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Theorem wwlksnextproplem2 27028

Description: Lemma 2 for wwlksnextprop 27030. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 20-Apr-2021.)

**Hypotheses**
Ref	Expression
wwlksnextprop.x	⊢ 𝑋 = ((𝑁 + 1) WWalksN 𝐺)
wwlksnextprop.e	⊢ 𝐸 = (Edg‘𝐺)

**Assertion**
Ref	Expression
wwlksnextproplem2	⊢ ((𝑊 ∈ 𝑋 ∧ 𝑁 ∈ ℕ₀) → {(lastS‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)), (lastS‘𝑊)} ∈ 𝐸)

Proof of Theorem wwlksnextproplem2 Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2760	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		wwlksnextprop.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
3	1, 2	wwlknp 26946	. . . 4 ⊢ (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))
4		fzonn0p1 12739	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ (0..^(𝑁 + 1)))
5	4	adantl 473	. . . . . . . . . 10 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ (0..^(𝑁 + 1)))
6		fveq2 6352	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑁 → (𝑊‘𝑖) = (𝑊‘𝑁))
7		oveq1 6820	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑁 → (𝑖 + 1) = (𝑁 + 1))
8	7	fveq2d 6356	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑁 → (𝑊‘(𝑖 + 1)) = (𝑊‘(𝑁 + 1)))
9	6, 8	preq12d 4420	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑁 → {(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} = {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))})
10	9	eleq1d 2824	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑁 → ({(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ↔ {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ 𝐸))
11	10	rspcv 3445	. . . . . . . . . 10 ⊢ (𝑁 ∈ (0..^(𝑁 + 1)) → (∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 → {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ 𝐸))
12	5, 11	syl 17	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 → {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ 𝐸))
13	12	imp 444	. . . . . . . 8 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ 𝐸)
14		simpll 807	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → 𝑊 ∈ Word (Vtx‘𝐺))
15		1zzd 11600	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → 1 ∈ ℤ)
16		lencl 13510	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (♯‘𝑊) ∈ ℕ₀)
17	16	nn0zd 11672	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (♯‘𝑊) ∈ ℤ)
18	17	ad2antrr 764	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (♯‘𝑊) ∈ ℤ)
19		peano2nn0 11525	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
20	19	nn0zd 11672	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℤ)
21	20	adantl 473	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (𝑁 + 1) ∈ ℤ)
22	15, 18, 21	3jca 1123	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (1 ∈ ℤ ∧ (♯‘𝑊) ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ))
23		nn0ge0 11510	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ₀ → 0 ≤ 𝑁)
24		1red 10247	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → 1 ∈ ℝ)
25		nn0re 11493	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
26	24, 25	addge02d 10808	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ₀ → (0 ≤ 𝑁 ↔ 1 ≤ (𝑁 + 1)))
27	23, 26	mpbid 222	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ₀ → 1 ≤ (𝑁 + 1))
28	27	adantl 473	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → 1 ≤ (𝑁 + 1))
29	19	nn0red 11544	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℝ)
30	29	lep1d 11147	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ≤ ((𝑁 + 1) + 1))
31		breq2 4808	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑊) = ((𝑁 + 1) + 1) → ((𝑁 + 1) ≤ (♯‘𝑊) ↔ (𝑁 + 1) ≤ ((𝑁 + 1) + 1)))
32	30, 31	syl5ibrcom 237	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ → ((♯‘𝑊) = ((𝑁 + 1) + 1) → (𝑁 + 1) ≤ (♯‘𝑊)))
33	32	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝑊) ∈ ℕ₀ → (𝑁 ∈ ℕ₀ → ((♯‘𝑊) = ((𝑁 + 1) + 1) → (𝑁 + 1) ≤ (♯‘𝑊))))
34	33	com23 86	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝑊) ∈ ℕ₀ → ((♯‘𝑊) = ((𝑁 + 1) + 1) → (𝑁 ∈ ℕ₀ → (𝑁 + 1) ≤ (♯‘𝑊))))
35	16, 34	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → ((♯‘𝑊) = ((𝑁 + 1) + 1) → (𝑁 ∈ ℕ₀ → (𝑁 + 1) ≤ (♯‘𝑊))))
36	35	imp31 447	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (𝑁 + 1) ≤ (♯‘𝑊))
37	28, 36	jca 555	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (1 ≤ (𝑁 + 1) ∧ (𝑁 + 1) ≤ (♯‘𝑊)))
38		elfz2 12526	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 + 1) ∈ (1...(♯‘𝑊)) ↔ ((1 ∈ ℤ ∧ (♯‘𝑊) ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ) ∧ (1 ≤ (𝑁 + 1) ∧ (𝑁 + 1) ≤ (♯‘𝑊))))
39	22, 37, 38	sylanbrc 701	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (𝑁 + 1) ∈ (1...(♯‘𝑊)))
40	14, 39	jca 555	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊))))
41		swrd0fvlsw 13643	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊))) → (lastS‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) = (𝑊‘((𝑁 + 1) − 1)))
42	40, 41	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (lastS‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) = (𝑊‘((𝑁 + 1) − 1)))
43		nn0cn 11494	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
44		1cnd 10248	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → 1 ∈ ℂ)
45	43, 44	pncand 10585	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 1) − 1) = 𝑁)
46	45	fveq2d 6356	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → (𝑊‘((𝑁 + 1) − 1)) = (𝑊‘𝑁))
47	46	adantl 473	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (𝑊‘((𝑁 + 1) − 1)) = (𝑊‘𝑁))
48	42, 47	eqtrd 2794	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (lastS‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) = (𝑊‘𝑁))
49		lsw 13538	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1)))
50	49	ad2antrr 764	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1)))
51		oveq1 6820	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑊) = ((𝑁 + 1) + 1) → ((♯‘𝑊) − 1) = (((𝑁 + 1) + 1) − 1))
52	51	fveq2d 6356	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑊) = ((𝑁 + 1) + 1) → (𝑊‘((♯‘𝑊) − 1)) = (𝑊‘(((𝑁 + 1) + 1) − 1)))
53	52	adantl 473	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → (𝑊‘((♯‘𝑊) − 1)) = (𝑊‘(((𝑁 + 1) + 1) − 1)))
54	19	nn0cnd 11545	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℂ)
55	54, 44	pncand 10585	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ₀ → (((𝑁 + 1) + 1) − 1) = (𝑁 + 1))
56	55	fveq2d 6356	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → (𝑊‘(((𝑁 + 1) + 1) − 1)) = (𝑊‘(𝑁 + 1)))
57	53, 56	sylan9eq 2814	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (𝑊‘((♯‘𝑊) − 1)) = (𝑊‘(𝑁 + 1)))
58	50, 57	eqtrd 2794	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (lastS‘𝑊) = (𝑊‘(𝑁 + 1)))
59	48, 58	preq12d 4420	. . . . . . . . . 10 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → {(lastS‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)), (lastS‘𝑊)} = {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))})
60	59	eleq1d 2824	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → ({(lastS‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)), (lastS‘𝑊)} ∈ 𝐸 ↔ {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ 𝐸))
61	60	adantr 472	. . . . . . . 8 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → ({(lastS‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)), (lastS‘𝑊)} ∈ 𝐸 ↔ {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ 𝐸))
62	13, 61	mpbird 247	. . . . . . 7 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → {(lastS‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)), (lastS‘𝑊)} ∈ 𝐸)
63	62	exp31 631	. . . . . 6 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 ∈ ℕ₀ → (∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 → {(lastS‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)), (lastS‘𝑊)} ∈ 𝐸)))
64	63	com23 86	. . . . 5 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → (∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 → (𝑁 ∈ ℕ₀ → {(lastS‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)), (lastS‘𝑊)} ∈ 𝐸)))
65	64	3impia 1110	. . . 4 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → (𝑁 ∈ ℕ₀ → {(lastS‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)), (lastS‘𝑊)} ∈ 𝐸))
66	3, 65	syl 17	. . 3 ⊢ (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑁 ∈ ℕ₀ → {(lastS‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)), (lastS‘𝑊)} ∈ 𝐸))
67		wwlksnextprop.x	. . 3 ⊢ 𝑋 = ((𝑁 + 1) WWalksN 𝐺)
68	66, 67	eleq2s 2857	. 2 ⊢ (𝑊 ∈ 𝑋 → (𝑁 ∈ ℕ₀ → {(lastS‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)), (lastS‘𝑊)} ∈ 𝐸))
69	68	imp 444	1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝑁 ∈ ℕ₀) → {(lastS‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)), (lastS‘𝑊)} ∈ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ∀wral 3050 {cpr 4323 ⟨cop 4327 class class class wbr 4804 ‘cfv 6049 (class class class)co 6813 0cc0 10128 1c1 10129 + caddc 10131 ≤ cle 10267 − cmin 10458 ℕ₀cn0 11484 ℤcz 11569 ...cfz 12519 ..^cfzo 12659 ♯chash 13311 Word cword 13477 lastSclsw 13478 substr csubstr 13481 Vtxcvtx 26073 Edgcedg 26138 WWalksN cwwlksn 26929

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-map 8025 df-pm 8026 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-card 8955 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-n0 11485 df-z 11570 df-uz 11880 df-fz 12520 df-fzo 12660 df-hash 13312 df-word 13485 df-lsw 13486 df-substr 13489 df-wwlks 26933 df-wwlksn 26934

This theorem is referenced by: wwlksnextprop 27030

W3C validator