Theorem coprimeprodsq 12148

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 9193	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ)
2		nn0z 9193	. . . . . . . 8 ⊢ (𝐶 ∈ ℕ0 → 𝐶 ∈ ℤ)
3		gcdcl 11866	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 gcd 𝐶) ∈ ℕ0)
4	1, 2, 3	syl2an 287	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝐴 gcd 𝐶) ∈ ℕ0)
5	4	3adant2 1001	. . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐴 gcd 𝐶) ∈ ℕ0)
6	5	3ad2ant1 1003	. . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐴 gcd 𝐶) ∈ ℕ0)
7	6	nn0cnd 9151	. . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐴 gcd 𝐶) ∈ ℂ)
8	7	sqvald 10558	. . 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 1014	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐶 ∈ ℕ0)
10	9	nn0cnd 9151	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐶 ∈ ℂ)
11		nn0cn 9106	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℂ)
12	11	3ad2ant1 1003	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → 𝐴 ∈ ℂ)
13	12	3ad2ant1 1003	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐴 ∈ ℂ)
14	10, 13	mulcomd 7902	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐶 · 𝐴) = (𝐴 · 𝐶))
15		simpl3 987	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → 𝐶 ∈ ℕ0)
16	15	nn0cnd 9151	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → 𝐶 ∈ ℂ)
17	16	sqvald 10558	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → (𝐶↑2) = (𝐶 · 𝐶))
18	17	eqeq1d 2166	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → ((𝐶↑2) = (𝐴 · 𝐵) ↔ (𝐶 · 𝐶) = (𝐴 · 𝐵)))
19	18	biimp3a 1327	. . . . . . 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 5845	. . . . . 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 1012	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐴 ∈ ℕ0)
22	21	nn0zd 9290	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐴 ∈ ℤ)
23	9	nn0zd 9290	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐶 ∈ ℤ)
24		mulgcd 11916	. . . . . . 7 ⊢ ((𝐶 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐶 · 𝐴) gcd (𝐶 · 𝐶)) = (𝐶 · (𝐴 gcd 𝐶)))
25	9, 22, 23, 24	syl3anc 1220	. . . . . 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 1013	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐵 ∈ ℤ)
27		mulgcd 11916	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐶 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 · 𝐶) gcd (𝐴 · 𝐵)) = (𝐴 · (𝐶 gcd 𝐵)))
28	21, 23, 26, 27	syl3anc 1220	. . . . . 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 2198	. . . . 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 5843	. . . 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 11918	. . . . 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 1220	. . . 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 9290	. . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐴 gcd 𝐶) ∈ ℤ)
34		gcdcl 11866	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐶 gcd 𝐵) ∈ ℕ0)
35	2, 34	sylan 281	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℕ0 ∧ 𝐵 ∈ ℤ) → (𝐶 gcd 𝐵) ∈ ℕ0)
36	35	ancoms 266	. . . . . . . 8 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐶 gcd 𝐵) ∈ ℕ0)
37	36	3adant1 1000	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐶 gcd 𝐵) ∈ ℕ0)
38	37	3ad2ant1 1003	. . . . . 6 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐶 gcd 𝐵) ∈ ℕ0)
39	38	nn0zd 9290	. . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐶 gcd 𝐵) ∈ ℤ)
40		mulgcd 11916	. . . . 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 1220	. . . 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 2198	. . 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 1005	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → 𝐶 ∈ ℤ)
44		gcdid 11886	. . . . . . . . . . . . . 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 5842	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((𝐶 gcd 𝐶) gcd 𝐵) = ((abs‘𝐶) gcd 𝐵))
47		simp2 983	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → 𝐵 ∈ ℤ)
48		gcdabs1 11889	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((abs‘𝐶) gcd 𝐵) = (𝐶 gcd 𝐵))
49	43, 47, 48	syl2anc 409	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((abs‘𝐶) gcd 𝐵) = (𝐶 gcd 𝐵))
50	46, 49	eqtrd 2190	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((𝐶 gcd 𝐶) gcd 𝐵) = (𝐶 gcd 𝐵))
51		gcdass 11915	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐶 gcd 𝐶) gcd 𝐵) = (𝐶 gcd (𝐶 gcd 𝐵)))
52	43, 43, 47, 51	syl3anc 1220	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((𝐶 gcd 𝐶) gcd 𝐵) = (𝐶 gcd (𝐶 gcd 𝐵)))
53	43, 47	gcdcomd 11874	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐶 gcd 𝐵) = (𝐵 gcd 𝐶))
54	50, 52, 53	3eqtr3d 2198	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐶 gcd (𝐶 gcd 𝐵)) = (𝐵 gcd 𝐶))
55	54	oveq2d 5843	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐴 gcd (𝐶 gcd (𝐶 gcd 𝐵))) = (𝐴 gcd (𝐵 gcd 𝐶)))
56	1	3ad2ant1 1003	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → 𝐴 ∈ ℤ)
57	37	nn0zd 9290	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐶 gcd 𝐵) ∈ ℤ)
58		gcdass 11915	. . . . . . . . . 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 1220	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((𝐴 gcd 𝐶) gcd (𝐶 gcd 𝐵)) = (𝐴 gcd (𝐶 gcd (𝐶 gcd 𝐵))))
60		gcdass 11915	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 gcd 𝐵) gcd 𝐶) = (𝐴 gcd (𝐵 gcd 𝐶)))
61	56, 47, 43, 60	syl3anc 1220	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((𝐴 gcd 𝐵) gcd 𝐶) = (𝐴 gcd (𝐵 gcd 𝐶)))
62	55, 59, 61	3eqtr4d 2200	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((𝐴 gcd 𝐶) gcd (𝐶 gcd 𝐵)) = ((𝐴 gcd 𝐵) gcd 𝐶))
63	62	eqeq1d 2166	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (((𝐴 gcd 𝐶) gcd (𝐶 gcd 𝐵)) = 1 ↔ ((𝐴 gcd 𝐵) gcd 𝐶) = 1))
64	63	biimpar 295	. . . . . 6 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → ((𝐴 gcd 𝐶) gcd (𝐶 gcd 𝐵)) = 1)
65	64	oveq2d 5843	. . . . 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 1002	. . . 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 7898	. . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐴 · 1) = 𝐴)
68	66, 67	eqtrd 2190	. . 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 2195	. 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 1187	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 103   ∧ w3a 963   = wceq 1335   ∈ wcel 2128  ‘cfv 5173  (class class class)co 5827  ℂcc 7733  1c1 7736   · cmul 7740  2c2 8890  ℕ₀cn0 9096  ℤcz 9173  ↑cexp 10428  abscabs 10909   gcd cgcd 11842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4082  ax-sep 4085  ax-nul 4093  ax-pow 4138  ax-pr 4172  ax-un 4396  ax-setind 4499  ax-iinf 4550  ax-cnex 7826  ax-resscn 7827  ax-1cn 7828  ax-1re 7829  ax-icn 7830  ax-addcl 7831  ax-addrcl 7832  ax-mulcl 7833  ax-mulrcl 7834  ax-addcom 7835  ax-mulcom 7836  ax-addass 7837  ax-mulass 7838  ax-distr 7839  ax-i2m1 7840  ax-0lt1 7841  ax-1rid 7842  ax-0id 7843  ax-rnegex 7844  ax-precex 7845  ax-cnre 7846  ax-pre-ltirr 7847  ax-pre-ltwlin 7848  ax-pre-lttrn 7849  ax-pre-apti 7850  ax-pre-ltadd 7851  ax-pre-mulgt0 7852  ax-pre-mulext 7853  ax-arch 7854  ax-caucvg 7855

This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-if 3507  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4029  df-mpt 4030  df-tr 4066  df-id 4256  df-po 4259  df-iso 4260  df-iord 4329  df-on 4331  df-ilim 4332  df-suc 4334  df-iom 4553  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-res 4601  df-ima 4602  df-iota 5138  df-fun 5175  df-fn 5176  df-f 5177  df-f1 5178  df-fo 5179  df-f1o 5180  df-fv 5181  df-riota 5783  df-ov 5830  df-oprab 5831  df-mpo 5832  df-1st 6091  df-2nd 6092  df-recs 6255  df-frec 6341  df-sup 6931  df-pnf 7917  df-mnf 7918  df-xr 7919  df-ltxr 7920  df-le 7921  df-sub 8053  df-neg 8054  df-reap 8455  df-ap 8462  df-div 8551  df-inn 8840  df-2 8898  df-3 8899  df-4 8900  df-n0 9097  df-z 9174  df-uz 9446  df-q 9536  df-rp 9568  df-fz 9920  df-fzo 10052  df-fl 10179  df-mod 10232  df-seqfrec 10355  df-exp 10429  df-cj 10754  df-re 10755  df-im 10756  df-rsqrt 10910  df-abs 10911  df-dvds 11696  df-gcd 11843

This theorem is referenced by:  coprimeprodsq2  12149  pythagtriplem6  12161