lighneallem4b - Mathbox for Alexander van der Vekens

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Theorem lighneallem4b 47483

Description: Lemma 2 for lighneallem4 47484. (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 12675	. . 3 ⊢ 2 ∈ ℤ
2	1	a1i 11	. 2 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 2 ∈ ℤ)
3		fzfid 14024	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → (0...(𝑀 − 1)) ∈ Fin)
4		neg1z 12679	. . . . . . 7 ⊢ -1 ∈ ℤ
5		elfznn0 13677	. . . . . . 7 ⊢ (𝑘 ∈ (0...(𝑀 − 1)) → 𝑘 ∈ ℕ₀)
6		zexpcl 14127	. . . . . . 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 12952	. . . . . . . . 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 14295	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → (𝐴↑𝑘) ∈ ℕ₀)
14	13	nn0zd 12665	. . . . 5 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → (𝐴↑𝑘) ∈ ℤ)
15	8, 14	zmulcld 12753	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → ((-1↑𝑘) · (𝐴↑𝑘)) ∈ ℤ)
16	3, 15	fsumzcl 15783	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) ∈ ℤ)
17	16	3adant3 1132	. 2 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) ∈ ℤ)
18		simp1 1136	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 𝐴 ∈ (ℤ_≥‘2))
19		3z 12676	. . . . 5 ⊢ 3 ∈ ℤ
20	19	a1i 11	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 3 ∈ ℤ)
21		eluzelz 12913	. . . . 5 ⊢ (𝑀 ∈ (ℤ_≥‘2) → 𝑀 ∈ ℤ)
22	21	3ad2ant2 1134	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 𝑀 ∈ ℤ)
23		eluz2 12909	. . . . . . 7 ⊢ (𝑀 ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 2 ≤ 𝑀))
24		2re 12367	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
25	24	a1i 11	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 2 ∈ ℝ)
26		zre 12643	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
27	25, 26	leloed 11433	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → (2 ≤ 𝑀 ↔ (2 < 𝑀 ∨ 2 = 𝑀)))
28		zltp1le 12693	. . . . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℤ → (2 < 𝑀 → (2 + 1) ≤ 𝑀))
31		df-3 12357	. . . . . . . . . . . . . . . 16 ⊢ 3 = (2 + 1)
32	31	breq1i 5173	. . . . . . . . . . . . . . 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 16418	. . . . . . . . . . . . . . 15 ⊢ 2 ∥ 2
37		breq2 5170	. . . . . . . . . . . . . . 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 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 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 1130	. . . . . . 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 1130	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 3 ≤ 𝑀)
49		eluz2 12909	. . . 4 ⊢ (𝑀 ∈ (ℤ_≥‘3) ↔ (3 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 3 ≤ 𝑀))
50	20, 22, 48, 49	syl3anbrc 1343	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 𝑀 ∈ (ℤ_≥‘3))
51		eluzelcn 12915	. . . . . . . 8 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ ℂ)
52	51	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 𝐴 ∈ ℂ)
53		eluz2nn 12949	. . . . . . . 8 ⊢ (𝑀 ∈ (ℤ_≥‘2) → 𝑀 ∈ ℕ)
54	53	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 𝑀 ∈ ℕ)
55		simp3 1138	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → ¬ 2 ∥ 𝑀)
56	52, 54, 55	oddpwp1fsum 16440	. . . . . 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 2746	. . . . 5 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → ((𝐴 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘))) = ((𝐴↑𝑀) + 1))
58		eluzge2nn0 12952	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (ℤ_≥‘2) → 𝑀 ∈ ℕ₀)
59	58	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → 𝑀 ∈ ℕ₀)
60	10, 59	nn0expcld 14295	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → (𝐴↑𝑀) ∈ ℕ₀)
61	60	nn0cnd 12615	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → (𝐴↑𝑀) ∈ ℂ)
62		peano2cn 11462	. . . . . . . 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 1132	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → ((𝐴↑𝑀) + 1) ∈ ℂ)
65	17	zcnd 12748	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) ∈ ℂ)
66		eluz2nn 12949	. . . . . . . . . 10 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ ℕ)
67	66	peano2nnd 12310	. . . . . . . . 9 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (𝐴 + 1) ∈ ℕ)
68	67	nncnd 12309	. . . . . . . 8 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (𝐴 + 1) ∈ ℂ)
69	67	nnne0d 12343	. . . . . . . 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 1133	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → ((𝐴 + 1) ∈ ℂ ∧ (𝐴 + 1) ≠ 0))
72		divmul 11952	. . . . . 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 1371	. . . . 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 2746	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) = (((𝐴↑𝑀) + 1) / (𝐴 + 1)))
76		lighneallem4a 47482	. . 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 1371	. 2 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 2 ≤ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)))
78		eluz2 12909	. 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 1343	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 846 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 < clt 11324 ≤ cle 11325 − cmin 11520 -cneg 11521 / cdiv 11947 ℕcn 12293 2c2 12348 3c3 12349 ℕ₀cn0 12553 ℤcz 12639 ℤ_≥cuz 12903 ...cfz 13567 ↑cexp 14112 Σcsu 15734 ∥ cdvds 16302

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-fz 13568 df-fzo 13712 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-sum 15735 df-dvds 16303

This theorem is referenced by: lighneallem4 47484

W3C validator