ X e. V ) /\ N e. ( ZZ>= ` 2 ) ) -> ( C ` N ) e. Fin )
30		rabfi 7737	. . . . 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 7737	. . . . 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 3767	. . . . 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 2651	. . . . . . . . . . . . 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 464	. . . . . . . . . . 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 411	. . . . . . . . 9 |- ( -. ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) \/ -. ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) )
40		ianor 486	. . . . . . . . 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 2868	. . . . . 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 3806	. . . . . 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 2507	. . . 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 12433	. . . 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 1226	. . 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 2499	. 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 25303	. . . . 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 475	. . . 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 5852	. . 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 25289	. . . . 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 469	. . . 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 5852	. . 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 6288	. 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 2498	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 366   /\ wa 367   /\ w3a 971   = wceq 1398   e. wcel 1823   =/= wne 2649   A.wral 2804   {crab 2808   _Vcvv 3106   u. cun 3459   i^i cin 3460   (/)c0 3783   class class class wbr 4439   |-> cmpt 4497   ` cfv 5570  (class class class)co 6270   |-> cmpt2 6272   Fincfn 7509   0cc0 9481   + caddc 9484   - cmin 9796   2c2 10581   NN0cn0 10791   ZZ>=cuz 11082   #chash 12387   lastS clsw 12519   USGrph cusg 24532   WWalksN cwwlkn 24880   ClWWalksN cclwwlkn 24951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12090  df-exp 12149  df-hash 12388  df-word 12526  df-usgra 24535  df-clwwlk 24953  df-clwwlkn 24954

This theorem is referenced by:  numclwwlk3  25311