wwlksnextproplem2 - Mathbox for Alexander van der Vekens

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Theorem wwlksnextproplem2 41116

Description: Lemma 2 for wwlkextprop 26272. (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 2610	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		wwlksnextprop.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
3	1, 2	wwlknp 41045	. . . 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 12411	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ (0..^(𝑁 + 1)))
5	4	adantl 481	. . . . . . . . . 10 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ (0..^(𝑁 + 1)))
6		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑁 → (𝑊‘𝑖) = (𝑊‘𝑁))
7		oveq1 6556	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑁 → (𝑖 + 1) = (𝑁 + 1))
8	7	fveq2d 6107	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑁 → (𝑊‘(𝑖 + 1)) = (𝑊‘(𝑁 + 1)))
9	6, 8	preq12d 4220	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑁 → {(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} = {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))})
10	9	eleq1d 2672	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑁 → ({(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ↔ {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ 𝐸))
11	10	rspcv 3278	. . . . . . . . . 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 786	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → 𝑊 ∈ Word (Vtx‘𝐺))
15		1zzd 11285	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → 1 ∈ ℤ)
16		lencl 13179	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (#‘𝑊) ∈ ℕ₀)
17	16	nn0zd 11356	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (#‘𝑊) ∈ ℤ)
18	17	ad2antrr 758	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (#‘𝑊) ∈ ℤ)
19		peano2nn0 11210	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
20	19	nn0zd 11356	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℤ)
21	20	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (𝑁 + 1) ∈ ℤ)
22	15, 18, 21	3jca 1235	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (1 ∈ ℤ ∧ (#‘𝑊) ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ))
23		nn0ge0 11195	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ₀ → 0 ≤ 𝑁)
24		1red 9934	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → 1 ∈ ℝ)
25		nn0re 11178	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
26	24, 25	addge02d 10495	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ₀ → (0 ≤ 𝑁 ↔ 1 ≤ (𝑁 + 1)))
27	23, 26	mpbid 221	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ₀ → 1 ≤ (𝑁 + 1))
28	27	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → 1 ≤ (𝑁 + 1))
29	19	nn0red 11229	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℝ)
30	29	lep1d 10834	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ≤ ((𝑁 + 1) + 1))
31		breq2 4587	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝑊) = ((𝑁 + 1) + 1) → ((𝑁 + 1) ≤ (#‘𝑊) ↔ (𝑁 + 1) ≤ ((𝑁 + 1) + 1)))
32	30, 31	syl5ibrcom 236	. . . . . . . . . . . . . . . . . . . 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 84	. . . . . . . . . . . . . . . . . 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 553	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (1 ≤ (𝑁 + 1) ∧ (𝑁 + 1) ≤ (#‘𝑊)))
38		elfz2 12204	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 + 1) ∈ (1...(#‘𝑊)) ↔ ((1 ∈ ℤ ∧ (#‘𝑊) ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ) ∧ (1 ≤ (𝑁 + 1) ∧ (𝑁 + 1) ≤ (#‘𝑊))))
39	22, 37, 38	sylanbrc 695	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (𝑁 + 1) ∈ (1...(#‘𝑊)))
40	14, 39	jca 553	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(#‘𝑊))))
41		swrd0fvlsw 13295	. . . . . . . . . . . . 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 11179	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
44		1cnd 9935	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → 1 ∈ ℂ)
45	43, 44	pncand 10272	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 1) − 1) = 𝑁)
46	45	fveq2d 6107	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → (𝑊‘((𝑁 + 1) − 1)) = (𝑊‘𝑁))
47	46	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (𝑊‘((𝑁 + 1) − 1)) = (𝑊‘𝑁))
48	42, 47	eqtrd 2644	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → ( lastS ‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) = (𝑊‘𝑁))
49		lsw 13204	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → ( lastS ‘𝑊) = (𝑊‘((#‘𝑊) − 1)))
50	49	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → ( lastS ‘𝑊) = (𝑊‘((#‘𝑊) − 1)))
51		oveq1 6556	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑊) = ((𝑁 + 1) + 1) → ((#‘𝑊) − 1) = (((𝑁 + 1) + 1) − 1))
52	51	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ ((#‘𝑊) = ((𝑁 + 1) + 1) → (𝑊‘((#‘𝑊) − 1)) = (𝑊‘(((𝑁 + 1) + 1) − 1)))
53	52	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (𝑊‘((#‘𝑊) − 1)) = (𝑊‘(((𝑁 + 1) + 1) − 1)))
54	19	nn0cnd 11230	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℂ)
55	54, 44	pncand 10272	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ₀ → (((𝑁 + 1) + 1) − 1) = (𝑁 + 1))
56	55	fveq2d 6107	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → (𝑊‘(((𝑁 + 1) + 1) − 1)) = (𝑊‘(𝑁 + 1)))
57	53, 56	sylan9eq 2664	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (𝑊‘((#‘𝑊) − 1)) = (𝑊‘(𝑁 + 1)))
58	50, 57	eqtrd 2644	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → ( lastS ‘𝑊) = (𝑊‘(𝑁 + 1)))
59	48, 58	preq12d 4220	. . . . . . . . . 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 2672	. . . . . . . . 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 480	. . . . . . . 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 246	. . . . . . 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 628	. . . . . 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 84	. . . . 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 1253	. . . 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 2706	. 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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {cpr 4127 ⟨cop 4131 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 ≤ cle 9954 − cmin 10145 ℕ₀cn0 11169 ℤcz 11254 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 lastS clsw 13147 substr csubstr 13150 Vtxcvtx 25673 Edgcedga 25792 WWalkSN cwwlksn 41029

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-lsw 13155 df-substr 13158 df-wwlks 41033 df-wwlksn 41034

This theorem is referenced by: wwlksnextprop 41118

W3C validator