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Theorem divgcdcoprmex 16803

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 488	. . . . 5 ⊢ ((𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) → 𝐵 ∈ ℤ)
2	1	anim2i 629	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ))
3		zeqzmulgcd 16647	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ∃𝑎 ∈ ℤ 𝐴 = (𝑎 · (𝐴 gcd 𝐵)))
4	2, 3	syl 18	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → ∃𝑎 ∈ ℤ 𝐴 = (𝑎 · (𝐴 gcd 𝐵)))
5	4	3adant3 1150	. 2 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → ∃𝑎 ∈ ℤ 𝐴 = (𝑎 · (𝐴 gcd 𝐵)))
6		zeqzmulgcd 16647	. . . . 5 ⊢ ((𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ) → ∃𝑏 ∈ ℤ 𝐵 = (𝑏 · (𝐵 gcd 𝐴)))
7	6	adantlr 728	. . . 4 ⊢ (((𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝐴 ∈ ℤ) → ∃𝑏 ∈ ℤ 𝐵 = (𝑏 · (𝐵 gcd 𝐴)))
8	7	ancoms 464	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → ∃𝑏 ∈ ℤ 𝐵 = (𝑏 · (𝐵 gcd 𝐴)))
9	8	3adant3 1150	. 2 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → ∃𝑏 ∈ ℤ 𝐵 = (𝑏 · (𝐵 gcd 𝐴)))
10		reeanv 3234	. . 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 12667	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ℤ → 𝑎 ∈ ℂ)
12	11	adantl 487	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) → 𝑎 ∈ ℂ)
13		gcdcl 16643	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 gcd 𝐵) ∈ ℕ₀)
14	2, 13	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝐴 gcd 𝐵) ∈ ℕ₀)
15	14	nn0cnd 12638	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝐴 gcd 𝐵) ∈ ℂ)
16	15	3adant3 1150	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 gcd 𝐵) ∈ ℂ)
17	16	adantr 486	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) → (𝐴 gcd 𝐵) ∈ ℂ)
18	12, 17	mulcomd 11301	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) → (𝑎 · (𝐴 gcd 𝐵)) = ((𝐴 gcd 𝐵) · 𝑎))
19		simp3 1156	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → 𝑀 = (𝐴 gcd 𝐵))
20	19	eqcomd 2766	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 gcd 𝐵) = 𝑀)
21	20	oveq1d 7423	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → ((𝐴 gcd 𝐵) · 𝑎) = (𝑀 · 𝑎))
22	21	adantr 486	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) → ((𝐴 gcd 𝐵) · 𝑎) = (𝑀 · 𝑎))
23	18, 22	eqtrd 2795	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) → (𝑎 · (𝐴 gcd 𝐵)) = (𝑀 · 𝑎))
24	23	ad2antrr 739	. . . . . . . 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 2764	. . . . . . . . . 10 ⊢ (𝐴 = (𝑎 · (𝐴 gcd 𝐵)) → (𝐴 = (𝑀 · 𝑎) ↔ (𝑎 · (𝐴 gcd 𝐵)) = (𝑀 · 𝑎)))
26	25	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 = (𝑎 · (𝐴 gcd 𝐵)) ∧ 𝐵 = (𝑏 · (𝐵 gcd 𝐴))) → (𝐴 = (𝑀 · 𝑎) ↔ (𝑎 · (𝐴 gcd 𝐵)) = (𝑀 · 𝑎)))
27	26	adantl 487	. . . . . . . 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 260	. . . . . . 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 490	. . . . . . . 8 ⊢ ((𝐴 = (𝑎 · (𝐴 gcd 𝐵)) ∧ 𝐵 = (𝑏 · (𝐵 gcd 𝐴))) → 𝐵 = (𝑏 · (𝐵 gcd 𝐴)))
30	2	ancomd 467	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ))
31		gcdcom 16650	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐵 gcd 𝐴) = (𝐴 gcd 𝐵))
32	30, 31	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝐵 gcd 𝐴) = (𝐴 gcd 𝐵))
33	32	3adant3 1150	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐵 gcd 𝐴) = (𝐴 gcd 𝐵))
34	33	oveq2d 7424	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝑏 · (𝐵 gcd 𝐴)) = (𝑏 · (𝐴 gcd 𝐵)))
35	34	adantr 486	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → (𝑏 · (𝐵 gcd 𝐴)) = (𝑏 · (𝐴 gcd 𝐵)))
36		zcn 12667	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ℤ → 𝑏 ∈ ℂ)
37	36	adantl 487	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → 𝑏 ∈ ℂ)
38	14	3adant3 1150	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 gcd 𝐵) ∈ ℕ₀)
39	38	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → (𝐴 gcd 𝐵) ∈ ℕ₀)
40	39	nn0cnd 12638	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → (𝐴 gcd 𝐵) ∈ ℂ)
41	37, 40	mulcomd 11301	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → (𝑏 · (𝐴 gcd 𝐵)) = ((𝐴 gcd 𝐵) · 𝑏))
42	20	adantr 486	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → (𝐴 gcd 𝐵) = 𝑀)
43	42	oveq1d 7423	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → ((𝐴 gcd 𝐵) · 𝑏) = (𝑀 · 𝑏))
44	35, 41, 43	3eqtrd 2799	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → (𝑏 · (𝐵 gcd 𝐴)) = (𝑀 · 𝑏))
45	44	adantlr 728	. . . . . . . 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 2817	. . . . . . 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 12667	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
48	47	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → 𝐴 ∈ ℂ)
49	48	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → 𝐴 ∈ ℂ)
50	12	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → 𝑎 ∈ ℂ)
51		simp1 1154	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → 𝐴 ∈ ℤ)
52	1	3ad2ant2 1152	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → 𝐵 ∈ ℤ)
53	51, 52	gcdcld 16645	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 gcd 𝐵) ∈ ℕ₀)
54	53	nn0cnd 12638	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 gcd 𝐵) ∈ ℂ)
55	54	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (𝐴 gcd 𝐵) ∈ ℂ)
56		gcdeq0 16654	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
57		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → 𝐵 = 0)
58	56, 57	biimtrdi 256	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) = 0 → 𝐵 = 0))
59	58	necon3d 2976	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 ≠ 0 → (𝐴 gcd 𝐵) ≠ 0))
60	59	impr 460	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝐴 gcd 𝐵) ≠ 0)
61	60	3adant3 1150	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 gcd 𝐵) ≠ 0)
62	61	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (𝐴 gcd 𝐵) ≠ 0)
63	49, 50, 55, 62	divmul3d 12096	. . . . . . . . . . 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))))
64	63	bicomd 226	. . . . . . . . . 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))
65		zcn 12667	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℂ)
66	65	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) → 𝐵 ∈ ℂ)
67	66	3ad2ant2 1152	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → 𝐵 ∈ ℂ)
68	67	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → 𝐵 ∈ ℂ)
69	36	adantl 487	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → 𝑏 ∈ ℂ)
70	68, 69, 55, 62	divmul3d 12096	. . . . . . . . . . 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))))
71	2	3adant3 1150	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ))
72		gcdcom 16650	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴))
73	71, 72	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴))
74	73	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴))
75	74	oveq2d 7424	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (𝑏 · (𝐴 gcd 𝐵)) = (𝑏 · (𝐵 gcd 𝐴)))
76	75	eqeq2d 2771	. . . . . . . . . . 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))))
77	70, 76	bitr2d 283	. . . . . . . . . 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))
78	64, 77	anbi12d 644	. . . . . . . . 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)))
79		3anass 1111	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ↔ (𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)))
80	79	biimpri 231	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0))
81	80	3adant3 1150	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0))
82		divgcdcoprm0 16802	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) → ((𝐴 / (𝐴 gcd 𝐵)) gcd (𝐵 / (𝐴 gcd 𝐵))) = 1)
83	81, 82	syl 18	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → ((𝐴 / (𝐴 gcd 𝐵)) gcd (𝐵 / (𝐴 gcd 𝐵))) = 1)
84		oveq12 7417	. . . . . . . . . . . 12 ⊢ (((𝐴 / (𝐴 gcd 𝐵)) = 𝑎 ∧ (𝐵 / (𝐴 gcd 𝐵)) = 𝑏) → ((𝐴 / (𝐴 gcd 𝐵)) gcd (𝐵 / (𝐴 gcd 𝐵))) = (𝑎 gcd 𝑏))
85	84	eqeq1d 2762	. . . . . . . . . . 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))
86	83, 85	syl5ibcom 248	. . . . . . . . . 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))
87	86	ad2antrr 739	. . . . . . . . 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))
88	78, 87	sylbid 243	. . . . . . . 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))
89	88	imp 412	. . . . . . 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)
90	28, 46, 89	3jca 1146	. . . . . 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))
91	90	ex 418	. . . . 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)))
92	91	reximdva 3175	. . . 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)))
93	92	reximdva 3175	. . 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)))
94	10, 93	biimtrrid 246	. 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)))
95	5, 9, 94	mp2and 712	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 setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∃wrex 3086 (class class class)co 7408 ℂcc 11169 0cc0 11171 1c1 11172 · cmul 11176 / cdiv 11942 ℕ₀cn0 12575 ℤcz 12662 gcd cgcd 16631

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-cnex 11227 ax-resscn 11228 ax-1cn 11229 ax-icn 11230 ax-addcl 11231 ax-addrcl 11232 ax-mulcl 11233 ax-mulrcl 11234 ax-mulcom 11235 ax-addass 11236 ax-mulass 11237 ax-distr 11238 ax-i2m1 11239 ax-1ne0 11240 ax-1rid 11241 ax-rnegex 11242 ax-rrecex 11243 ax-cnre 11244 ax-pre-lttri 11245 ax-pre-lttrn 11246 ax-pre-ltadd 11247 ax-pre-mulgt0 11248 ax-pre-sup 11249

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6293 df-ord 6354 df-on 6355 df-lim 6356 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7861 df-2nd 7985 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-er 8695 df-en 8952 df-dom 8953 df-sdom 8954 df-sup 9412 df-inf 9413 df-pnf 11316 df-mnf 11317 df-xr 11318 df-ltxr 11319 df-le 11320 df-sub 11514 df-neg 11515 df-div 11943 df-nn 12305 df-2 12374 df-3 12375 df-n0 12576 df-z 12663 df-uz 12935 df-rp 13090 df-fl 13900 df-mod 13978 df-seq 14113 df-exp 14173 df-cj 15233 df-re 15234 df-im 15235 df-sqrt 15369 df-abs 15370 df-dvds 16390 df-gcd 16632

This theorem is used by: cncongr1 16804

W3C validator