Theorem 2sqlem10 16158

Description: Lemma for 2sq . Every factor of a "proper" sum of two squares (where the summands are coprime) is a sum of two squares. (Contributed by Mario Carneiro, 19-Jun-2015.)

Hypotheses

Ref	Expression
2sq.1	⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))
2sqlem7.2	⊢ 𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))}

Assertion

Ref	Expression
2sqlem10	⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴) → 𝐵 ∈ 𝑆)

Distinct variable groups:   𝑥,𝑤,𝑦,𝑧   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦   𝑥,𝑆,𝑦,𝑧   𝑥,𝑌,𝑦

Allowed substitution hints:   𝐴(𝑤)   𝐵(𝑧,𝑤)   𝑆(𝑤)   𝑌(𝑧,𝑤)

Proof of Theorem 2sqlem10

Dummy variables 𝑎 𝑏 𝑛 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 4128	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑏 ∥ 𝑎 ↔ 𝐵 ∥ 𝑎))
2		eleq1 2301	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑏 ∈ 𝑆 ↔ 𝐵 ∈ 𝑆))
3	1, 2	imbi12d 234	. . . . 5 ⊢ (𝑏 = 𝐵 → ((𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ (𝐵 ∥ 𝑎 → 𝐵 ∈ 𝑆)))
4	3	ralbidv 2550	. . . 4 ⊢ (𝑏 = 𝐵 → (∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑎 ∈ 𝑌 (𝐵 ∥ 𝑎 → 𝐵 ∈ 𝑆)))
5		oveq2 6083	. . . . . 6 ⊢ (𝑚 = 1 → (1...𝑚) = (1...1))
6	5	raleqdv 2755	. . . . 5 ⊢ (𝑚 = 1 → (∀𝑏 ∈ (1...𝑚)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑏 ∈ (1...1)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)))
7		oveq2 6083	. . . . . 6 ⊢ (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
8	7	raleqdv 2755	. . . . 5 ⊢ (𝑚 = 𝑛 → (∀𝑏 ∈ (1...𝑚)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)))
9		oveq2 6083	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (1...𝑚) = (1...(𝑛 + 1)))
10	9	raleqdv 2755	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (∀𝑏 ∈ (1...𝑚)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑏 ∈ (1...(𝑛 + 1))∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)))
11		oveq2 6083	. . . . . 6 ⊢ (𝑚 = 𝐵 → (1...𝑚) = (1...𝐵))
12	11	raleqdv 2755	. . . . 5 ⊢ (𝑚 = 𝐵 → (∀𝑏 ∈ (1...𝑚)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑏 ∈ (1...𝐵)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)))
13		elfz1eq 10418	. . . . . . . . 9 ⊢ (𝑏 ∈ (1...1) → 𝑏 = 1)
14		1z 9649	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
15		zgz 13130	. . . . . . . . . . . 12 ⊢ (1 ∈ ℤ → 1 ∈ ℤ[i])
16	14, 15	ax-mp 5	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ[i]
17		sq1 11048	. . . . . . . . . . . 12 ⊢ (1↑2) = 1
18	17	eqcomi 2242	. . . . . . . . . . 11 ⊢ 1 = (1↑2)
19		fveq2 5690	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → (abs‘𝑥) = (abs‘1))
20		abs1 11816	. . . . . . . . . . . . . 14 ⊢ (abs‘1) = 1
21	19, 20	eqtrdi 2287	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (abs‘𝑥) = 1)
22	21	oveq1d 6090	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → ((abs‘𝑥)↑2) = (1↑2))
23	22	rspceeqv 2948	. . . . . . . . . . 11 ⊢ ((1 ∈ ℤ[i] ∧ 1 = (1↑2)) → ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2))
24	16, 18, 23	mp2an 430	. . . . . . . . . 10 ⊢ ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2)
25		2sq.1	. . . . . . . . . . 11 ⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))
26	25	2sqlem1 16147	. . . . . . . . . 10 ⊢ (1 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2))
27	24, 26	mpbir 146	. . . . . . . . 9 ⊢ 1 ∈ 𝑆
28	13, 27	eqeltrdi 2329	. . . . . . . 8 ⊢ (𝑏 ∈ (1...1) → 𝑏 ∈ 𝑆)
29	28	a1d 22	. . . . . . 7 ⊢ (𝑏 ∈ (1...1) → (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆))
30	29	ralrimivw 2624	. . . . . 6 ⊢ (𝑏 ∈ (1...1) → ∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆))
31	30	rgen 2603	. . . . 5 ⊢ ∀𝑏 ∈ (1...1)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)
32		2sqlem7.2	. . . . . . . . . . . . 13 ⊢ 𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))}
33		simplr 533	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) ∧ (𝑚 ∈ 𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆))
34		nncn 9291	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
35	34	ad2antrr 492	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) ∧ (𝑚 ∈ 𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → 𝑛 ∈ ℂ)
36		ax-1cn 8262	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
37		pncan 8522	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
38	35, 36, 37	sylancl 417	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) ∧ (𝑚 ∈ 𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → ((𝑛 + 1) − 1) = 𝑛)
39	38	oveq2d 6091	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) ∧ (𝑚 ∈ 𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → (1...((𝑛 + 1) − 1)) = (1...𝑛))
40	39	raleqdv 2755	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) ∧ (𝑚 ∈ 𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → (∀𝑏 ∈ (1...((𝑛 + 1) − 1))∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)))
41	33, 40	mpbird 167	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) ∧ (𝑚 ∈ 𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → ∀𝑏 ∈ (1...((𝑛 + 1) − 1))∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆))
42		simprr 537	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) ∧ (𝑚 ∈ 𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → (𝑛 + 1) ∥ 𝑚)
43		peano2nn 9295	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
44	43	ad2antrr 492	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) ∧ (𝑚 ∈ 𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → (𝑛 + 1) ∈ ℕ)
45		simprl 535	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) ∧ (𝑚 ∈ 𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → 𝑚 ∈ 𝑌)
46	25, 32, 41, 42, 44, 45	2sqlem9 16157	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) ∧ (𝑚 ∈ 𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → (𝑛 + 1) ∈ 𝑆)
47	46	expr 375	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) ∧ 𝑚 ∈ 𝑌) → ((𝑛 + 1) ∥ 𝑚 → (𝑛 + 1) ∈ 𝑆))
48	47	ralrimiva 2623	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) → ∀𝑚 ∈ 𝑌 ((𝑛 + 1) ∥ 𝑚 → (𝑛 + 1) ∈ 𝑆))
49	48	ex 115	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) → ∀𝑚 ∈ 𝑌 ((𝑛 + 1) ∥ 𝑚 → (𝑛 + 1) ∈ 𝑆)))
50		breq2 4129	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑚 → ((𝑛 + 1) ∥ 𝑎 ↔ (𝑛 + 1) ∥ 𝑚))
51	50	imbi1d 231	. . . . . . . . . 10 ⊢ (𝑎 = 𝑚 → (((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆) ↔ ((𝑛 + 1) ∥ 𝑚 → (𝑛 + 1) ∈ 𝑆)))
52	51	cbvralvw 2790	. . . . . . . . 9 ⊢ (∀𝑎 ∈ 𝑌 ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆) ↔ ∀𝑚 ∈ 𝑌 ((𝑛 + 1) ∥ 𝑚 → (𝑛 + 1) ∈ 𝑆))
53	49, 52	imbitrrdi 162	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) → ∀𝑎 ∈ 𝑌 ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆)))
54		breq1 4128	. . . . . . . . . . . 12 ⊢ (𝑏 = (𝑛 + 1) → (𝑏 ∥ 𝑎 ↔ (𝑛 + 1) ∥ 𝑎))
55		eleq1 2301	. . . . . . . . . . . 12 ⊢ (𝑏 = (𝑛 + 1) → (𝑏 ∈ 𝑆 ↔ (𝑛 + 1) ∈ 𝑆))
56	54, 55	imbi12d 234	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑛 + 1) → ((𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆)))
57	56	ralbidv 2550	. . . . . . . . . 10 ⊢ (𝑏 = (𝑛 + 1) → (∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑎 ∈ 𝑌 ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆)))
58	57	ralsng 3745	. . . . . . . . 9 ⊢ ((𝑛 + 1) ∈ ℕ → (∀𝑏 ∈ {(𝑛 + 1)}∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑎 ∈ 𝑌 ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆)))
59	43, 58	syl 14	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (∀𝑏 ∈ {(𝑛 + 1)}∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑎 ∈ 𝑌 ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆)))
60	53, 59	sylibrd 169	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) → ∀𝑏 ∈ {(𝑛 + 1)}∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)))
61	60	ancld 325	. . . . . 6 ⊢ (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) → (∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ∧ ∀𝑏 ∈ {(𝑛 + 1)}∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆))))
62		elnnuz 9938	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ≥‘1))
63		fzsuc 10453	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ≥‘1) → (1...(𝑛 + 1)) = ((1...𝑛) ∪ {(𝑛 + 1)}))
64	62, 63	sylbi 121	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (1...(𝑛 + 1)) = ((1...𝑛) ∪ {(𝑛 + 1)}))
65	64	raleqdv 2755	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...(𝑛 + 1))∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑏 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)))
66		ralunb 3410	. . . . . . 7 ⊢ (∀𝑏 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ (∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ∧ ∀𝑏 ∈ {(𝑛 + 1)}∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)))
67	65, 66	bitrdi 196	. . . . . 6 ⊢ (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...(𝑛 + 1))∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ (∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ∧ ∀𝑏 ∈ {(𝑛 + 1)}∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆))))
68	61, 67	sylibrd 169	. . . . 5 ⊢ (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) → ∀𝑏 ∈ (1...(𝑛 + 1))∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)))
69	6, 8, 10, 12, 31, 68	nnind 9299	. . . 4 ⊢ (𝐵 ∈ ℕ → ∀𝑏 ∈ (1...𝐵)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆))
70		elfz1end 10439	. . . . 5 ⊢ (𝐵 ∈ ℕ ↔ 𝐵 ∈ (1...𝐵))
71	70	biimpi 120	. . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ (1...𝐵))
72	4, 69, 71	rspcdva 2934	. . 3 ⊢ (𝐵 ∈ ℕ → ∀𝑎 ∈ 𝑌 (𝐵 ∥ 𝑎 → 𝐵 ∈ 𝑆))
73		breq2 4129	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝐵 ∥ 𝑎 ↔ 𝐵 ∥ 𝐴))
74	73	imbi1d 231	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝐵 ∥ 𝑎 → 𝐵 ∈ 𝑆) ↔ (𝐵 ∥ 𝐴 → 𝐵 ∈ 𝑆)))
75	74	rspcv 2925	. . 3 ⊢ (𝐴 ∈ 𝑌 → (∀𝑎 ∈ 𝑌 (𝐵 ∥ 𝑎 → 𝐵 ∈ 𝑆) → (𝐵 ∥ 𝐴 → 𝐵 ∈ 𝑆)))
76	72, 75	syl5 32	. 2 ⊢ (𝐴 ∈ 𝑌 → (𝐵 ∈ ℕ → (𝐵 ∥ 𝐴 → 𝐵 ∈ 𝑆)))
77	76	3imp 1224	1 ⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴) → 𝐵 ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∃wrex 2529   ∪ cun 3218  {csn 3705   class class class wbr 4125   ↦ cmpt 4187  ran crn 4770  ‘cfv 5372  (class class class)co 6075  ℂcc 8167  1c1 8170   + caddc 8172   − cmin 8487  ℕcn 9283  2c2 9334  ℤcz 9623  ℤ_≥cuz 9900  ...cfz 10390  ↑cexp 10953  abscabs 11741   ∥ cdvds 12532   gcd cgcd 12708  ℤ[i]cgz 13126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-2o 6678  df-er 6797  df-en 7013  df-sup 7314  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-fz 10391  df-fzo 10528  df-fl 10683  df-mod 10738  df-seqfrec 10863  df-exp 10954  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-dvds 12533  df-gcd 12709  df-prm 12864  df-gz 13127

This theorem is referenced by: (None)