Theorem coprimeprodsq 12256

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 9272	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ)
2		nn0z 9272	. . . . . . . 8 ⊢ (𝐶 ∈ ℕ0 → 𝐶 ∈ ℤ)
3		gcdcl 11966	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 gcd 𝐶) ∈ ℕ0)
4	1, 2, 3	syl2an 289	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝐴 gcd 𝐶) ∈ ℕ0)
5	4	3adant2 1016	. . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐴 gcd 𝐶) ∈ ℕ0)
6	5	3ad2ant1 1018	. . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐴 gcd 𝐶) ∈ ℕ0)
7	6	nn0cnd 9230	. . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐴 gcd 𝐶) ∈ ℂ)
8	7	sqvald 10650	. . 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 1029	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐶 ∈ ℕ0)
10	9	nn0cnd 9230	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐶 ∈ ℂ)
11		nn0cn 9185	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℂ)
12	11	3ad2ant1 1018	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → 𝐴 ∈ ℂ)
13	12	3ad2ant1 1018	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐴 ∈ ℂ)
14	10, 13	mulcomd 7978	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐶 · 𝐴) = (𝐴 · 𝐶))
15		simpl3 1002	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → 𝐶 ∈ ℕ0)
16	15	nn0cnd 9230	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → 𝐶 ∈ ℂ)
17	16	sqvald 10650	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → (𝐶↑2) = (𝐶 · 𝐶))
18	17	eqeq1d 2186	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → ((𝐶↑2) = (𝐴 · 𝐵) ↔ (𝐶 · 𝐶) = (𝐴 · 𝐵)))
19	18	biimp3a 1345	. . . . . . 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 5892	. . . . . 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 1027	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐴 ∈ ℕ0)
22	21	nn0zd 9372	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐴 ∈ ℤ)
23	9	nn0zd 9372	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐶 ∈ ℤ)
24		mulgcd 12016	. . . . . . 7 ⊢ ((𝐶 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐶 · 𝐴) gcd (𝐶 · 𝐶)) = (𝐶 · (𝐴 gcd 𝐶)))
25	9, 22, 23, 24	syl3anc 1238	. . . . . 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 1028	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐵 ∈ ℤ)
27		mulgcd 12016	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐶 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 · 𝐶) gcd (𝐴 · 𝐵)) = (𝐴 · (𝐶 gcd 𝐵)))
28	21, 23, 26, 27	syl3anc 1238	. . . . . 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 2218	. . . . 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 5890	. . . 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 12018	. . . . 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 1238	. . . 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 9372	. . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐴 gcd 𝐶) ∈ ℤ)
34		gcdcl 11966	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐶 gcd 𝐵) ∈ ℕ0)
35	2, 34	sylan 283	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℕ0 ∧ 𝐵 ∈ ℤ) → (𝐶 gcd 𝐵) ∈ ℕ0)
36	35	ancoms 268	. . . . . . . 8 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐶 gcd 𝐵) ∈ ℕ0)
37	36	3adant1 1015	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐶 gcd 𝐵) ∈ ℕ0)
38	37	3ad2ant1 1018	. . . . . 6 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐶 gcd 𝐵) ∈ ℕ0)
39	38	nn0zd 9372	. . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐶 gcd 𝐵) ∈ ℤ)
40		mulgcd 12016	. . . . 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 1238	. . . 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 2218	. . 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 1020	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → 𝐶 ∈ ℤ)
44		gcdid 11986	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ ℤ → (𝐶 gcd 𝐶) = (abs‘𝐶))
45	43, 44	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐶 gcd 𝐶) = (abs‘𝐶))
46	45	oveq1d 5889	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((𝐶 gcd 𝐶) gcd 𝐵) = ((abs‘𝐶) gcd 𝐵))
47		simp2 998	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → 𝐵 ∈ ℤ)
48		gcdabs1 11989	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((abs‘𝐶) gcd 𝐵) = (𝐶 gcd 𝐵))
49	43, 47, 48	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((abs‘𝐶) gcd 𝐵) = (𝐶 gcd 𝐵))
50	46, 49	eqtrd 2210	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((𝐶 gcd 𝐶) gcd 𝐵) = (𝐶 gcd 𝐵))
51		gcdass 12015	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐶 gcd 𝐶) gcd 𝐵) = (𝐶 gcd (𝐶 gcd 𝐵)))
52	43, 43, 47, 51	syl3anc 1238	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((𝐶 gcd 𝐶) gcd 𝐵) = (𝐶 gcd (𝐶 gcd 𝐵)))
53	43, 47	gcdcomd 11974	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐶 gcd 𝐵) = (𝐵 gcd 𝐶))
54	50, 52, 53	3eqtr3d 2218	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐶 gcd (𝐶 gcd 𝐵)) = (𝐵 gcd 𝐶))
55	54	oveq2d 5890	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐴 gcd (𝐶 gcd (𝐶 gcd 𝐵))) = (𝐴 gcd (𝐵 gcd 𝐶)))
56	1	3ad2ant1 1018	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → 𝐴 ∈ ℤ)
57	37	nn0zd 9372	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐶 gcd 𝐵) ∈ ℤ)
58		gcdass 12015	. . . . . . . . . 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 1238	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((𝐴 gcd 𝐶) gcd (𝐶 gcd 𝐵)) = (𝐴 gcd (𝐶 gcd (𝐶 gcd 𝐵))))
60		gcdass 12015	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 gcd 𝐵) gcd 𝐶) = (𝐴 gcd (𝐵 gcd 𝐶)))
61	56, 47, 43, 60	syl3anc 1238	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((𝐴 gcd 𝐵) gcd 𝐶) = (𝐴 gcd (𝐵 gcd 𝐶)))
62	55, 59, 61	3eqtr4d 2220	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((𝐴 gcd 𝐶) gcd (𝐶 gcd 𝐵)) = ((𝐴 gcd 𝐵) gcd 𝐶))
63	62	eqeq1d 2186	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (((𝐴 gcd 𝐶) gcd (𝐶 gcd 𝐵)) = 1 ↔ ((𝐴 gcd 𝐵) gcd 𝐶) = 1))
64	63	biimpar 297	. . . . . 6 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → ((𝐴 gcd 𝐶) gcd (𝐶 gcd 𝐵)) = 1)
65	64	oveq2d 5890	. . . . 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 1017	. . . 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 7973	. . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐴 · 1) = 𝐴)
68	66, 67	eqtrd 2210	. . 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 2215	. 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 1205	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 104   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  ‘cfv 5216  (class class class)co 5874  ℂcc 7808  1c1 7811   · cmul 7815  2c2 8969  ℕ₀cn0 9175  ℤcz 9252  ↑cexp 10518  abscabs 11005   gcd cgcd 11942

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-sup 6982  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-reap 8531  df-ap 8538  df-div 8629  df-inn 8919  df-2 8977  df-3 8978  df-4 8979  df-n0 9176  df-z 9253  df-uz 9528  df-q 9619  df-rp 9653  df-fz 10008  df-fzo 10142  df-fl 10269  df-mod 10322  df-seqfrec 10445  df-exp 10519  df-cj 10850  df-re 10851  df-im 10852  df-rsqrt 11006  df-abs 11007  df-dvds 11794  df-gcd 11943

This theorem is referenced by:  coprimeprodsq2  12257  pythagtriplem6  12269