Theorem coprimeprodsq 16726

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 12499	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ)
2		nn0z 12499	. . . . . . . 8 ⊢ (𝐶 ∈ ℕ0 → 𝐶 ∈ ℤ)
3		gcdcl 16423	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 gcd 𝐶) ∈ ℕ0)
4	1, 2, 3	syl2an 596	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝐴 gcd 𝐶) ∈ ℕ0)
5	4	3adant2 1131	. . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐴 gcd 𝐶) ∈ ℕ0)
6	5	3ad2ant1 1133	. . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐴 gcd 𝐶) ∈ ℕ0)
7	6	nn0cnd 12450	. . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐴 gcd 𝐶) ∈ ℂ)
8	7	sqvald 14056	. . 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 1206	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐶 ∈ ℕ0)
10	9	nn0cnd 12450	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐶 ∈ ℂ)
11		nn0cn 12397	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℂ)
12	11	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → 𝐴 ∈ ℂ)
13	12	3ad2ant1 1133	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐴 ∈ ℂ)
14	10, 13	mulcomd 11139	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐶 · 𝐴) = (𝐴 · 𝐶))
15		simpl3 1194	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → 𝐶 ∈ ℕ0)
16	15	nn0cnd 12450	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → 𝐶 ∈ ℂ)
17	16	sqvald 14056	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → (𝐶↑2) = (𝐶 · 𝐶))
18	17	eqeq1d 2733	. . . . . . . 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 7370	. . . . . 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 1204	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐴 ∈ ℕ0)
22	21	nn0zd 12500	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐴 ∈ ℤ)
23	9	nn0zd 12500	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐶 ∈ ℤ)
24		mulgcd 16465	. . . . . . 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 1205	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐵 ∈ ℤ)
27		mulgcd 16465	. . . . . . 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 2774	. . . . 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 7368	. . . 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 16467	. . . . 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 12500	. . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐴 gcd 𝐶) ∈ ℤ)
34		gcdcl 16423	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐶 gcd 𝐵) ∈ ℕ0)
35	2, 34	sylan 580	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℕ0 ∧ 𝐵 ∈ ℤ) → (𝐶 gcd 𝐵) ∈ ℕ0)
36	35	ancoms 458	. . . . . . . 8 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐶 gcd 𝐵) ∈ ℕ0)
37	36	3adant1 1130	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐶 gcd 𝐵) ∈ ℕ0)
38	37	3ad2ant1 1133	. . . . . 6 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐶 gcd 𝐵) ∈ ℕ0)
39	38	nn0zd 12500	. . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐶 gcd 𝐵) ∈ ℤ)
40		mulgcd 16465	. . . . 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 2774	. . 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 1135	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → 𝐶 ∈ ℤ)
44		gcdid 16444	. . . . . . . . . . . . . 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 7367	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((𝐶 gcd 𝐶) gcd 𝐵) = ((abs‘𝐶) gcd 𝐵))
47		simp2 1137	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → 𝐵 ∈ ℤ)
48		gcdabs1 16446	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((abs‘𝐶) gcd 𝐵) = (𝐶 gcd 𝐵))
49	43, 47, 48	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((abs‘𝐶) gcd 𝐵) = (𝐶 gcd 𝐵))
50	46, 49	eqtrd 2766	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((𝐶 gcd 𝐶) gcd 𝐵) = (𝐶 gcd 𝐵))
51		gcdass 16464	. . . . . . . . . . . 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 16431	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐶 gcd 𝐵) = (𝐵 gcd 𝐶))
54	50, 52, 53	3eqtr3d 2774	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐶 gcd (𝐶 gcd 𝐵)) = (𝐵 gcd 𝐶))
55	54	oveq2d 7368	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐴 gcd (𝐶 gcd (𝐶 gcd 𝐵))) = (𝐴 gcd (𝐵 gcd 𝐶)))
56	1	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → 𝐴 ∈ ℤ)
57	37	nn0zd 12500	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐶 gcd 𝐵) ∈ ℤ)
58		gcdass 16464	. . . . . . . . . 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 16464	. . . . . . . . . 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 2776	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((𝐴 gcd 𝐶) gcd (𝐶 gcd 𝐵)) = ((𝐴 gcd 𝐵) gcd 𝐶))
63	62	eqeq1d 2733	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (((𝐴 gcd 𝐶) gcd (𝐶 gcd 𝐵)) = 1 ↔ ((𝐴 gcd 𝐵) gcd 𝐶) = 1))
64	63	biimpar 477	. . . . . 6 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → ((𝐴 gcd 𝐶) gcd (𝐶 gcd 𝐵)) = 1)
65	64	oveq2d 7368	. . . . 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 1132	. . . 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	mulridd 11135	. . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐴 · 1) = 𝐴)
68	66, 67	eqtrd 2766	. . 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 2771	. 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 1121	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 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ‘cfv 6487  (class class class)co 7352  ℂcc 11010  1c1 11013   · cmul 11017  2c2 12186  ℕ₀cn0 12387  ℤcz 12474  ↑cexp 13974  abscabs 15147   gcd cgcd 16411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11068  ax-resscn 11069  ax-1cn 11070  ax-icn 11071  ax-addcl 11072  ax-addrcl 11073  ax-mulcl 11074  ax-mulrcl 11075  ax-mulcom 11076  ax-addass 11077  ax-mulass 11078  ax-distr 11079  ax-i2m1 11080  ax-1ne0 11081  ax-1rid 11082  ax-rnegex 11083  ax-rrecex 11084  ax-cnre 11085  ax-pre-lttri 11086  ax-pre-lttrn 11087  ax-pre-ltadd 11088  ax-pre-mulgt0 11089  ax-pre-sup 11090

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-er 8628  df-en 8876  df-dom 8877  df-sdom 8878  df-sup 9332  df-inf 9333  df-pnf 11154  df-mnf 11155  df-xr 11156  df-ltxr 11157  df-le 11158  df-sub 11352  df-neg 11353  df-div 11781  df-nn 12132  df-2 12194  df-3 12195  df-n0 12388  df-z 12475  df-uz 12739  df-rp 12897  df-fl 13702  df-mod 13780  df-seq 13915  df-exp 13975  df-cj 15012  df-re 15013  df-im 15014  df-sqrt 15148  df-abs 15149  df-dvds 16170  df-gcd 16412

This theorem is referenced by:  coprimeprodsq2  16727  pythagtriplem6  16739  flt4lem4  42748