Theorem 2sqlem10 15212

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 4032	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑏 ∥ 𝑎 ↔ 𝐵 ∥ 𝑎))
2		eleq1 2256	. . . . . 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 2494	. . . 4 ⊢ (𝑏 = 𝐵 → (∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑎 ∈ 𝑌 (𝐵 ∥ 𝑎 → 𝐵 ∈ 𝑆)))
5		oveq2 5926	. . . . . 6 ⊢ (𝑚 = 1 → (1...𝑚) = (1...1))
6	5	raleqdv 2696	. . . . 5 ⊢ (𝑚 = 1 → (∀𝑏 ∈ (1...𝑚)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑏 ∈ (1...1)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)))
7		oveq2 5926	. . . . . 6 ⊢ (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
8	7	raleqdv 2696	. . . . 5 ⊢ (𝑚 = 𝑛 → (∀𝑏 ∈ (1...𝑚)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)))
9		oveq2 5926	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (1...𝑚) = (1...(𝑛 + 1)))
10	9	raleqdv 2696	. . . . 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 5926	. . . . . 6 ⊢ (𝑚 = 𝐵 → (1...𝑚) = (1...𝐵))
12	11	raleqdv 2696	. . . . 5 ⊢ (𝑚 = 𝐵 → (∀𝑏 ∈ (1...𝑚)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑏 ∈ (1...𝐵)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)))
13		elfz1eq 10101	. . . . . . . . 9 ⊢ (𝑏 ∈ (1...1) → 𝑏 = 1)
14		1z 9343	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
15		zgz 12511	. . . . . . . . . . . 12 ⊢ (1 ∈ ℤ → 1 ∈ ℤ[i])
16	14, 15	ax-mp 5	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ[i]
17		sq1 10704	. . . . . . . . . . . 12 ⊢ (1↑2) = 1
18	17	eqcomi 2197	. . . . . . . . . . 11 ⊢ 1 = (1↑2)
19		fveq2 5554	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → (abs‘𝑥) = (abs‘1))
20		abs1 11216	. . . . . . . . . . . . . 14 ⊢ (abs‘1) = 1
21	19, 20	eqtrdi 2242	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (abs‘𝑥) = 1)
22	21	oveq1d 5933	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → ((abs‘𝑥)↑2) = (1↑2))
23	22	rspceeqv 2882	. . . . . . . . . . 11 ⊢ ((1 ∈ ℤ[i] ∧ 1 = (1↑2)) → ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2))
24	16, 18, 23	mp2an 426	. . . . . . . . . 10 ⊢ ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2)
25		2sq.1	. . . . . . . . . . 11 ⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))
26	25	2sqlem1 15201	. . . . . . . . . 10 ⊢ (1 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2))
27	24, 26	mpbir 146	. . . . . . . . 9 ⊢ 1 ∈ 𝑆
28	13, 27	eqeltrdi 2284	. . . . . . . 8 ⊢ (𝑏 ∈ (1...1) → 𝑏 ∈ 𝑆)
29	28	a1d 22	. . . . . . 7 ⊢ (𝑏 ∈ (1...1) → (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆))
30	29	ralrimivw 2568	. . . . . 6 ⊢ (𝑏 ∈ (1...1) → ∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆))
31	30	rgen 2547	. . . . 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 528	. . . . . . . . . . . . . 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 8990	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
35	34	ad2antrr 488	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) ∧ (𝑚 ∈ 𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → 𝑛 ∈ ℂ)
36		ax-1cn 7965	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
37		pncan 8225	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
38	35, 36, 37	sylancl 413	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) ∧ (𝑚 ∈ 𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → ((𝑛 + 1) − 1) = 𝑛)
39	38	oveq2d 5934	. . . . . . . . . . . . . . 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 2696	. . . . . . . . . . . . . 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 531	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) ∧ (𝑚 ∈ 𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → (𝑛 + 1) ∥ 𝑚)
43		peano2nn 8994	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
44	43	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) ∧ (𝑚 ∈ 𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → (𝑛 + 1) ∈ ℕ)
45		simprl 529	. . . . . . . . . . . . 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 15211	. . . . . . . . . . . 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 2567	. . . . . . . . . 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 4033	. . . . . . . . . . 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 2730	. . . . . . . . 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 4032	. . . . . . . . . . . 12 ⊢ (𝑏 = (𝑛 + 1) → (𝑏 ∥ 𝑎 ↔ (𝑛 + 1) ∥ 𝑎))
55		eleq1 2256	. . . . . . . . . . . 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 2494	. . . . . . . . . 10 ⊢ (𝑏 = (𝑛 + 1) → (∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑎 ∈ 𝑌 ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆)))
58	57	ralsng 3658	. . . . . . . . 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 9629	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ≥‘1))
63		fzsuc 10135	. . . . . . . . 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 2696	. . . . . . 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 3340	. . . . . . 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 8998	. . . 4 ⊢ (𝐵 ∈ ℕ → ∀𝑏 ∈ (1...𝐵)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆))
70		elfz1end 10121	. . . . 5 ⊢ (𝐵 ∈ ℕ ↔ 𝐵 ∈ (1...𝐵))
71	70	biimpi 120	. . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ (1...𝐵))
72	4, 69, 71	rspcdva 2869	. . 3 ⊢ (𝐵 ∈ ℕ → ∀𝑎 ∈ 𝑌 (𝐵 ∥ 𝑎 → 𝐵 ∈ 𝑆))
73		breq2 4033	. . . . 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 2860	. . 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 1195	1 ⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴) → 𝐵 ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  {cab 2179  ∀wral 2472  ∃wrex 2473   ∪ cun 3151  {csn 3618   class class class wbr 4029   ↦ cmpt 4090  ran crn 4660  ‘cfv 5254  (class class class)co 5918  ℂcc 7870  1c1 7873   + caddc 7875   − cmin 8190  ℕcn 8982  2c2 9033  ℤcz 9317  ℤ_≥cuz 9592  ...cfz 10074  ↑cexp 10609  abscabs 11141   ∥ cdvds 11930   gcd cgcd 12079  ℤ[i]cgz 12507

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-1o 6469  df-2o 6470  df-er 6587  df-en 6795  df-sup 7043  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-dvds 11931  df-gcd 12080  df-prm 12246  df-gz 12508

This theorem is referenced by: (None)