Theorem wlkiswwlks1 29812

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 29566	. 2 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃 ≠ ∅)
2		eqid 2729	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2729	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4	2, 3	upgriswlk 29586	. . 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 484	. . . . . 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 14448	. . . . . . . 8 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → 𝑃 ∈ Word (Vtx‘𝐺))
7	6	3ad2ant2 1134	. . . . . . 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 727	. . . . . 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 29047	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
10	3	uhgrfun 29011	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
11		funfn 6512	. . . . . . . . . . . . . . . . . . 19 ⊢ (Fun (iEdg‘𝐺) ↔ (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
12	11	biimpi 216	. . . . . . . . . . . . . . . . . 18 ⊢ (Fun (iEdg‘𝐺) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
13	9, 10, 12	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
14	13	ad2antlr 727	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph) ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
15		wrdsymbcl 14434	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → (𝐹‘𝑖) ∈ dom (iEdg‘𝐺))
16	15	ad4ant14 752	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph) ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → (𝐹‘𝑖) ∈ dom (iEdg‘𝐺))
17		fnfvelrn 7014	. . . . . . . . . . . . . . . 16 ⊢ (((iEdg‘𝐺) Fn dom (iEdg‘𝐺) ∧ (𝐹‘𝑖) ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘(𝐹‘𝑖)) ∈ ran (iEdg‘𝐺))
18	14, 16, 17	syl2anc 584	. . . . . . . . . . . . . . 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 28994	. . . . . . . . . . . . . . 15 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
20	18, 19	eleqtrrdi 2839	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph) ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹‘𝑖)) ∈ (Edg‘𝐺))
21		eleq1 2816	. . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . 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 247	. . . . . . . . . . . . 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 3141	. . . . . . . . . . . 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 412	. . . . . . . . . . 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 86	. . . . . . . . . 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 1117	. . . . . . . . 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 407	. . . . . . . 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 14440	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ Word dom (iEdg‘𝐺) → (♯‘𝐹) ∈ ℕ0)
30		ffz0hash 14354	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (♯‘𝑃) = ((♯‘𝐹) + 1))
31	30	ex 412	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) ∈ ℕ0 → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → (♯‘𝑃) = ((♯‘𝐹) + 1)))
32		oveq1 7356	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑃) = ((♯‘𝐹) + 1) → ((♯‘𝑃) − 1) = (((♯‘𝐹) + 1) − 1))
33		nn0cn 12394	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐹) ∈ ℕ0 → (♯‘𝐹) ∈ ℂ)
34		pncan1 11544	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐹) ∈ ℂ → (((♯‘𝐹) + 1) − 1) = (♯‘𝐹))
35	33, 34	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝐹) ∈ ℕ0 → (((♯‘𝐹) + 1) − 1) = (♯‘𝐹))
36	32, 35	sylan9eqr 2786	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ (♯‘𝑃) = ((♯‘𝐹) + 1)) → ((♯‘𝑃) − 1) = (♯‘𝐹))
37	36	ex 412	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) ∈ ℕ0 → ((♯‘𝑃) = ((♯‘𝐹) + 1) → ((♯‘𝑃) − 1) = (♯‘𝐹)))
38	31, 37	syld 47	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐹) ∈ ℕ0 → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → ((♯‘𝑃) − 1) = (♯‘𝐹)))
39	29, 38	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ Word dom (iEdg‘𝐺) → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → ((♯‘𝑃) − 1) = (♯‘𝐹)))
40	39	imp 406	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → ((♯‘𝑃) − 1) = (♯‘𝐹))
41	40	oveq2d 7365	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (0..^((♯‘𝑃) − 1)) = (0..^(♯‘𝐹)))
42	41	raleqdv 3289	. . . . . . . . . 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 1132	. . . . . . . . 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 481	. . . . . . . 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 257	. . . . . . 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 480	. . . . . 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 2729	. . . . . . 7 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
48	2, 47	iswwlks 29781	. . . . . 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 1344	. . . . 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 412	. . . 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 412	. . 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 240	. 2 ⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 → (𝑃 ≠ ∅ → 𝑃 ∈ (WWalks‘𝐺))))
53	1, 52	mpdi 45	1 ⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 → 𝑃 ∈ (WWalks‘𝐺)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∅c0 4284  {cpr 4579   class class class wbr 5092  dom cdm 5619  ran crn 5620  Fun wfun 6476   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349  ℂcc 11007  0cc0 11009  1c1 11010   + caddc 11012   − cmin 11347  ℕ₀cn0 12384  ...cfz 13410  ..^cfzo 13557  ♯chash 14237  Word cword 14420  Vtxcvtx 28941  iEdgciedg 28942  Edgcedg 28992  UHGraphcuhgr 29001  UPGraphcupgr 29025  Walkscwlks 29542  WWalkscwwlks 29770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-er 8625  df-map 8755  df-pm 8756  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-dju 9797  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-n0 12385  df-xnn0 12458  df-z 12472  df-uz 12736  df-fz 13411  df-fzo 13558  df-hash 14238  df-word 14421  df-edg 28993  df-uhgr 29003  df-upgr 29027  df-wlks 29545  df-wwlks 29775

This theorem is referenced by:  wlklnwwlkln1  29813  wlkiswwlks  29821  wlkiswwlkupgr  29823  elwspths2spth  29912