Theorem wlkiswwlks1 29896

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 29653	. 2 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃 ≠ ∅)
2		eqid 2734	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2734	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4	2, 3	upgriswlk 29673	. . 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 14575	. . . . . . . 8 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → 𝑃 ∈ Word (Vtx‘𝐺))
7	6	3ad2ant2 1133	. . . . . . 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 29133	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
10	3	uhgrfun 29097	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
11		funfn 6597	. . . . . . . . . . . . . . . . . . 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 14561	. . . . . . . . . . . . . . . . 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 7099	. . . . . . . . . . . . . . . 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 29080	. . . . . . . . . . . . . . 15 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
20	18, 19	eleqtrrdi 2849	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph) ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹‘𝑖)) ∈ (Edg‘𝐺))
21		eleq1 2826	. . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . 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 3164	. . . . . . . . . . . 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 1116	. . . . . . . . 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 14567	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ Word dom (iEdg‘𝐺) → (♯‘𝐹) ∈ ℕ0)
30		ffz0hash 14482	. . . . . . . . . . . . . . . 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 7437	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑃) = ((♯‘𝐹) + 1) → ((♯‘𝑃) − 1) = (((♯‘𝐹) + 1) − 1))
33		nn0cn 12533	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐹) ∈ ℕ0 → (♯‘𝐹) ∈ ℂ)
34		pncan1 11684	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐹) ∈ ℂ → (((♯‘𝐹) + 1) − 1) = (♯‘𝐹))
35	33, 34	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝐹) ∈ ℕ0 → (((♯‘𝐹) + 1) − 1) = (♯‘𝐹))
36	32, 35	sylan9eqr 2796	. . . . . . . . . . . . . . . 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 7446	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (0..^((♯‘𝑃) − 1)) = (0..^(♯‘𝐹)))
42	41	raleqdv 3323	. . . . . . . . . 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 1131	. . . . . . . . 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 2734	. . . . . . 7 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
48	2, 47	iswwlks 29865	. . . . . 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 1342	. . . . 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 1536   ∈ wcel 2105   ≠ wne 2937  ∀wral 3058  ∅c0 4338  {cpr 4632   class class class wbr 5147  dom cdm 5688  ran crn 5689  Fun wfun 6556   Fn wfn 6557  ⟶wf 6558  ‘cfv 6562  (class class class)co 7430  ℂcc 11150  0cc0 11152  1c1 11153   + caddc 11155   − cmin 11489  ℕ₀cn0 12523  ...cfz 13543  ..^cfzo 13690  ♯chash 14365  Word cword 14548  Vtxcvtx 29027  iEdgciedg 29028  Edgcedg 29078  UHGraphcuhgr 29087  UPGraphcupgr 29111  Walkscwlks 29628  WWalkscwwlks 29854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229

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 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-n0 12524  df-xnn0 12597  df-z 12611  df-uz 12876  df-fz 13544  df-fzo 13691  df-hash 14366  df-word 14549  df-edg 29079  df-uhgr 29089  df-upgr 29113  df-wlks 29631  df-wwlks 29859

This theorem is referenced by:  wlklnwwlkln1  29897  wlkiswwlks  29905  wlkiswwlkupgr  29907  elwspths2spth  29996