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Theorem clwwlkextfrlem1 26603

Description: Lemma for numclwwlk2lem1 26629. (Contributed by Alexander van der Vekens, 3-Oct-2018.)

**Assertion**
Ref	Expression
clwwlkextfrlem1	⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → (((𝑊 ++ ⟨“𝑍”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁) ≠ 𝑋))

**Proof of Theorem clwwlkextfrlem1**
Step	Hyp	Ref	Expression
1		wwlknimpb 26232	. . . 4 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)))
2		simprll 798	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → 𝑊 ∈ Word 𝑉)
3		s1cl 13235	. . . . . . . . . . 11 ⊢ (𝑍 ∈ 𝑉 → ⟨“𝑍”⟩ ∈ Word 𝑉)
4	3	3ad2ant3 1077	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) → ⟨“𝑍”⟩ ∈ Word 𝑉)
5	4	adantr 480	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → ⟨“𝑍”⟩ ∈ Word 𝑉)
6		nnnn0 11176	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
7		nn0p1gt0 11199	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → 0 < (𝑁 + 1))
8	6, 7	syl 17	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 0 < (𝑁 + 1))
9	8	3ad2ant2 1076	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) → 0 < (𝑁 + 1))
10	9	adantr 480	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → 0 < (𝑁 + 1))
11		breq2 4587	. . . . . . . . . . . 12 ⊢ ((#‘𝑊) = (𝑁 + 1) → (0 < (#‘𝑊) ↔ 0 < (𝑁 + 1)))
12	11	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → (0 < (#‘𝑊) ↔ 0 < (𝑁 + 1)))
13	12	ad2antrl 760	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → (0 < (#‘𝑊) ↔ 0 < (𝑁 + 1)))
14	10, 13	mpbird 246	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → 0 < (#‘𝑊))
15		ccatfv0 13220	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑍”⟩ ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → ((𝑊 ++ ⟨“𝑍”⟩)‘0) = (𝑊‘0))
16	2, 5, 14, 15	syl3anc 1318	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → ((𝑊 ++ ⟨“𝑍”⟩)‘0) = (𝑊‘0))
17		simprl 790	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → (𝑊‘0) = 𝑋)
18	17	adantl 481	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → (𝑊‘0) = 𝑋)
19	16, 18	eqtrd 2644	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → ((𝑊 ++ ⟨“𝑍”⟩)‘0) = 𝑋)
20		oveq1 6556	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝑊) = (𝑁 + 1) → ((#‘𝑊) − 1) = ((𝑁 + 1) − 1))
21	20	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → ((#‘𝑊) − 1) = ((𝑁 + 1) − 1))
22	21	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1))) → ((#‘𝑊) − 1) = ((𝑁 + 1) − 1))
23		nncn 10905	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
24		pncan1 10333	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) − 1) = 𝑁)
25	23, 24	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) − 1) = 𝑁)
26	25	3ad2ant2 1076	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) → ((𝑁 + 1) − 1) = 𝑁)
27	26	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1))) → ((𝑁 + 1) − 1) = 𝑁)
28	22, 27	eqtr2d 2645	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1))) → 𝑁 = ((#‘𝑊) − 1))
29	28	fveq2d 6107	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1))) → ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁) = ((𝑊 ++ ⟨“𝑍”⟩)‘((#‘𝑊) − 1)))
30		simprl 790	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1))) → 𝑊 ∈ Word 𝑉)
31	4	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1))) → ⟨“𝑍”⟩ ∈ Word 𝑉)
32	9	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1))) → 0 < (𝑁 + 1))
33	12	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1))) → (0 < (#‘𝑊) ↔ 0 < (𝑁 + 1)))
34	32, 33	mpbird 246	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1))) → 0 < (#‘𝑊))
35		hashneq0 13016	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑊 ∈ Word 𝑉 → (0 < (#‘𝑊) ↔ 𝑊 ≠ ∅))
36	35	ad2antrl 760	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1))) → (0 < (#‘𝑊) ↔ 𝑊 ≠ ∅))
37	34, 36	mpbid 221	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1))) → 𝑊 ≠ ∅)
38		ccatval1lsw 13221	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑍”⟩ ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → ((𝑊 ++ ⟨“𝑍”⟩)‘((#‘𝑊) − 1)) = ( lastS ‘𝑊))
39	30, 31, 37, 38	syl3anc 1318	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1))) → ((𝑊 ++ ⟨“𝑍”⟩)‘((#‘𝑊) − 1)) = ( lastS ‘𝑊))
40	29, 39	eqtr2d 2645	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1))) → ( lastS ‘𝑊) = ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁))
41	40	neeq1d 2841	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1))) → (( lastS ‘𝑊) ≠ 𝑋 ↔ ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁) ≠ 𝑋))
42	41	biimpd 218	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1))) → (( lastS ‘𝑊) ≠ 𝑋 → ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁) ≠ 𝑋))
43	42	ex 449	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → (( lastS ‘𝑊) ≠ 𝑋 → ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁) ≠ 𝑋)))
44	43	com13 86	. . . . . . . . . 10 ⊢ (( lastS ‘𝑊) ≠ 𝑋 → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) → ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁) ≠ 𝑋)))
45	44	adantl 481	. . . . . . . . 9 ⊢ (((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) → ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁) ≠ 𝑋)))
46	45	com13 86	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → (((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋) → ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁) ≠ 𝑋)))
47	46	imp32 448	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁) ≠ 𝑋)
48	19, 47	jca 553	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → (((𝑊 ++ ⟨“𝑍”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁) ≠ 𝑋))
49	48	expcom 450	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) → (((𝑊 ++ ⟨“𝑍”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁) ≠ 𝑋)))
50	49	exp32 629	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → ((𝑊‘0) = 𝑋 → (( lastS ‘𝑊) ≠ 𝑋 → ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) → (((𝑊 ++ ⟨“𝑍”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁) ≠ 𝑋)))))
51	1, 50	syl 17	. . 3 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑊‘0) = 𝑋 → (( lastS ‘𝑊) ≠ 𝑋 → ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) → (((𝑊 ++ ⟨“𝑍”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁) ≠ 𝑋)))))
52	51	3imp 1249	. 2 ⊢ ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋) → ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) → (((𝑊 ++ ⟨“𝑍”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁) ≠ 𝑋)))
53	52	impcom 445	1 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → (((𝑊 ++ ⟨“𝑍”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁) ≠ 𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 − cmin 10145 ℕcn 10897 ℕ₀cn0 11169 #chash 12979 Word cword 13146 lastS clsw 13147 ++ cconcat 13148 ⟨“cs1 13149 WWalksN cwwlkn 26206

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-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-lsw 13155 df-concat 13156 df-s1 13157 df-wwlk 26207 df-wwlkn 26208

This theorem is referenced by: numclwwlk2lem1 26629

W3C validator