divgcdcoprmex - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > divgcdcoprmex		GIF version

Theorem divgcdcoprmex 12645

Description: Integers divided by gcd are coprime (see ProofWiki "Integers Divided by GCD are Coprime", 11-Jul-2021, https://proofwiki.org/wiki/Integers_Divided_by_GCD_are_Coprime): Any pair of integers, not both zero, can be reduced to a pair of coprime ones by dividing them by their gcd. (Contributed by AV, 12-Jul-2021.)

**Assertion**
Ref	Expression
divgcdcoprmex	⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ (𝐴 = (𝑀 · 𝑎) ∧ 𝐵 = (𝑀 · 𝑏) ∧ (𝑎 gcd 𝑏) = 1))

Distinct variable groups: 𝐴,𝑎,𝑏 𝐵,𝑎,𝑏 𝑀,𝑎,𝑏

**Proof of Theorem divgcdcoprmex**
Step	Hyp	Ref	Expression
1		simpl 109	. . . . 5 ⊢ ((𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) → 𝐵 ∈ ℤ)
2	1	anim2i 342	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ))
3		zeqzmulgcd 12512	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ∃𝑎 ∈ ℤ 𝐴 = (𝑎 · (𝐴 gcd 𝐵)))
4	2, 3	syl 14	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → ∃𝑎 ∈ ℤ 𝐴 = (𝑎 · (𝐴 gcd 𝐵)))
5	4	3adant3 1041	. 2 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → ∃𝑎 ∈ ℤ 𝐴 = (𝑎 · (𝐴 gcd 𝐵)))
6		zeqzmulgcd 12512	. . . . 5 ⊢ ((𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ) → ∃𝑏 ∈ ℤ 𝐵 = (𝑏 · (𝐵 gcd 𝐴)))
7	6	adantlr 477	. . . 4 ⊢ (((𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝐴 ∈ ℤ) → ∃𝑏 ∈ ℤ 𝐵 = (𝑏 · (𝐵 gcd 𝐴)))
8	7	ancoms 268	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → ∃𝑏 ∈ ℤ 𝐵 = (𝑏 · (𝐵 gcd 𝐴)))
9	8	3adant3 1041	. 2 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → ∃𝑏 ∈ ℤ 𝐵 = (𝑏 · (𝐵 gcd 𝐴)))
10		reeanv 2701	. . 3 ⊢ (∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ (𝐴 = (𝑎 · (𝐴 gcd 𝐵)) ∧ 𝐵 = (𝑏 · (𝐵 gcd 𝐴))) ↔ (∃𝑎 ∈ ℤ 𝐴 = (𝑎 · (𝐴 gcd 𝐵)) ∧ ∃𝑏 ∈ ℤ 𝐵 = (𝑏 · (𝐵 gcd 𝐴))))
11		zcn 9467	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ℤ → 𝑎 ∈ ℂ)
12	11	adantl 277	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) → 𝑎 ∈ ℂ)
13		gcdcl 12508	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 gcd 𝐵) ∈ ℕ₀)
14	2, 13	syl 14	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝐴 gcd 𝐵) ∈ ℕ₀)
15	14	nn0cnd 9440	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝐴 gcd 𝐵) ∈ ℂ)
16	15	3adant3 1041	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 gcd 𝐵) ∈ ℂ)
17	16	adantr 276	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) → (𝐴 gcd 𝐵) ∈ ℂ)
18	12, 17	mulcomd 8184	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) → (𝑎 · (𝐴 gcd 𝐵)) = ((𝐴 gcd 𝐵) · 𝑎))
19		simp3 1023	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → 𝑀 = (𝐴 gcd 𝐵))
20	19	eqcomd 2235	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 gcd 𝐵) = 𝑀)
21	20	oveq1d 6025	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → ((𝐴 gcd 𝐵) · 𝑎) = (𝑀 · 𝑎))
22	21	adantr 276	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) → ((𝐴 gcd 𝐵) · 𝑎) = (𝑀 · 𝑎))
23	18, 22	eqtrd 2262	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) → (𝑎 · (𝐴 gcd 𝐵)) = (𝑀 · 𝑎))
24	23	ad2antrr 488	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) ∧ (𝐴 = (𝑎 · (𝐴 gcd 𝐵)) ∧ 𝐵 = (𝑏 · (𝐵 gcd 𝐴)))) → (𝑎 · (𝐴 gcd 𝐵)) = (𝑀 · 𝑎))
25		eqeq1 2236	. . . . . . . . . 10 ⊢ (𝐴 = (𝑎 · (𝐴 gcd 𝐵)) → (𝐴 = (𝑀 · 𝑎) ↔ (𝑎 · (𝐴 gcd 𝐵)) = (𝑀 · 𝑎)))
26	25	adantr 276	. . . . . . . . 9 ⊢ ((𝐴 = (𝑎 · (𝐴 gcd 𝐵)) ∧ 𝐵 = (𝑏 · (𝐵 gcd 𝐴))) → (𝐴 = (𝑀 · 𝑎) ↔ (𝑎 · (𝐴 gcd 𝐵)) = (𝑀 · 𝑎)))
27	26	adantl 277	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) ∧ (𝐴 = (𝑎 · (𝐴 gcd 𝐵)) ∧ 𝐵 = (𝑏 · (𝐵 gcd 𝐴)))) → (𝐴 = (𝑀 · 𝑎) ↔ (𝑎 · (𝐴 gcd 𝐵)) = (𝑀 · 𝑎)))
28	24, 27	mpbird 167	. . . . . . 7 ⊢ (((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) ∧ (𝐴 = (𝑎 · (𝐴 gcd 𝐵)) ∧ 𝐵 = (𝑏 · (𝐵 gcd 𝐴)))) → 𝐴 = (𝑀 · 𝑎))
29		simpr 110	. . . . . . . 8 ⊢ ((𝐴 = (𝑎 · (𝐴 gcd 𝐵)) ∧ 𝐵 = (𝑏 · (𝐵 gcd 𝐴))) → 𝐵 = (𝑏 · (𝐵 gcd 𝐴)))
30	2	ancomd 267	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ))
31		gcdcom 12515	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐵 gcd 𝐴) = (𝐴 gcd 𝐵))
32	30, 31	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝐵 gcd 𝐴) = (𝐴 gcd 𝐵))
33	32	3adant3 1041	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐵 gcd 𝐴) = (𝐴 gcd 𝐵))
34	33	oveq2d 6026	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝑏 · (𝐵 gcd 𝐴)) = (𝑏 · (𝐴 gcd 𝐵)))
35	34	adantr 276	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → (𝑏 · (𝐵 gcd 𝐴)) = (𝑏 · (𝐴 gcd 𝐵)))
36		zcn 9467	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ℤ → 𝑏 ∈ ℂ)
37	36	adantl 277	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → 𝑏 ∈ ℂ)
38	14	3adant3 1041	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 gcd 𝐵) ∈ ℕ₀)
39	38	adantr 276	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → (𝐴 gcd 𝐵) ∈ ℕ₀)
40	39	nn0cnd 9440	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → (𝐴 gcd 𝐵) ∈ ℂ)
41	37, 40	mulcomd 8184	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → (𝑏 · (𝐴 gcd 𝐵)) = ((𝐴 gcd 𝐵) · 𝑏))
42	20	adantr 276	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → (𝐴 gcd 𝐵) = 𝑀)
43	42	oveq1d 6025	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → ((𝐴 gcd 𝐵) · 𝑏) = (𝑀 · 𝑏))
44	35, 41, 43	3eqtrd 2266	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → (𝑏 · (𝐵 gcd 𝐴)) = (𝑀 · 𝑏))
45	44	adantlr 477	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (𝑏 · (𝐵 gcd 𝐴)) = (𝑀 · 𝑏))
46	29, 45	sylan9eqr 2284	. . . . . . 7 ⊢ (((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) ∧ (𝐴 = (𝑎 · (𝐴 gcd 𝐵)) ∧ 𝐵 = (𝑏 · (𝐵 gcd 𝐴)))) → 𝐵 = (𝑀 · 𝑏))
47		zcn 9467	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
48	47	3ad2ant1 1042	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → 𝐴 ∈ ℂ)
49	48	ad2antrr 488	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → 𝐴 ∈ ℂ)
50	12	adantr 276	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → 𝑎 ∈ ℂ)
51		simp1 1021	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → 𝐴 ∈ ℤ)
52	1	3ad2ant2 1043	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → 𝐵 ∈ ℤ)
53	51, 52	gcdcld 12510	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 gcd 𝐵) ∈ ℕ₀)
54	53	nn0cnd 9440	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 gcd 𝐵) ∈ ℂ)
55	54	ad2antrr 488	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (𝐴 gcd 𝐵) ∈ ℂ)
56		gcdeq0 12519	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
57		simpr 110	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → 𝐵 = 0)
58	56, 57	biimtrdi 163	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) = 0 → 𝐵 = 0))
59	58	necon3d 2444	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 ≠ 0 → (𝐴 gcd 𝐵) ≠ 0))
60	59	impr 379	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝐴 gcd 𝐵) ≠ 0)
61	60	3adant3 1041	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 gcd 𝐵) ≠ 0)
62	61	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (𝐴 gcd 𝐵) ≠ 0)
63	38	ad2antrr 488	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (𝐴 gcd 𝐵) ∈ ℕ₀)
64	63	nn0zd 9583	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (𝐴 gcd 𝐵) ∈ ℤ)
65		0zd 9474	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → 0 ∈ ℤ)
66		zapne 9537	. . . . . . . . . . . . . 14 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 0 ∈ ℤ) → ((𝐴 gcd 𝐵) # 0 ↔ (𝐴 gcd 𝐵) ≠ 0))
67	64, 65, 66	syl2anc 411	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → ((𝐴 gcd 𝐵) # 0 ↔ (𝐴 gcd 𝐵) ≠ 0))
68	62, 67	mpbird 167	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (𝐴 gcd 𝐵) # 0)
69	49, 50, 55, 68	divmulap3d 8988	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → ((𝐴 / (𝐴 gcd 𝐵)) = 𝑎 ↔ 𝐴 = (𝑎 · (𝐴 gcd 𝐵))))
70	69	bicomd 141	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (𝐴 = (𝑎 · (𝐴 gcd 𝐵)) ↔ (𝐴 / (𝐴 gcd 𝐵)) = 𝑎))
71		zcn 9467	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℂ)
72	71	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) → 𝐵 ∈ ℂ)
73	72	3ad2ant2 1043	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → 𝐵 ∈ ℂ)
74	73	ad2antrr 488	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → 𝐵 ∈ ℂ)
75	36	adantl 277	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → 𝑏 ∈ ℂ)
76	74, 75, 55, 68	divmulap3d 8988	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → ((𝐵 / (𝐴 gcd 𝐵)) = 𝑏 ↔ 𝐵 = (𝑏 · (𝐴 gcd 𝐵))))
77	2	3adant3 1041	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ))
78		gcdcom 12515	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴))
79	77, 78	syl 14	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴))
80	79	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴))
81	80	oveq2d 6026	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (𝑏 · (𝐴 gcd 𝐵)) = (𝑏 · (𝐵 gcd 𝐴)))
82	81	eqeq2d 2241	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (𝐵 = (𝑏 · (𝐴 gcd 𝐵)) ↔ 𝐵 = (𝑏 · (𝐵 gcd 𝐴))))
83	76, 82	bitr2d 189	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (𝐵 = (𝑏 · (𝐵 gcd 𝐴)) ↔ (𝐵 / (𝐴 gcd 𝐵)) = 𝑏))
84	70, 83	anbi12d 473	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → ((𝐴 = (𝑎 · (𝐴 gcd 𝐵)) ∧ 𝐵 = (𝑏 · (𝐵 gcd 𝐴))) ↔ ((𝐴 / (𝐴 gcd 𝐵)) = 𝑎 ∧ (𝐵 / (𝐴 gcd 𝐵)) = 𝑏)))
85		3anass 1006	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ↔ (𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)))
86	85	biimpri 133	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0))
87	86	3adant3 1041	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0))
88		divgcdcoprm0 12644	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) → ((𝐴 / (𝐴 gcd 𝐵)) gcd (𝐵 / (𝐴 gcd 𝐵))) = 1)
89	87, 88	syl 14	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → ((𝐴 / (𝐴 gcd 𝐵)) gcd (𝐵 / (𝐴 gcd 𝐵))) = 1)
90		oveq12 6019	. . . . . . . . . . . 12 ⊢ (((𝐴 / (𝐴 gcd 𝐵)) = 𝑎 ∧ (𝐵 / (𝐴 gcd 𝐵)) = 𝑏) → ((𝐴 / (𝐴 gcd 𝐵)) gcd (𝐵 / (𝐴 gcd 𝐵))) = (𝑎 gcd 𝑏))
91	90	eqeq1d 2238	. . . . . . . . . . 11 ⊢ (((𝐴 / (𝐴 gcd 𝐵)) = 𝑎 ∧ (𝐵 / (𝐴 gcd 𝐵)) = 𝑏) → (((𝐴 / (𝐴 gcd 𝐵)) gcd (𝐵 / (𝐴 gcd 𝐵))) = 1 ↔ (𝑎 gcd 𝑏) = 1))
92	89, 91	syl5ibcom 155	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (((𝐴 / (𝐴 gcd 𝐵)) = 𝑎 ∧ (𝐵 / (𝐴 gcd 𝐵)) = 𝑏) → (𝑎 gcd 𝑏) = 1))
93	92	ad2antrr 488	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (((𝐴 / (𝐴 gcd 𝐵)) = 𝑎 ∧ (𝐵 / (𝐴 gcd 𝐵)) = 𝑏) → (𝑎 gcd 𝑏) = 1))
94	84, 93	sylbid 150	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → ((𝐴 = (𝑎 · (𝐴 gcd 𝐵)) ∧ 𝐵 = (𝑏 · (𝐵 gcd 𝐴))) → (𝑎 gcd 𝑏) = 1))
95	94	imp 124	. . . . . . 7 ⊢ (((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) ∧ (𝐴 = (𝑎 · (𝐴 gcd 𝐵)) ∧ 𝐵 = (𝑏 · (𝐵 gcd 𝐴)))) → (𝑎 gcd 𝑏) = 1)
96	28, 46, 95	3jca 1201	. . . . . 6 ⊢ (((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) ∧ (𝐴 = (𝑎 · (𝐴 gcd 𝐵)) ∧ 𝐵 = (𝑏 · (𝐵 gcd 𝐴)))) → (𝐴 = (𝑀 · 𝑎) ∧ 𝐵 = (𝑀 · 𝑏) ∧ (𝑎 gcd 𝑏) = 1))
97	96	ex 115	. . . . 5 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → ((𝐴 = (𝑎 · (𝐴 gcd 𝐵)) ∧ 𝐵 = (𝑏 · (𝐵 gcd 𝐴))) → (𝐴 = (𝑀 · 𝑎) ∧ 𝐵 = (𝑀 · 𝑏) ∧ (𝑎 gcd 𝑏) = 1)))
98	97	reximdva 2632	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) → (∃𝑏 ∈ ℤ (𝐴 = (𝑎 · (𝐴 gcd 𝐵)) ∧ 𝐵 = (𝑏 · (𝐵 gcd 𝐴))) → ∃𝑏 ∈ ℤ (𝐴 = (𝑀 · 𝑎) ∧ 𝐵 = (𝑀 · 𝑏) ∧ (𝑎 gcd 𝑏) = 1)))
99	98	reximdva 2632	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ (𝐴 = (𝑎 · (𝐴 gcd 𝐵)) ∧ 𝐵 = (𝑏 · (𝐵 gcd 𝐴))) → ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ (𝐴 = (𝑀 · 𝑎) ∧ 𝐵 = (𝑀 · 𝑏) ∧ (𝑎 gcd 𝑏) = 1)))
100	10, 99	biimtrrid 153	. 2 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → ((∃𝑎 ∈ ℤ 𝐴 = (𝑎 · (𝐴 gcd 𝐵)) ∧ ∃𝑏 ∈ ℤ 𝐵 = (𝑏 · (𝐵 gcd 𝐴))) → ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ (𝐴 = (𝑀 · 𝑎) ∧ 𝐵 = (𝑀 · 𝑏) ∧ (𝑎 gcd 𝑏) = 1)))
101	5, 9, 100	mp2and 433	1 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ (𝐴 = (𝑀 · 𝑎) ∧ 𝐵 = (𝑀 · 𝑏) ∧ (𝑎 gcd 𝑏) = 1))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1002 = wceq 1395 ∈ wcel 2200 ≠ wne 2400 ∃wrex 2509 class class class wbr 4083 (class class class)co 6010 ℂcc 8013 0cc0 8015 1c1 8016 · cmul 8020 # cap 8744 / cdiv 8835 ℕ₀cn0 9385 ℤcz 9462 gcd cgcd 12495

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4259 ax-pr 4294 ax-un 4525 ax-setind 4630 ax-iinf 4681 ax-cnex 8106 ax-resscn 8107 ax-1cn 8108 ax-1re 8109 ax-icn 8110 ax-addcl 8111 ax-addrcl 8112 ax-mulcl 8113 ax-mulrcl 8114 ax-addcom 8115 ax-mulcom 8116 ax-addass 8117 ax-mulass 8118 ax-distr 8119 ax-i2m1 8120 ax-0lt1 8121 ax-1rid 8122 ax-0id 8123 ax-rnegex 8124 ax-precex 8125 ax-cnre 8126 ax-pre-ltirr 8127 ax-pre-ltwlin 8128 ax-pre-lttrn 8129 ax-pre-apti 8130 ax-pre-ltadd 8131 ax-pre-mulgt0 8132 ax-pre-mulext 8133 ax-arch 8134 ax-caucvg 8135

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4385 df-po 4388 df-iso 4389 df-iord 4458 df-on 4460 df-ilim 4461 df-suc 4463 df-iom 4684 df-xp 4726 df-rel 4727 df-cnv 4728 df-co 4729 df-dm 4730 df-rn 4731 df-res 4732 df-ima 4733 df-iota 5281 df-fun 5323 df-fn 5324 df-f 5325 df-f1 5326 df-fo 5327 df-f1o 5328 df-fv 5329 df-riota 5963 df-ov 6013 df-oprab 6014 df-mpo 6015 df-1st 6295 df-2nd 6296 df-recs 6462 df-frec 6548 df-sup 7167 df-pnf 8199 df-mnf 8200 df-xr 8201 df-ltxr 8202 df-le 8203 df-sub 8335 df-neg 8336 df-reap 8738 df-ap 8745 df-div 8836 df-inn 9127 df-2 9185 df-3 9186 df-4 9187 df-n0 9386 df-z 9463 df-uz 9739 df-q 9832 df-rp 9867 df-fz 10222 df-fzo 10356 df-fl 10507 df-mod 10562 df-seqfrec 10687 df-exp 10778 df-cj 11374 df-re 11375 df-im 11376 df-rsqrt 11530 df-abs 11531 df-dvds 12320 df-gcd 12496

This theorem is referenced by: cncongr1 12646

W3C validator