Theorem wlkiswwlks1 30156

Description: The sequence of vertices in a walk is a walk as word in a pseudograph. (Contributed by Alexander van der Vekens, 20-Jul-2018.) (Revised by AV, 9-Apr-2021.)

Assertion

Ref	Expression
wlkiswwlks1	⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 → 𝑃 ∈ (WWalks‘𝐺)))

Proof of Theorem wlkiswwlks1

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wlkn0 29910	. 2 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃 ≠ ∅)
2		eqid 2769	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2769	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4	2, 3	upgriswlk 29930	. . 3 ⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
5		simpr 489	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})) ∧ 𝑃 ≠ ∅) → 𝑃 ≠ ∅)
6		ffz0iswrd 14577	. . . . . . . 8 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → 𝑃 ∈ Word (Vtx‘𝐺))
7	6	3ad2ant2 1150	. . . . . . 7 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → 𝑃 ∈ Word (Vtx‘𝐺))
8	7	ad2antlr 739	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})) ∧ 𝑃 ≠ ∅) → 𝑃 ∈ Word (Vtx‘𝐺))
9		upgruhgr 29392	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
10	3	uhgrfun 29356	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
11		funfn 6567	. . . . . . . . . . . . . . . . . . 19 ⊢ (Fun (iEdg‘𝐺) ↔ (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
12	11	biimpi 219	. . . . . . . . . . . . . . . . . 18 ⊢ (Fun (iEdg‘𝐺) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
13	9, 10, 12	3syl 19	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
14	13	ad2antlr 739	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph) ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
15		wrdsymbcl 14563	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → (𝐹‘𝑖) ∈ dom (iEdg‘𝐺))
16	15	ad4ant14 764	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph) ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → (𝐹‘𝑖) ∈ dom (iEdg‘𝐺))
17		fnfvelrn 7076	. . . . . . . . . . . . . . . 16 ⊢ (((iEdg‘𝐺) Fn dom (iEdg‘𝐺) ∧ (𝐹‘𝑖) ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘(𝐹‘𝑖)) ∈ ran (iEdg‘𝐺))
18	14, 16, 17	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph) ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹‘𝑖)) ∈ ran (iEdg‘𝐺))
19		edgval 29339	. . . . . . . . . . . . . . 15 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
20	18, 19	eleqtrrdi 2880	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph) ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹‘𝑖)) ∈ (Edg‘𝐺))
21		eleq1 2857	. . . . . . . . . . . . . . 15 ⊢ ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} = ((iEdg‘𝐺)‘(𝐹‘𝑖)) → ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ((iEdg‘𝐺)‘(𝐹‘𝑖)) ∈ (Edg‘𝐺)))
22	21	eqcoms 2777	. . . . . . . . . . . . . 14 ⊢ (((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} → ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ((iEdg‘𝐺)‘(𝐹‘𝑖)) ∈ (Edg‘𝐺)))
23	20, 22	syl5ibrcom 250	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph) ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → (((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
24	23	ralimdva 3183	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph) → (∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} → ∀𝑖 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
25	24	ex 417	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (𝐺 ∈ UPGraph → (∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} → ∀𝑖 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))))
26	25	com23 87	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} → (𝐺 ∈ UPGraph → ∀𝑖 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))))
27	26	3impia 1133	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → (𝐺 ∈ UPGraph → ∀𝑖 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
28	27	impcom 412	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})) → ∀𝑖 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))
29		lencl 14569	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ Word dom (iEdg‘𝐺) → (♯‘𝐹) ∈ ℕ0)
30		ffz0hash 14483	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (♯‘𝑃) = ((♯‘𝐹) + 1))
31	30	ex 417	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) ∈ ℕ0 → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → (♯‘𝑃) = ((♯‘𝐹) + 1)))
32		oveq1 7418	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑃) = ((♯‘𝐹) + 1) → ((♯‘𝑃) − 1) = (((♯‘𝐹) + 1) − 1))
33		nn0cn 12513	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐹) ∈ ℕ0 → (♯‘𝐹) ∈ ℂ)
34		pncan1 11637	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐹) ∈ ℂ → (((♯‘𝐹) + 1) − 1) = (♯‘𝐹))
35	33, 34	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝐹) ∈ ℕ0 → (((♯‘𝐹) + 1) − 1) = (♯‘𝐹))
36	32, 35	sylan9eqr 2826	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ (♯‘𝑃) = ((♯‘𝐹) + 1)) → ((♯‘𝑃) − 1) = (♯‘𝐹))
37	36	ex 417	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) ∈ ℕ0 → ((♯‘𝑃) = ((♯‘𝐹) + 1) → ((♯‘𝑃) − 1) = (♯‘𝐹)))
38	31, 37	syld 48	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐹) ∈ ℕ0 → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → ((♯‘𝑃) − 1) = (♯‘𝐹)))
39	29, 38	syl 18	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ Word dom (iEdg‘𝐺) → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → ((♯‘𝑃) − 1) = (♯‘𝐹)))
40	39	imp 411	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → ((♯‘𝑃) − 1) = (♯‘𝐹))
41	40	oveq2d 7427	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (0..^((♯‘𝑃) − 1)) = (0..^(♯‘𝐹)))
42	41	raleqdv 3329	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
43	42	3adant3 1148	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
44	43	adantl 486	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
45	28, 44	mpbird 260	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})) → ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))
46	45	adantr 485	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})) ∧ 𝑃 ≠ ∅) → ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))
47		eqid 2769	. . . . . . 7 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
48	2, 47	iswwlks 30125	. . . . . 6 ⊢ (𝑃 ∈ (WWalks‘𝐺) ↔ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
49	5, 8, 46, 48	syl3anbrc 1360	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})) ∧ 𝑃 ≠ ∅) → 𝑃 ∈ (WWalks‘𝐺))
50	49	ex 417	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})) → (𝑃 ≠ ∅ → 𝑃 ∈ (WWalks‘𝐺)))
51	50	ex 417	. . 3 ⊢ (𝐺 ∈ UPGraph → ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → (𝑃 ≠ ∅ → 𝑃 ∈ (WWalks‘𝐺))))
52	4, 51	sylbid 243	. 2 ⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 → (𝑃 ≠ ∅ → 𝑃 ∈ (WWalks‘𝐺))))
53	1, 52	mpdi 46	1 ⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 → 𝑃 ∈ (WWalks‘𝐺)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∅c0 4294  {cpr 4596   class class class wbr 5113  dom cdm 5662  ran crn 5663  Fun wfun 6531   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411  ℂcc 11097  0cc0 11099  1c1 11100   + caddc 11102   − cmin 11440  ℕ₀cn0 12503  ...cfz 13534  ..^cfzo 13681  ♯chash 14365  Word cword 14549  Vtxcvtx 29286  iEdgciedg 29287  Edgcedg 29337  UHGraphcuhgr 29346  UPGraphcupgr 29370  Walkscwlks 29886  WWalkscwwlks 30114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ifp 1077  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-oadd 8456  df-er 8693  df-map 8825  df-pm 8826  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-dju 9886  df-card 9924  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-2 12302  df-n0 12504  df-xnn0 12577  df-z 12591  df-uz 12862  df-fz 13535  df-fzo 13682  df-hash 14366  df-word 14550  df-edg 29338  df-uhgr 29348  df-upgr 29372  df-wlks 29889  df-wwlks 30119

This theorem is referenced by:  wlklnwwlkln1  30157  wlkiswwlks  30165  wlkiswwlkupgr  30167  elwspths2spth  30259