Theorem 2sqlem10 15944

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 4096	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑏 ∥ 𝑎 ↔ 𝐵 ∥ 𝑎))
2		eleq1 2294	. . . . . 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 2533	. . . 4 ⊢ (𝑏 = 𝐵 → (∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑎 ∈ 𝑌 (𝐵 ∥ 𝑎 → 𝐵 ∈ 𝑆)))
5		oveq2 6036	. . . . . 6 ⊢ (𝑚 = 1 → (1...𝑚) = (1...1))
6	5	raleqdv 2737	. . . . 5 ⊢ (𝑚 = 1 → (∀𝑏 ∈ (1...𝑚)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑏 ∈ (1...1)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)))
7		oveq2 6036	. . . . . 6 ⊢ (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
8	7	raleqdv 2737	. . . . 5 ⊢ (𝑚 = 𝑛 → (∀𝑏 ∈ (1...𝑚)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)))
9		oveq2 6036	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (1...𝑚) = (1...(𝑛 + 1)))
10	9	raleqdv 2737	. . . . 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 6036	. . . . . 6 ⊢ (𝑚 = 𝐵 → (1...𝑚) = (1...𝐵))
12	11	raleqdv 2737	. . . . 5 ⊢ (𝑚 = 𝐵 → (∀𝑏 ∈ (1...𝑚)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑏 ∈ (1...𝐵)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)))
13		elfz1eq 10332	. . . . . . . . 9 ⊢ (𝑏 ∈ (1...1) → 𝑏 = 1)
14		1z 9566	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
15		zgz 13026	. . . . . . . . . . . 12 ⊢ (1 ∈ ℤ → 1 ∈ ℤ[i])
16	14, 15	ax-mp 5	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ[i]
17		sq1 10958	. . . . . . . . . . . 12 ⊢ (1↑2) = 1
18	17	eqcomi 2235	. . . . . . . . . . 11 ⊢ 1 = (1↑2)
19		fveq2 5648	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → (abs‘𝑥) = (abs‘1))
20		abs1 11712	. . . . . . . . . . . . . 14 ⊢ (abs‘1) = 1
21	19, 20	eqtrdi 2280	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (abs‘𝑥) = 1)
22	21	oveq1d 6043	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → ((abs‘𝑥)↑2) = (1↑2))
23	22	rspceeqv 2929	. . . . . . . . . . 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 15933	. . . . . . . . . 10 ⊢ (1 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2))
27	24, 26	mpbir 146	. . . . . . . . 9 ⊢ 1 ∈ 𝑆
28	13, 27	eqeltrdi 2322	. . . . . . . 8 ⊢ (𝑏 ∈ (1...1) → 𝑏 ∈ 𝑆)
29	28	a1d 22	. . . . . . 7 ⊢ (𝑏 ∈ (1...1) → (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆))
30	29	ralrimivw 2607	. . . . . 6 ⊢ (𝑏 ∈ (1...1) → ∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆))
31	30	rgen 2586	. . . . 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 529	. . . . . . . . . . . . . 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 9210	. . . . . . . . . . . . . . . . . 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 8185	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
37		pncan 8444	. . . . . . . . . . . . . . . . 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 6044	. . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . 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 533	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) ∧ (𝑚 ∈ 𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → (𝑛 + 1) ∥ 𝑚)
43		peano2nn 9214	. . . . . . . . . . . . . 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 531	. . . . . . . . . . . . 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 15943	. . . . . . . . . . . 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 2606	. . . . . . . . . 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 4097	. . . . . . . . . . 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 2772	. . . . . . . . 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 4096	. . . . . . . . . . . 12 ⊢ (𝑏 = (𝑛 + 1) → (𝑏 ∥ 𝑎 ↔ (𝑛 + 1) ∥ 𝑎))
55		eleq1 2294	. . . . . . . . . . . 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 2533	. . . . . . . . . 10 ⊢ (𝑏 = (𝑛 + 1) → (∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑎 ∈ 𝑌 ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆)))
58	57	ralsng 3713	. . . . . . . . 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 9854	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ≥‘1))
63		fzsuc 10366	. . . . . . . . 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 2737	. . . . . . 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 3390	. . . . . . 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 9218	. . . 4 ⊢ (𝐵 ∈ ℕ → ∀𝑏 ∈ (1...𝐵)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆))
70		elfz1end 10352	. . . . 5 ⊢ (𝐵 ∈ ℕ ↔ 𝐵 ∈ (1...𝐵))
71	70	biimpi 120	. . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ (1...𝐵))
72	4, 69, 71	rspcdva 2916	. . 3 ⊢ (𝐵 ∈ ℕ → ∀𝑎 ∈ 𝑌 (𝐵 ∥ 𝑎 → 𝐵 ∈ 𝑆))
73		breq2 4097	. . . . 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 2907	. . 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 1220	1 ⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴) → 𝐵 ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2202  {cab 2217  ∀wral 2511  ∃wrex 2512   ∪ cun 3199  {csn 3673   class class class wbr 4093   ↦ cmpt 4155  ran crn 4732  ‘cfv 5333  (class class class)co 6028  ℂcc 8090  1c1 8093   + caddc 8095   − cmin 8409  ℕcn 9202  2c2 9253  ℤcz 9540  ℤ_≥cuz 9816  ...cfz 10305  ↑cexp 10863  abscabs 11637   ∥ cdvds 12428   gcd cgcd 12604  ℤ[i]cgz 13022

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-2o 6626  df-er 6745  df-en 6953  df-sup 7243  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-q 9915  df-rp 9950  df-fz 10306  df-fzo 10440  df-fl 10593  df-mod 10648  df-seqfrec 10773  df-exp 10864  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-dvds 12429  df-gcd 12605  df-prm 12760  df-gz 13023

This theorem is referenced by: (None)