lighneallem4b - Mathbox for Alexander van der Vekens

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Theorem lighneallem4b 45061

Description: Lemma 2 for lighneallem4 45062. (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 12352	. . 3 ⊢ 2 ∈ ℤ
2	1	a1i 11	. 2 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 2 ∈ ℤ)
3		fzfid 13693	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → (0...(𝑀 − 1)) ∈ Fin)
4		neg1z 12356	. . . . . . 7 ⊢ -1 ∈ ℤ
5		elfznn0 13349	. . . . . . 7 ⊢ (𝑘 ∈ (0...(𝑀 − 1)) → 𝑘 ∈ ℕ₀)
6		zexpcl 13797	. . . . . . 7 ⊢ ((-1 ∈ ℤ ∧ 𝑘 ∈ ℕ₀) → (-1↑𝑘) ∈ ℤ)
7	4, 5, 6	sylancr 587	. . . . . 6 ⊢ (𝑘 ∈ (0...(𝑀 − 1)) → (-1↑𝑘) ∈ ℤ)
8	7	adantl 482	. . . . 5 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → (-1↑𝑘) ∈ ℤ)
9		eluzge2nn0 12627	. . . . . . . . 9 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ ℕ₀)
10	9	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → 𝐴 ∈ ℕ₀)
11	10	adantr 481	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → 𝐴 ∈ ℕ₀)
12	5	adantl 482	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → 𝑘 ∈ ℕ₀)
13	11, 12	nn0expcld 13961	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → (𝐴↑𝑘) ∈ ℕ₀)
14	13	nn0zd 12424	. . . . 5 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → (𝐴↑𝑘) ∈ ℤ)
15	8, 14	zmulcld 12432	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → ((-1↑𝑘) · (𝐴↑𝑘)) ∈ ℤ)
16	3, 15	fsumzcl 15447	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) ∈ ℤ)
17	16	3adant3 1131	. 2 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) ∈ ℤ)
18		simp1 1135	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 𝐴 ∈ (ℤ_≥‘2))
19		3z 12353	. . . . 5 ⊢ 3 ∈ ℤ
20	19	a1i 11	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 3 ∈ ℤ)
21		eluzelz 12592	. . . . 5 ⊢ (𝑀 ∈ (ℤ_≥‘2) → 𝑀 ∈ ℤ)
22	21	3ad2ant2 1133	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 𝑀 ∈ ℤ)
23		eluz2 12588	. . . . . . 7 ⊢ (𝑀 ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 2 ≤ 𝑀))
24		2re 12047	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
25	24	a1i 11	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 2 ∈ ℝ)
26		zre 12323	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
27	25, 26	leloed 11118	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → (2 ≤ 𝑀 ↔ (2 < 𝑀 ∨ 2 = 𝑀)))
28		zltp1le 12370	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (2 < 𝑀 ↔ (2 + 1) ≤ 𝑀))
29	1, 28	mpan 687	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℤ → (2 < 𝑀 ↔ (2 + 1) ≤ 𝑀))
30	29	biimpd 228	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℤ → (2 < 𝑀 → (2 + 1) ≤ 𝑀))
31		df-3 12037	. . . . . . . . . . . . . . . 16 ⊢ 3 = (2 + 1)
32	31	breq1i 5081	. . . . . . . . . . . . . . 15 ⊢ (3 ≤ 𝑀 ↔ (2 + 1) ≤ 𝑀)
33	30, 32	syl6ibr 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 16079	. . . . . . . . . . . . . . 15 ⊢ 2 ∥ 2
37		breq2 5078	. . . . . . . . . . . . . . 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 854	. . . . . . . . . . 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 407	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 2 ≤ 𝑀) → (¬ 2 ∥ 𝑀 → 3 ≤ 𝑀))
45	44	3adant1 1129	. . . . . . 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 407	. . . . 5 ⊢ ((𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 3 ≤ 𝑀)
48	47	3adant1 1129	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 3 ≤ 𝑀)
49		eluz2 12588	. . . 4 ⊢ (𝑀 ∈ (ℤ_≥‘3) ↔ (3 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 3 ≤ 𝑀))
50	20, 22, 48, 49	syl3anbrc 1342	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 𝑀 ∈ (ℤ_≥‘3))
51		eluzelcn 12594	. . . . . . . 8 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ ℂ)
52	51	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 𝐴 ∈ ℂ)
53		eluz2nn 12624	. . . . . . . 8 ⊢ (𝑀 ∈ (ℤ_≥‘2) → 𝑀 ∈ ℕ)
54	53	3ad2ant2 1133	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 𝑀 ∈ ℕ)
55		simp3 1137	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → ¬ 2 ∥ 𝑀)
56	52, 54, 55	oddpwp1fsum 16101	. . . . . 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 2744	. . . . 5 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → ((𝐴 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘))) = ((𝐴↑𝑀) + 1))
58		eluzge2nn0 12627	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (ℤ_≥‘2) → 𝑀 ∈ ℕ₀)
59	58	adantl 482	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → 𝑀 ∈ ℕ₀)
60	10, 59	nn0expcld 13961	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → (𝐴↑𝑀) ∈ ℕ₀)
61	60	nn0cnd 12295	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → (𝐴↑𝑀) ∈ ℂ)
62		peano2cn 11147	. . . . . . . 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 1131	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → ((𝐴↑𝑀) + 1) ∈ ℂ)
65	17	zcnd 12427	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) ∈ ℂ)
66		eluz2nn 12624	. . . . . . . . . 10 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ ℕ)
67	66	peano2nnd 11990	. . . . . . . . 9 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (𝐴 + 1) ∈ ℕ)
68	67	nncnd 11989	. . . . . . . 8 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (𝐴 + 1) ∈ ℂ)
69	67	nnne0d 12023	. . . . . . . 8 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (𝐴 + 1) ≠ 0)
70	68, 69	jca 512	. . . . . . 7 ⊢ (𝐴 ∈ (ℤ_≥‘2) → ((𝐴 + 1) ∈ ℂ ∧ (𝐴 + 1) ≠ 0))
71	70	3ad2ant1 1132	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → ((𝐴 + 1) ∈ ℂ ∧ (𝐴 + 1) ≠ 0))
72		divmul 11636	. . . . . 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 1370	. . . . 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 256	. . . 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 2744	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) = (((𝐴↑𝑀) + 1) / (𝐴 + 1)))
76		lighneallem4a 45060	. . 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 1370	. 2 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 2 ≤ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)))
78		eluz2 12588	. 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 1342	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 396 ∨ wo 844 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 < clt 11009 ≤ cle 11010 − cmin 11205 -cneg 11206 / cdiv 11632 ℕcn 11973 2c2 12028 3c3 12029 ℕ₀cn0 12233 ℤcz 12319 ℤ_≥cuz 12582 ...cfz 13239 ↑cexp 13782 Σcsu 15397 ∥ cdvds 15963

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-sum 15398 df-dvds 15964

This theorem is referenced by: lighneallem4 45062

W3C validator