lighneallem4b - Mathbox for Alexander van der Vekens

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Theorem lighneallem4b 46872

Description: Lemma 2 for lighneallem4 46873. (Contributed by AV, 16-Aug-2021.)

**Assertion**
Ref	Expression
lighneallem4b	⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) ∈ (ℤ_≥‘2))

Distinct variable groups: 𝐴,𝑘 𝑘,𝑀

**Proof of Theorem lighneallem4b**
Step	Hyp	Ref	Expression
1		2z 12616	. . 3 ⊢ 2 ∈ ℤ
2	1	a1i 11	. 2 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 2 ∈ ℤ)
3		fzfid 13962	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → (0...(𝑀 − 1)) ∈ Fin)
4		neg1z 12620	. . . . . . 7 ⊢ -1 ∈ ℤ
5		elfznn0 13618	. . . . . . 7 ⊢ (𝑘 ∈ (0...(𝑀 − 1)) → 𝑘 ∈ ℕ₀)
6		zexpcl 14065	. . . . . . 7 ⊢ ((-1 ∈ ℤ ∧ 𝑘 ∈ ℕ₀) → (-1↑𝑘) ∈ ℤ)
7	4, 5, 6	sylancr 586	. . . . . 6 ⊢ (𝑘 ∈ (0...(𝑀 − 1)) → (-1↑𝑘) ∈ ℤ)
8	7	adantl 481	. . . . 5 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → (-1↑𝑘) ∈ ℤ)
9		eluzge2nn0 12893	. . . . . . . . 9 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ ℕ₀)
10	9	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → 𝐴 ∈ ℕ₀)
11	10	adantr 480	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → 𝐴 ∈ ℕ₀)
12	5	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → 𝑘 ∈ ℕ₀)
13	11, 12	nn0expcld 14232	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → (𝐴↑𝑘) ∈ ℕ₀)
14	13	nn0zd 12606	. . . . 5 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → (𝐴↑𝑘) ∈ ℤ)
15	8, 14	zmulcld 12694	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → ((-1↑𝑘) · (𝐴↑𝑘)) ∈ ℤ)
16	3, 15	fsumzcl 15705	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) ∈ ℤ)
17	16	3adant3 1130	. 2 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) ∈ ℤ)
18		simp1 1134	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 𝐴 ∈ (ℤ_≥‘2))
19		3z 12617	. . . . 5 ⊢ 3 ∈ ℤ
20	19	a1i 11	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 3 ∈ ℤ)
21		eluzelz 12854	. . . . 5 ⊢ (𝑀 ∈ (ℤ_≥‘2) → 𝑀 ∈ ℤ)
22	21	3ad2ant2 1132	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 𝑀 ∈ ℤ)
23		eluz2 12850	. . . . . . 7 ⊢ (𝑀 ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 2 ≤ 𝑀))
24		2re 12308	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
25	24	a1i 11	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 2 ∈ ℝ)
26		zre 12584	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
27	25, 26	leloed 11379	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → (2 ≤ 𝑀 ↔ (2 < 𝑀 ∨ 2 = 𝑀)))
28		zltp1le 12634	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (2 < 𝑀 ↔ (2 + 1) ≤ 𝑀))
29	1, 28	mpan 689	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℤ → (2 < 𝑀 ↔ (2 + 1) ≤ 𝑀))
30	29	biimpd 228	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℤ → (2 < 𝑀 → (2 + 1) ≤ 𝑀))
31		df-3 12298	. . . . . . . . . . . . . . . 16 ⊢ 3 = (2 + 1)
32	31	breq1i 5149	. . . . . . . . . . . . . . 15 ⊢ (3 ≤ 𝑀 ↔ (2 + 1) ≤ 𝑀)
33	30, 32	imbitrrdi 251	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℤ → (2 < 𝑀 → 3 ≤ 𝑀))
34	33	a1i 11	. . . . . . . . . . . . 13 ⊢ (¬ 2 ∥ 𝑀 → (𝑀 ∈ ℤ → (2 < 𝑀 → 3 ≤ 𝑀)))
35	34	com13 88	. . . . . . . . . . . 12 ⊢ (2 < 𝑀 → (𝑀 ∈ ℤ → (¬ 2 ∥ 𝑀 → 3 ≤ 𝑀)))
36		z2even 16338	. . . . . . . . . . . . . . 15 ⊢ 2 ∥ 2
37		breq2 5146	. . . . . . . . . . . . . . 15 ⊢ (2 = 𝑀 → (2 ∥ 2 ↔ 2 ∥ 𝑀))
38	36, 37	mpbii 232	. . . . . . . . . . . . . 14 ⊢ (2 = 𝑀 → 2 ∥ 𝑀)
39	38	pm2.24d 151	. . . . . . . . . . . . 13 ⊢ (2 = 𝑀 → (¬ 2 ∥ 𝑀 → 3 ≤ 𝑀))
40	39	a1d 25	. . . . . . . . . . . 12 ⊢ (2 = 𝑀 → (𝑀 ∈ ℤ → (¬ 2 ∥ 𝑀 → 3 ≤ 𝑀)))
41	35, 40	jaoi 856	. . . . . . . . . . 11 ⊢ ((2 < 𝑀 ∨ 2 = 𝑀) → (𝑀 ∈ ℤ → (¬ 2 ∥ 𝑀 → 3 ≤ 𝑀)))
42	41	com12 32	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → ((2 < 𝑀 ∨ 2 = 𝑀) → (¬ 2 ∥ 𝑀 → 3 ≤ 𝑀)))
43	27, 42	sylbid 239	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (2 ≤ 𝑀 → (¬ 2 ∥ 𝑀 → 3 ≤ 𝑀)))
44	43	imp 406	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 2 ≤ 𝑀) → (¬ 2 ∥ 𝑀 → 3 ≤ 𝑀))
45	44	3adant1 1128	. . . . . . 7 ⊢ ((2 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 2 ≤ 𝑀) → (¬ 2 ∥ 𝑀 → 3 ≤ 𝑀))
46	23, 45	sylbi 216	. . . . . 6 ⊢ (𝑀 ∈ (ℤ_≥‘2) → (¬ 2 ∥ 𝑀 → 3 ≤ 𝑀))
47	46	imp 406	. . . . 5 ⊢ ((𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 3 ≤ 𝑀)
48	47	3adant1 1128	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 3 ≤ 𝑀)
49		eluz2 12850	. . . 4 ⊢ (𝑀 ∈ (ℤ_≥‘3) ↔ (3 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 3 ≤ 𝑀))
50	20, 22, 48, 49	syl3anbrc 1341	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 𝑀 ∈ (ℤ_≥‘3))
51		eluzelcn 12856	. . . . . . . 8 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ ℂ)
52	51	3ad2ant1 1131	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 𝐴 ∈ ℂ)
53		eluz2nn 12890	. . . . . . . 8 ⊢ (𝑀 ∈ (ℤ_≥‘2) → 𝑀 ∈ ℕ)
54	53	3ad2ant2 1132	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 𝑀 ∈ ℕ)
55		simp3 1136	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → ¬ 2 ∥ 𝑀)
56	52, 54, 55	oddpwp1fsum 16360	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → ((𝐴↑𝑀) + 1) = ((𝐴 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘))))
57	56	eqcomd 2733	. . . . 5 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → ((𝐴 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘))) = ((𝐴↑𝑀) + 1))
58		eluzge2nn0 12893	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (ℤ_≥‘2) → 𝑀 ∈ ℕ₀)
59	58	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → 𝑀 ∈ ℕ₀)
60	10, 59	nn0expcld 14232	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → (𝐴↑𝑀) ∈ ℕ₀)
61	60	nn0cnd 12556	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → (𝐴↑𝑀) ∈ ℂ)
62		peano2cn 11408	. . . . . . . 8 ⊢ ((𝐴↑𝑀) ∈ ℂ → ((𝐴↑𝑀) + 1) ∈ ℂ)
63	61, 62	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → ((𝐴↑𝑀) + 1) ∈ ℂ)
64	63	3adant3 1130	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → ((𝐴↑𝑀) + 1) ∈ ℂ)
65	17	zcnd 12689	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) ∈ ℂ)
66		eluz2nn 12890	. . . . . . . . . 10 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ ℕ)
67	66	peano2nnd 12251	. . . . . . . . 9 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (𝐴 + 1) ∈ ℕ)
68	67	nncnd 12250	. . . . . . . 8 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (𝐴 + 1) ∈ ℂ)
69	67	nnne0d 12284	. . . . . . . 8 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (𝐴 + 1) ≠ 0)
70	68, 69	jca 511	. . . . . . 7 ⊢ (𝐴 ∈ (ℤ_≥‘2) → ((𝐴 + 1) ∈ ℂ ∧ (𝐴 + 1) ≠ 0))
71	70	3ad2ant1 1131	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → ((𝐴 + 1) ∈ ℂ ∧ (𝐴 + 1) ≠ 0))
72		divmul 11897	. . . . . 6 ⊢ ((((𝐴↑𝑀) + 1) ∈ ℂ ∧ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) ∈ ℂ ∧ ((𝐴 + 1) ∈ ℂ ∧ (𝐴 + 1) ≠ 0)) → ((((𝐴↑𝑀) + 1) / (𝐴 + 1)) = Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) ↔ ((𝐴 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘))) = ((𝐴↑𝑀) + 1)))
73	64, 65, 71, 72	syl3anc 1369	. . . . 5 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → ((((𝐴↑𝑀) + 1) / (𝐴 + 1)) = Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) ↔ ((𝐴 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘))) = ((𝐴↑𝑀) + 1)))
74	57, 73	mpbird 257	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → (((𝐴↑𝑀) + 1) / (𝐴 + 1)) = Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)))
75	74	eqcomd 2733	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) = (((𝐴↑𝑀) + 1) / (𝐴 + 1)))
76		lighneallem4a 46871	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘3) ∧ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) = (((𝐴↑𝑀) + 1) / (𝐴 + 1))) → 2 ≤ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)))
77	18, 50, 75, 76	syl3anc 1369	. 2 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 2 ≤ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)))
78		eluz2 12850	. 2 ⊢ (Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) ∈ ℤ ∧ 2 ≤ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘))))
79	2, 17, 77, 78	syl3anbrc 1341	1 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) ∈ (ℤ_≥‘2))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 846 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 class class class wbr 5142 ‘cfv 6542 (class class class)co 7414 ℂcc 11128 ℝcr 11129 0cc0 11130 1c1 11131 + caddc 11133 · cmul 11135 < clt 11270 ≤ cle 11271 − cmin 11466 -cneg 11467 / cdiv 11893 ℕcn 12234 2c2 12289 3c3 12290 ℕ₀cn0 12494 ℤcz 12580 ℤ_≥cuz 12844 ...cfz 13508 ↑cexp 14050 Σcsu 15656 ∥ cdvds 16222

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9656 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9457 df-oi 9525 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-n0 12495 df-z 12581 df-uz 12845 df-rp 12999 df-fz 13509 df-fzo 13652 df-seq 13991 df-exp 14051 df-hash 14314 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-clim 15456 df-sum 15657 df-dvds 16223

This theorem is referenced by: lighneallem4 46873

W3C validator