Theorem coprimeprodsq 16342

Description: If three numbers are coprime, and the square of one is the product of the other two, then there is a formula for the other two in terms of gcd and square. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
coprimeprodsq	⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → ((𝐶↑2) = (𝐴 · 𝐵) → 𝐴 = ((𝐴 gcd 𝐶)↑2)))

Proof of Theorem coprimeprodsq

Step	Hyp	Ref	Expression
1		nn0z 12183	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ)
2		nn0z 12183	. . . . . . . 8 ⊢ (𝐶 ∈ ℕ0 → 𝐶 ∈ ℤ)
3		gcdcl 16046	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 gcd 𝐶) ∈ ℕ0)
4	1, 2, 3	syl2an 599	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝐴 gcd 𝐶) ∈ ℕ0)
5	4	3adant2 1133	. . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐴 gcd 𝐶) ∈ ℕ0)
6	5	3ad2ant1 1135	. . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐴 gcd 𝐶) ∈ ℕ0)
7	6	nn0cnd 12135	. . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐴 gcd 𝐶) ∈ ℂ)
8	7	sqvald 13696	. . 3 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → ((𝐴 gcd 𝐶)↑2) = ((𝐴 gcd 𝐶) · (𝐴 gcd 𝐶)))
9		simp13 1207	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐶 ∈ ℕ0)
10	9	nn0cnd 12135	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐶 ∈ ℂ)
11		nn0cn 12083	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℂ)
12	11	3ad2ant1 1135	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → 𝐴 ∈ ℂ)
13	12	3ad2ant1 1135	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐴 ∈ ℂ)
14	10, 13	mulcomd 10837	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐶 · 𝐴) = (𝐴 · 𝐶))
15		simpl3 1195	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → 𝐶 ∈ ℕ0)
16	15	nn0cnd 12135	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → 𝐶 ∈ ℂ)
17	16	sqvald 13696	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → (𝐶↑2) = (𝐶 · 𝐶))
18	17	eqeq1d 2736	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → ((𝐶↑2) = (𝐴 · 𝐵) ↔ (𝐶 · 𝐶) = (𝐴 · 𝐵)))
19	18	biimp3a 1471	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐶 · 𝐶) = (𝐴 · 𝐵))
20	14, 19	oveq12d 7220	. . . . . 6 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → ((𝐶 · 𝐴) gcd (𝐶 · 𝐶)) = ((𝐴 · 𝐶) gcd (𝐴 · 𝐵)))
21		simp11 1205	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐴 ∈ ℕ0)
22	21	nn0zd 12263	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐴 ∈ ℤ)
23	9	nn0zd 12263	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐶 ∈ ℤ)
24		mulgcd 16089	. . . . . . 7 ⊢ ((𝐶 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐶 · 𝐴) gcd (𝐶 · 𝐶)) = (𝐶 · (𝐴 gcd 𝐶)))
25	9, 22, 23, 24	syl3anc 1373	. . . . . 6 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → ((𝐶 · 𝐴) gcd (𝐶 · 𝐶)) = (𝐶 · (𝐴 gcd 𝐶)))
26		simp12 1206	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐵 ∈ ℤ)
27		mulgcd 16089	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐶 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 · 𝐶) gcd (𝐴 · 𝐵)) = (𝐴 · (𝐶 gcd 𝐵)))
28	21, 23, 26, 27	syl3anc 1373	. . . . . 6 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → ((𝐴 · 𝐶) gcd (𝐴 · 𝐵)) = (𝐴 · (𝐶 gcd 𝐵)))
29	20, 25, 28	3eqtr3d 2782	. . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐶 · (𝐴 gcd 𝐶)) = (𝐴 · (𝐶 gcd 𝐵)))
30	29	oveq2d 7218	. . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → ((𝐴 · (𝐴 gcd 𝐶)) gcd (𝐶 · (𝐴 gcd 𝐶))) = ((𝐴 · (𝐴 gcd 𝐶)) gcd (𝐴 · (𝐶 gcd 𝐵))))
31		mulgcdr 16091	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ (𝐴 gcd 𝐶) ∈ ℕ0) → ((𝐴 · (𝐴 gcd 𝐶)) gcd (𝐶 · (𝐴 gcd 𝐶))) = ((𝐴 gcd 𝐶) · (𝐴 gcd 𝐶)))
32	22, 23, 6, 31	syl3anc 1373	. . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → ((𝐴 · (𝐴 gcd 𝐶)) gcd (𝐶 · (𝐴 gcd 𝐶))) = ((𝐴 gcd 𝐶) · (𝐴 gcd 𝐶)))
33	6	nn0zd 12263	. . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐴 gcd 𝐶) ∈ ℤ)
34		gcdcl 16046	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐶 gcd 𝐵) ∈ ℕ0)
35	2, 34	sylan 583	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℕ0 ∧ 𝐵 ∈ ℤ) → (𝐶 gcd 𝐵) ∈ ℕ0)
36	35	ancoms 462	. . . . . . . 8 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐶 gcd 𝐵) ∈ ℕ0)
37	36	3adant1 1132	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐶 gcd 𝐵) ∈ ℕ0)
38	37	3ad2ant1 1135	. . . . . 6 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐶 gcd 𝐵) ∈ ℕ0)
39	38	nn0zd 12263	. . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐶 gcd 𝐵) ∈ ℤ)
40		mulgcd 16089	. . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝐴 gcd 𝐶) ∈ ℤ ∧ (𝐶 gcd 𝐵) ∈ ℤ) → ((𝐴 · (𝐴 gcd 𝐶)) gcd (𝐴 · (𝐶 gcd 𝐵))) = (𝐴 · ((𝐴 gcd 𝐶) gcd (𝐶 gcd 𝐵))))
41	21, 33, 39, 40	syl3anc 1373	. . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → ((𝐴 · (𝐴 gcd 𝐶)) gcd (𝐴 · (𝐶 gcd 𝐵))) = (𝐴 · ((𝐴 gcd 𝐶) gcd (𝐶 gcd 𝐵))))
42	30, 32, 41	3eqtr3d 2782	. . 3 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → ((𝐴 gcd 𝐶) · (𝐴 gcd 𝐶)) = (𝐴 · ((𝐴 gcd 𝐶) gcd (𝐶 gcd 𝐵))))
43	2	3ad2ant3 1137	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → 𝐶 ∈ ℤ)
44		gcdid 16067	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ ℤ → (𝐶 gcd 𝐶) = (abs‘𝐶))
45	43, 44	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐶 gcd 𝐶) = (abs‘𝐶))
46	45	oveq1d 7217	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((𝐶 gcd 𝐶) gcd 𝐵) = ((abs‘𝐶) gcd 𝐵))
47		simp2 1139	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → 𝐵 ∈ ℤ)
48		gcdabs1 16069	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((abs‘𝐶) gcd 𝐵) = (𝐶 gcd 𝐵))
49	43, 47, 48	syl2anc 587	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((abs‘𝐶) gcd 𝐵) = (𝐶 gcd 𝐵))
50	46, 49	eqtrd 2774	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((𝐶 gcd 𝐶) gcd 𝐵) = (𝐶 gcd 𝐵))
51		gcdass 16088	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐶 gcd 𝐶) gcd 𝐵) = (𝐶 gcd (𝐶 gcd 𝐵)))
52	43, 43, 47, 51	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((𝐶 gcd 𝐶) gcd 𝐵) = (𝐶 gcd (𝐶 gcd 𝐵)))
53	43, 47	gcdcomd 16054	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐶 gcd 𝐵) = (𝐵 gcd 𝐶))
54	50, 52, 53	3eqtr3d 2782	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐶 gcd (𝐶 gcd 𝐵)) = (𝐵 gcd 𝐶))
55	54	oveq2d 7218	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐴 gcd (𝐶 gcd (𝐶 gcd 𝐵))) = (𝐴 gcd (𝐵 gcd 𝐶)))
56	1	3ad2ant1 1135	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → 𝐴 ∈ ℤ)
57	37	nn0zd 12263	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐶 gcd 𝐵) ∈ ℤ)
58		gcdass 16088	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ (𝐶 gcd 𝐵) ∈ ℤ) → ((𝐴 gcd 𝐶) gcd (𝐶 gcd 𝐵)) = (𝐴 gcd (𝐶 gcd (𝐶 gcd 𝐵))))
59	56, 43, 57, 58	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((𝐴 gcd 𝐶) gcd (𝐶 gcd 𝐵)) = (𝐴 gcd (𝐶 gcd (𝐶 gcd 𝐵))))
60		gcdass 16088	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 gcd 𝐵) gcd 𝐶) = (𝐴 gcd (𝐵 gcd 𝐶)))
61	56, 47, 43, 60	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((𝐴 gcd 𝐵) gcd 𝐶) = (𝐴 gcd (𝐵 gcd 𝐶)))
62	55, 59, 61	3eqtr4d 2784	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((𝐴 gcd 𝐶) gcd (𝐶 gcd 𝐵)) = ((𝐴 gcd 𝐵) gcd 𝐶))
63	62	eqeq1d 2736	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (((𝐴 gcd 𝐶) gcd (𝐶 gcd 𝐵)) = 1 ↔ ((𝐴 gcd 𝐵) gcd 𝐶) = 1))
64	63	biimpar 481	. . . . . 6 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → ((𝐴 gcd 𝐶) gcd (𝐶 gcd 𝐵)) = 1)
65	64	oveq2d 7218	. . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → (𝐴 · ((𝐴 gcd 𝐶) gcd (𝐶 gcd 𝐵))) = (𝐴 · 1))
66	65	3adant3 1134	. . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐴 · ((𝐴 gcd 𝐶) gcd (𝐶 gcd 𝐵))) = (𝐴 · 1))
67	13	mulid1d 10833	. . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐴 · 1) = 𝐴)
68	66, 67	eqtrd 2774	. . 3 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐴 · ((𝐴 gcd 𝐶) gcd (𝐶 gcd 𝐵))) = 𝐴)
69	8, 42, 68	3eqtrrd 2779	. 2 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐴 = ((𝐴 gcd 𝐶)↑2))
70	69	3expia 1123	1 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → ((𝐶↑2) = (𝐴 · 𝐵) → 𝐴 = ((𝐴 gcd 𝐶)↑2)))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ‘cfv 6369  (class class class)co 7202  ℂcc 10710  1c1 10713   · cmul 10717  2c2 11868  ℕ₀cn0 12073  ℤcz 12159  ↑cexp 13618  abscabs 14780   gcd cgcd 16034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512  ax-cnex 10768  ax-resscn 10769  ax-1cn 10770  ax-icn 10771  ax-addcl 10772  ax-addrcl 10773  ax-mulcl 10774  ax-mulrcl 10775  ax-mulcom 10776  ax-addass 10777  ax-mulass 10778  ax-distr 10779  ax-i2m1 10780  ax-1ne0 10781  ax-1rid 10782  ax-rnegex 10783  ax-rrecex 10784  ax-cnre 10785  ax-pre-lttri 10786  ax-pre-lttrn 10787  ax-pre-ltadd 10788  ax-pre-mulgt0 10789  ax-pre-sup 10790

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-ord 6205  df-on 6206  df-lim 6207  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-riota 7159  df-ov 7205  df-oprab 7206  df-mpo 7207  df-om 7634  df-2nd 7751  df-wrecs 8036  df-recs 8097  df-rdg 8135  df-er 8380  df-en 8616  df-dom 8617  df-sdom 8618  df-sup 9047  df-inf 9048  df-pnf 10852  df-mnf 10853  df-xr 10854  df-ltxr 10855  df-le 10856  df-sub 11047  df-neg 11048  df-div 11473  df-nn 11814  df-2 11876  df-3 11877  df-n0 12074  df-z 12160  df-uz 12422  df-rp 12570  df-fl 13350  df-mod 13426  df-seq 13558  df-exp 13619  df-cj 14645  df-re 14646  df-im 14647  df-sqrt 14781  df-abs 14782  df-dvds 15797  df-gcd 16035

This theorem is referenced by:  coprimeprodsq2  16343  pythagtriplem6  16355  flt4lem4  40141