lighneallem4b - Mathbox for Alexander van der Vekens

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Theorem lighneallem4b 47998

Description: Lemma 2 for lighneallem4 47999. (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 12537	. . 3 ⊢ 2 ∈ ℤ
2	1	a1i 11	. 2 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 2 ∈ ℤ)
3		fzfid 13910	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → (0...(𝑀 − 1)) ∈ Fin)
4		neg1z 12541	. . . . . . 7 ⊢ -1 ∈ ℤ
5		elfznn0 13550	. . . . . . 7 ⊢ (𝑘 ∈ (0...(𝑀 − 1)) → 𝑘 ∈ ℕ₀)
6		zexpcl 14013	. . . . . . 7 ⊢ ((-1 ∈ ℤ ∧ 𝑘 ∈ ℕ₀) → (-1↑𝑘) ∈ ℤ)
7	4, 5, 6	sylancr 588	. . . . . 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 12819	. . . . . . . . 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 14183	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → (𝐴↑𝑘) ∈ ℕ₀)
14	13	nn0zd 12527	. . . . 5 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → (𝐴↑𝑘) ∈ ℤ)
15	8, 14	zmulcld 12616	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → ((-1↑𝑘) · (𝐴↑𝑘)) ∈ ℤ)
16	3, 15	fsumzcl 15672	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) ∈ ℤ)
17	16	3adant3 1133	. 2 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) ∈ ℤ)
18		simp1 1137	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 𝐴 ∈ (ℤ_≥‘2))
19		3z 12538	. . . . 5 ⊢ 3 ∈ ℤ
20	19	a1i 11	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 3 ∈ ℤ)
21		eluzelz 12775	. . . . 5 ⊢ (𝑀 ∈ (ℤ_≥‘2) → 𝑀 ∈ ℤ)
22	21	3ad2ant2 1135	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 𝑀 ∈ ℤ)
23		eluz2 12771	. . . . . . 7 ⊢ (𝑀 ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 2 ≤ 𝑀))
24		2re 12233	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
25	24	a1i 11	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 2 ∈ ℝ)
26		zre 12506	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
27	25, 26	leloed 11290	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → (2 ≤ 𝑀 ↔ (2 < 𝑀 ∨ 2 = 𝑀)))
28		zltp1le 12555	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (2 < 𝑀 ↔ (2 + 1) ≤ 𝑀))
29	1, 28	mpan 691	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℤ → (2 < 𝑀 ↔ (2 + 1) ≤ 𝑀))
30	29	biimpd 229	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℤ → (2 < 𝑀 → (2 + 1) ≤ 𝑀))
31		df-3 12223	. . . . . . . . . . . . . . . 16 ⊢ 3 = (2 + 1)
32	31	breq1i 5107	. . . . . . . . . . . . . . 15 ⊢ (3 ≤ 𝑀 ↔ (2 + 1) ≤ 𝑀)
33	30, 32	imbitrrdi 252	. . . . . . . . . . . . . 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 16311	. . . . . . . . . . . . . . 15 ⊢ 2 ∥ 2
37		breq2 5104	. . . . . . . . . . . . . . 15 ⊢ (2 = 𝑀 → (2 ∥ 2 ↔ 2 ∥ 𝑀))
38	36, 37	mpbii 233	. . . . . . . . . . . . . 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 858	. . . . . . . . . . 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 240	. . . . . . . . 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 1131	. . . . . . 7 ⊢ ((2 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 2 ≤ 𝑀) → (¬ 2 ∥ 𝑀 → 3 ≤ 𝑀))
46	23, 45	sylbi 217	. . . . . 6 ⊢ (𝑀 ∈ (ℤ_≥‘2) → (¬ 2 ∥ 𝑀 → 3 ≤ 𝑀))
47	46	imp 406	. . . . 5 ⊢ ((𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 3 ≤ 𝑀)
48	47	3adant1 1131	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 3 ≤ 𝑀)
49		eluz2 12771	. . . 4 ⊢ (𝑀 ∈ (ℤ_≥‘3) ↔ (3 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 3 ≤ 𝑀))
50	20, 22, 48, 49	syl3anbrc 1345	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 𝑀 ∈ (ℤ_≥‘3))
51		eluzelcn 12777	. . . . . . . 8 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ ℂ)
52	51	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 𝐴 ∈ ℂ)
53		eluz2nn 12815	. . . . . . . 8 ⊢ (𝑀 ∈ (ℤ_≥‘2) → 𝑀 ∈ ℕ)
54	53	3ad2ant2 1135	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 𝑀 ∈ ℕ)
55		simp3 1139	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → ¬ 2 ∥ 𝑀)
56	52, 54, 55	oddpwp1fsum 16333	. . . . . 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 2743	. . . . 5 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → ((𝐴 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘))) = ((𝐴↑𝑀) + 1))
58		eluzge2nn0 12819	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (ℤ_≥‘2) → 𝑀 ∈ ℕ₀)
59	58	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → 𝑀 ∈ ℕ₀)
60	10, 59	nn0expcld 14183	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → (𝐴↑𝑀) ∈ ℕ₀)
61	60	nn0cnd 12478	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → (𝐴↑𝑀) ∈ ℂ)
62		peano2cn 11319	. . . . . . . 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 1133	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → ((𝐴↑𝑀) + 1) ∈ ℂ)
65	17	zcnd 12611	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) ∈ ℂ)
66		eluz2nn 12815	. . . . . . . . . 10 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ ℕ)
67	66	peano2nnd 12176	. . . . . . . . 9 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (𝐴 + 1) ∈ ℕ)
68	67	nncnd 12175	. . . . . . . 8 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (𝐴 + 1) ∈ ℂ)
69	67	nnne0d 12209	. . . . . . . 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 1134	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → ((𝐴 + 1) ∈ ℂ ∧ (𝐴 + 1) ≠ 0))
72		divmul 11813	. . . . . 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 1374	. . . . 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 2743	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) = (((𝐴↑𝑀) + 1) / (𝐴 + 1)))
76		lighneallem4a 47997	. . 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 1374	. 2 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 2 ≤ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)))
78		eluz2 12771	. 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 1345	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 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 class class class wbr 5100 ‘cfv 6502 (class class class)co 7370 ℂcc 11038 ℝcr 11039 0cc0 11040 1c1 11041 + caddc 11043 · cmul 11045 < clt 11180 ≤ cle 11181 − cmin 11378 -cneg 11379 / cdiv 11808 ℕcn 12159 2c2 12214 3c3 12215 ℕ₀cn0 12415 ℤcz 12502 ℤ_≥cuz 12765 ...cfz 13437 ↑cexp 13998 Σcsu 15623 ∥ cdvds 16193

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-inf2 9564 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117 ax-pre-sup 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-se 5588 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-isom 6511 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-om 7821 df-1st 7945 df-2nd 7946 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-1o 8409 df-er 8647 df-en 8898 df-dom 8899 df-sdom 8900 df-fin 8901 df-sup 9359 df-oi 9429 df-card 9865 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186 df-sub 11380 df-neg 11381 df-div 11809 df-nn 12160 df-2 12222 df-3 12223 df-4 12224 df-n0 12416 df-z 12503 df-uz 12766 df-rp 12920 df-fz 13438 df-fzo 13585 df-seq 13939 df-exp 13999 df-hash 14268 df-cj 15036 df-re 15037 df-im 15038 df-sqrt 15172 df-abs 15173 df-clim 15425 df-sum 15624 df-dvds 16194

This theorem is referenced by: lighneallem4 47999

W3C validator