Theorem coprimeprodsq 16706

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 12548	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ)
2		nn0z 12548	. . . . . . . 8 ⊢ (𝐶 ∈ ℕ0 → 𝐶 ∈ ℤ)
3		gcdcl 16412	. . . . . . . 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 12499	. . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐴 gcd 𝐶) ∈ ℂ)
8	7	sqvald 14073	. . 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 1205	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐶 ∈ ℕ0)
10	9	nn0cnd 12499	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐶 ∈ ℂ)
11		nn0cn 12447	. . . . . . . . . 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 11200	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐶 · 𝐴) = (𝐴 · 𝐶))
15		simpl3 1193	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → 𝐶 ∈ ℕ0)
16	15	nn0cnd 12499	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → 𝐶 ∈ ℂ)
17	16	sqvald 14073	. . . . . . . . 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 1469	. . . . . . 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 7395	. . . . . 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 1203	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐴 ∈ ℕ0)
22	21	nn0zd 12549	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐴 ∈ ℤ)
23	9	nn0zd 12549	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐶 ∈ ℤ)
24		mulgcd 16455	. . . . . . 7 ⊢ ((𝐶 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐶 · 𝐴) gcd (𝐶 · 𝐶)) = (𝐶 · (𝐴 gcd 𝐶)))
25	9, 22, 23, 24	syl3anc 1371	. . . . . 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 1204	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → 𝐵 ∈ ℤ)
27		mulgcd 16455	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐶 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 · 𝐶) gcd (𝐴 · 𝐵)) = (𝐴 · (𝐶 gcd 𝐵)))
28	21, 23, 26, 27	syl3anc 1371	. . . . . 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 2779	. . . . 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 7393	. . . 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 16457	. . . . 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 1371	. . . 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 12549	. . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐴 gcd 𝐶) ∈ ℤ)
34		gcdcl 16412	. . . . . . . . . 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 459	. . . . . . . 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 12549	. . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐶 gcd 𝐵) ∈ ℤ)
40		mulgcd 16455	. . . . 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 1371	. . . 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 2779	. . 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 16433	. . . . . . . . . . . . . 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 7392	. . . . . . . . . . . 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 16435	. . . . . . . . . . . . 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 2771	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((𝐶 gcd 𝐶) gcd 𝐵) = (𝐶 gcd 𝐵))
51		gcdass 16454	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐶 gcd 𝐶) gcd 𝐵) = (𝐶 gcd (𝐶 gcd 𝐵)))
52	43, 43, 47, 51	syl3anc 1371	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((𝐶 gcd 𝐶) gcd 𝐵) = (𝐶 gcd (𝐶 gcd 𝐵)))
53	43, 47	gcdcomd 16420	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐶 gcd 𝐵) = (𝐵 gcd 𝐶))
54	50, 52, 53	3eqtr3d 2779	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐶 gcd (𝐶 gcd 𝐵)) = (𝐵 gcd 𝐶))
55	54	oveq2d 7393	. . . . . . . . 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 12549	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → (𝐶 gcd 𝐵) ∈ ℤ)
58		gcdass 16454	. . . . . . . . . 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 1371	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((𝐴 gcd 𝐶) gcd (𝐶 gcd 𝐵)) = (𝐴 gcd (𝐶 gcd (𝐶 gcd 𝐵))))
60		gcdass 16454	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 gcd 𝐵) gcd 𝐶) = (𝐴 gcd (𝐵 gcd 𝐶)))
61	56, 47, 43, 60	syl3anc 1371	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) → ((𝐴 gcd 𝐵) gcd 𝐶) = (𝐴 gcd (𝐵 gcd 𝐶)))
62	55, 59, 61	3eqtr4d 2781	. . . . . . . 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 478	. . . . . 6 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1) → ((𝐴 gcd 𝐶) gcd (𝐶 gcd 𝐵)) = 1)
65	64	oveq2d 7393	. . . . 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 11196	. . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ0) ∧ ((𝐴 gcd 𝐵) gcd 𝐶) = 1 ∧ (𝐶↑2) = (𝐴 · 𝐵)) → (𝐴 · 1) = 𝐴)
68	66, 67	eqtrd 2771	. . 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 2776	. 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ‘cfv 6516  (class class class)co 7377  ℂcc 11073  1c1 11076   · cmul 11080  2c2 12232  ℕ₀cn0 12437  ℤcz 12523  ↑cexp 13992  abscabs 15146   gcd cgcd 16400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3364  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-pss 3947  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-tr 5243  df-id 5551  df-eprel 5557  df-po 5565  df-so 5566  df-fr 5608  df-we 5610  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-pred 6273  df-ord 6340  df-on 6341  df-lim 6342  df-suc 6343  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7333  df-ov 7380  df-oprab 7381  df-mpo 7382  df-om 7823  df-2nd 7942  df-frecs 8232  df-wrecs 8263  df-recs 8337  df-rdg 8376  df-er 8670  df-en 8906  df-dom 8907  df-sdom 8908  df-sup 9402  df-inf 9403  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11411  df-neg 11412  df-div 11837  df-nn 12178  df-2 12240  df-3 12241  df-n0 12438  df-z 12524  df-uz 12788  df-rp 12940  df-fl 13722  df-mod 13800  df-seq 13932  df-exp 13993  df-cj 15011  df-re 15012  df-im 15013  df-sqrt 15147  df-abs 15148  df-dvds 16163  df-gcd 16401

This theorem is referenced by:  coprimeprodsq2  16707  pythagtriplem6  16719  flt4lem4  41078