Theorem 2sqlem10 15804

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 4086	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑏 ∥ 𝑎 ↔ 𝐵 ∥ 𝑎))
2		eleq1 2292	. . . . . 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 2530	. . . 4 ⊢ (𝑏 = 𝐵 → (∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑎 ∈ 𝑌 (𝐵 ∥ 𝑎 → 𝐵 ∈ 𝑆)))
5		oveq2 6009	. . . . . 6 ⊢ (𝑚 = 1 → (1...𝑚) = (1...1))
6	5	raleqdv 2734	. . . . 5 ⊢ (𝑚 = 1 → (∀𝑏 ∈ (1...𝑚)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑏 ∈ (1...1)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)))
7		oveq2 6009	. . . . . 6 ⊢ (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
8	7	raleqdv 2734	. . . . 5 ⊢ (𝑚 = 𝑛 → (∀𝑏 ∈ (1...𝑚)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)))
9		oveq2 6009	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (1...𝑚) = (1...(𝑛 + 1)))
10	9	raleqdv 2734	. . . . 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 6009	. . . . . 6 ⊢ (𝑚 = 𝐵 → (1...𝑚) = (1...𝐵))
12	11	raleqdv 2734	. . . . 5 ⊢ (𝑚 = 𝐵 → (∀𝑏 ∈ (1...𝑚)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑏 ∈ (1...𝐵)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)))
13		elfz1eq 10231	. . . . . . . . 9 ⊢ (𝑏 ∈ (1...1) → 𝑏 = 1)
14		1z 9472	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
15		zgz 12896	. . . . . . . . . . . 12 ⊢ (1 ∈ ℤ → 1 ∈ ℤ[i])
16	14, 15	ax-mp 5	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ[i]
17		sq1 10855	. . . . . . . . . . . 12 ⊢ (1↑2) = 1
18	17	eqcomi 2233	. . . . . . . . . . 11 ⊢ 1 = (1↑2)
19		fveq2 5627	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → (abs‘𝑥) = (abs‘1))
20		abs1 11583	. . . . . . . . . . . . . 14 ⊢ (abs‘1) = 1
21	19, 20	eqtrdi 2278	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (abs‘𝑥) = 1)
22	21	oveq1d 6016	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → ((abs‘𝑥)↑2) = (1↑2))
23	22	rspceeqv 2925	. . . . . . . . . . 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 15793	. . . . . . . . . 10 ⊢ (1 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2))
27	24, 26	mpbir 146	. . . . . . . . 9 ⊢ 1 ∈ 𝑆
28	13, 27	eqeltrdi 2320	. . . . . . . 8 ⊢ (𝑏 ∈ (1...1) → 𝑏 ∈ 𝑆)
29	28	a1d 22	. . . . . . 7 ⊢ (𝑏 ∈ (1...1) → (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆))
30	29	ralrimivw 2604	. . . . . 6 ⊢ (𝑏 ∈ (1...1) → ∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆))
31	30	rgen 2583	. . . . 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 9118	. . . . . . . . . . . . . . . . . 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 8092	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
37		pncan 8352	. . . . . . . . . . . . . . . . 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 6017	. . . . . . . . . . . . . . 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 2734	. . . . . . . . . . . . . 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 9122	. . . . . . . . . . . . . 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 15803	. . . . . . . . . . . 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 2603	. . . . . . . . . 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 4087	. . . . . . . . . . 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 2769	. . . . . . . . 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 4086	. . . . . . . . . . . 12 ⊢ (𝑏 = (𝑛 + 1) → (𝑏 ∥ 𝑎 ↔ (𝑛 + 1) ∥ 𝑎))
55		eleq1 2292	. . . . . . . . . . . 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 2530	. . . . . . . . . 10 ⊢ (𝑏 = (𝑛 + 1) → (∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑎 ∈ 𝑌 ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆)))
58	57	ralsng 3706	. . . . . . . . 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 9759	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ≥‘1))
63		fzsuc 10265	. . . . . . . . 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 2734	. . . . . . 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 3385	. . . . . . 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 9126	. . . 4 ⊢ (𝐵 ∈ ℕ → ∀𝑏 ∈ (1...𝐵)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆))
70		elfz1end 10251	. . . . 5 ⊢ (𝐵 ∈ ℕ ↔ 𝐵 ∈ (1...𝐵))
71	70	biimpi 120	. . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ (1...𝐵))
72	4, 69, 71	rspcdva 2912	. . 3 ⊢ (𝐵 ∈ ℕ → ∀𝑎 ∈ 𝑌 (𝐵 ∥ 𝑎 → 𝐵 ∈ 𝑆))
73		breq2 4087	. . . . 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 2903	. . 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 1217	1 ⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴) → 𝐵 ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  {cab 2215  ∀wral 2508  ∃wrex 2509   ∪ cun 3195  {csn 3666   class class class wbr 4083   ↦ cmpt 4145  ran crn 4720  ‘cfv 5318  (class class class)co 6001  ℂcc 7997  1c1 8000   + caddc 8002   − cmin 8317  ℕcn 9110  2c2 9161  ℤcz 9446  ℤ_≥cuz 9722  ...cfz 10204  ↑cexp 10760  abscabs 11508   ∥ cdvds 12298   gcd cgcd 12474  ℤ[i]cgz 12892

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-1o 6562  df-2o 6563  df-er 6680  df-en 6888  df-sup 7151  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-fz 10205  df-fzo 10339  df-fl 10490  df-mod 10545  df-seqfrec 10670  df-exp 10761  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-dvds 12299  df-gcd 12475  df-prm 12630  df-gz 12893

This theorem is referenced by: (None)