lighneallem4b - Mathbox for Alexander van der Vekens

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Theorem lighneallem4b 41526

Description: Lemma 2 for lighneallem4 41527. (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 11409	. . 3 ⊢ 2 ∈ ℤ
2	1	a1i 11	. 2 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 2 ∈ ℤ)
3		fzfid 12772	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → (0...(𝑀 − 1)) ∈ Fin)
4		neg1z 11413	. . . . . . 7 ⊢ -1 ∈ ℤ
5		elfznn0 12433	. . . . . . 7 ⊢ (𝑘 ∈ (0...(𝑀 − 1)) → 𝑘 ∈ ℕ₀)
6		zexpcl 12875	. . . . . . 7 ⊢ ((-1 ∈ ℤ ∧ 𝑘 ∈ ℕ₀) → (-1↑𝑘) ∈ ℤ)
7	4, 5, 6	sylancr 695	. . . . . 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 11727	. . . . . . . . 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 13031	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → (𝐴↑𝑘) ∈ ℕ₀)
14	13	nn0zd 11480	. . . . 5 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → (𝐴↑𝑘) ∈ ℤ)
15	8, 14	zmulcld 11488	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) ∧ 𝑘 ∈ (0...(𝑀 − 1))) → ((-1↑𝑘) · (𝐴↑𝑘)) ∈ ℤ)
16	3, 15	fsumzcl 14466	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) ∈ ℤ)
17	16	3adant3 1081	. 2 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) ∈ ℤ)
18		simp1 1061	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 𝐴 ∈ (ℤ_≥‘2))
19		3z 11410	. . . . 5 ⊢ 3 ∈ ℤ
20	19	a1i 11	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 3 ∈ ℤ)
21		eluzelz 11697	. . . . 5 ⊢ (𝑀 ∈ (ℤ_≥‘2) → 𝑀 ∈ ℤ)
22	21	3ad2ant2 1083	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 𝑀 ∈ ℤ)
23		eluz2 11693	. . . . . . 7 ⊢ (𝑀 ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 2 ≤ 𝑀))
24		2re 11090	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
25	24	a1i 11	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 2 ∈ ℝ)
26		zre 11381	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
27	25, 26	leloed 10180	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → (2 ≤ 𝑀 ↔ (2 < 𝑀 ∨ 2 = 𝑀)))
28		zltp1le 11427	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (2 < 𝑀 ↔ (2 + 1) ≤ 𝑀))
29	1, 28	mpan 706	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℤ → (2 < 𝑀 ↔ (2 + 1) ≤ 𝑀))
30	29	biimpd 219	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℤ → (2 < 𝑀 → (2 + 1) ≤ 𝑀))
31		df-3 11080	. . . . . . . . . . . . . . . 16 ⊢ 3 = (2 + 1)
32	31	breq1i 4660	. . . . . . . . . . . . . . 15 ⊢ (3 ≤ 𝑀 ↔ (2 + 1) ≤ 𝑀)
33	30, 32	syl6ibr 242	. . . . . . . . . . . . . 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 15106	. . . . . . . . . . . . . . 15 ⊢ 2 ∥ 2
37		breq2 4657	. . . . . . . . . . . . . . 15 ⊢ (2 = 𝑀 → (2 ∥ 2 ↔ 2 ∥ 𝑀))
38	36, 37	mpbii 223	. . . . . . . . . . . . . 14 ⊢ (2 = 𝑀 → 2 ∥ 𝑀)
39	38	pm2.24d 147	. . . . . . . . . . . . 13 ⊢ (2 = 𝑀 → (¬ 2 ∥ 𝑀 → 3 ≤ 𝑀))
40	39	a1d 25	. . . . . . . . . . . 12 ⊢ (2 = 𝑀 → (𝑀 ∈ ℤ → (¬ 2 ∥ 𝑀 → 3 ≤ 𝑀)))
41	35, 40	jaoi 394	. . . . . . . . . . 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 230	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (2 ≤ 𝑀 → (¬ 2 ∥ 𝑀 → 3 ≤ 𝑀)))
44	43	imp 445	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 2 ≤ 𝑀) → (¬ 2 ∥ 𝑀 → 3 ≤ 𝑀))
45	44	3adant1 1079	. . . . . . 7 ⊢ ((2 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 2 ≤ 𝑀) → (¬ 2 ∥ 𝑀 → 3 ≤ 𝑀))
46	23, 45	sylbi 207	. . . . . 6 ⊢ (𝑀 ∈ (ℤ_≥‘2) → (¬ 2 ∥ 𝑀 → 3 ≤ 𝑀))
47	46	imp 445	. . . . 5 ⊢ ((𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 3 ≤ 𝑀)
48	47	3adant1 1079	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 3 ≤ 𝑀)
49		eluz2 11693	. . . 4 ⊢ (𝑀 ∈ (ℤ_≥‘3) ↔ (3 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 3 ≤ 𝑀))
50	20, 22, 48, 49	syl3anbrc 1246	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 𝑀 ∈ (ℤ_≥‘3))
51		eluzelcn 11699	. . . . . . . 8 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ ℂ)
52	51	3ad2ant1 1082	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 𝐴 ∈ ℂ)
53		eluz2nn 11726	. . . . . . . 8 ⊢ (𝑀 ∈ (ℤ_≥‘2) → 𝑀 ∈ ℕ)
54	53	3ad2ant2 1083	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 𝑀 ∈ ℕ)
55		simp3 1063	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → ¬ 2 ∥ 𝑀)
56	52, 54, 55	oddpwp1fsum 15115	. . . . . 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 2628	. . . . 5 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → ((𝐴 + 1) · Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘))) = ((𝐴↑𝑀) + 1))
58		eluzge2nn0 11727	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (ℤ_≥‘2) → 𝑀 ∈ ℕ₀)
59	58	adantl 482	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → 𝑀 ∈ ℕ₀)
60	10, 59	nn0expcld 13031	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → (𝐴↑𝑀) ∈ ℕ₀)
61	60	nn0cnd 11353	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2)) → (𝐴↑𝑀) ∈ ℂ)
62		peano2cn 10208	. . . . . . . 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 1081	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → ((𝐴↑𝑀) + 1) ∈ ℂ)
65	17	zcnd 11483	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) ∈ ℂ)
66		eluz2nn 11726	. . . . . . . . . 10 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ ℕ)
67	66	peano2nnd 11037	. . . . . . . . 9 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (𝐴 + 1) ∈ ℕ)
68	67	nncnd 11036	. . . . . . . 8 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (𝐴 + 1) ∈ ℂ)
69	67	nnne0d 11065	. . . . . . . 8 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (𝐴 + 1) ≠ 0)
70	68, 69	jca 554	. . . . . . 7 ⊢ (𝐴 ∈ (ℤ_≥‘2) → ((𝐴 + 1) ∈ ℂ ∧ (𝐴 + 1) ≠ 0))
71	70	3ad2ant1 1082	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → ((𝐴 + 1) ∈ ℂ ∧ (𝐴 + 1) ≠ 0))
72		divmul 10688	. . . . . 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 1326	. . . . 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 247	. . . 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 2628	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) = (((𝐴↑𝑀) + 1) / (𝐴 + 1)))
76		lighneallem4a 41525	. . 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 1326	. 2 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ (ℤ_≥‘2) ∧ ¬ 2 ∥ 𝑀) → 2 ≤ Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)))
78		eluz2 11693	. 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 1246	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 196 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 < clt 10074 ≤ cle 10075 − cmin 10266 -cneg 10267 / cdiv 10684 ℕcn 11020 2c2 11070 3c3 11071 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ...cfz 12326 ↑cexp 12860 Σcsu 14416 ∥ cdvds 14983

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-dvds 14984

This theorem is referenced by: lighneallem4 41527

W3C validator