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Theorem divgcdcoprmex 16583

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 482	. . . . 5 ⊢ ((𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) → 𝐵 ∈ ℤ)
2	1	anim2i 617	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ))
3		zeqzmulgcd 16427	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ∃𝑎 ∈ ℤ 𝐴 = (𝑎 · (𝐴 gcd 𝐵)))
4	2, 3	syl 17	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → ∃𝑎 ∈ ℤ 𝐴 = (𝑎 · (𝐴 gcd 𝐵)))
5	4	3adant3 1132	. 2 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → ∃𝑎 ∈ ℤ 𝐴 = (𝑎 · (𝐴 gcd 𝐵)))
6		zeqzmulgcd 16427	. . . . 5 ⊢ ((𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ) → ∃𝑏 ∈ ℤ 𝐵 = (𝑏 · (𝐵 gcd 𝐴)))
7	6	adantlr 715	. . . 4 ⊢ (((𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝐴 ∈ ℤ) → ∃𝑏 ∈ ℤ 𝐵 = (𝑏 · (𝐵 gcd 𝐴)))
8	7	ancoms 458	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → ∃𝑏 ∈ ℤ 𝐵 = (𝑏 · (𝐵 gcd 𝐴)))
9	8	3adant3 1132	. 2 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → ∃𝑏 ∈ ℤ 𝐵 = (𝑏 · (𝐵 gcd 𝐴)))
10		reeanv 3204	. . 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 12479	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ℤ → 𝑎 ∈ ℂ)
12	11	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) → 𝑎 ∈ ℂ)
13		gcdcl 16423	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 gcd 𝐵) ∈ ℕ₀)
14	2, 13	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝐴 gcd 𝐵) ∈ ℕ₀)
15	14	nn0cnd 12450	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝐴 gcd 𝐵) ∈ ℂ)
16	15	3adant3 1132	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 gcd 𝐵) ∈ ℂ)
17	16	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) → (𝐴 gcd 𝐵) ∈ ℂ)
18	12, 17	mulcomd 11139	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) → (𝑎 · (𝐴 gcd 𝐵)) = ((𝐴 gcd 𝐵) · 𝑎))
19		simp3 1138	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → 𝑀 = (𝐴 gcd 𝐵))
20	19	eqcomd 2737	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 gcd 𝐵) = 𝑀)
21	20	oveq1d 7367	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → ((𝐴 gcd 𝐵) · 𝑎) = (𝑀 · 𝑎))
22	21	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) → ((𝐴 gcd 𝐵) · 𝑎) = (𝑀 · 𝑎))
23	18, 22	eqtrd 2766	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) → (𝑎 · (𝐴 gcd 𝐵)) = (𝑀 · 𝑎))
24	23	ad2antrr 726	. . . . . . . 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 2735	. . . . . . . . . 10 ⊢ (𝐴 = (𝑎 · (𝐴 gcd 𝐵)) → (𝐴 = (𝑀 · 𝑎) ↔ (𝑎 · (𝐴 gcd 𝐵)) = (𝑀 · 𝑎)))
26	25	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 = (𝑎 · (𝐴 gcd 𝐵)) ∧ 𝐵 = (𝑏 · (𝐵 gcd 𝐴))) → (𝐴 = (𝑀 · 𝑎) ↔ (𝑎 · (𝐴 gcd 𝐵)) = (𝑀 · 𝑎)))
27	26	adantl 481	. . . . . . . 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 257	. . . . . . 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 484	. . . . . . . 8 ⊢ ((𝐴 = (𝑎 · (𝐴 gcd 𝐵)) ∧ 𝐵 = (𝑏 · (𝐵 gcd 𝐴))) → 𝐵 = (𝑏 · (𝐵 gcd 𝐴)))
30	2	ancomd 461	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ))
31		gcdcom 16430	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐵 gcd 𝐴) = (𝐴 gcd 𝐵))
32	30, 31	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝐵 gcd 𝐴) = (𝐴 gcd 𝐵))
33	32	3adant3 1132	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐵 gcd 𝐴) = (𝐴 gcd 𝐵))
34	33	oveq2d 7368	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝑏 · (𝐵 gcd 𝐴)) = (𝑏 · (𝐴 gcd 𝐵)))
35	34	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → (𝑏 · (𝐵 gcd 𝐴)) = (𝑏 · (𝐴 gcd 𝐵)))
36		zcn 12479	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ℤ → 𝑏 ∈ ℂ)
37	36	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → 𝑏 ∈ ℂ)
38	14	3adant3 1132	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 gcd 𝐵) ∈ ℕ₀)
39	38	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → (𝐴 gcd 𝐵) ∈ ℕ₀)
40	39	nn0cnd 12450	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → (𝐴 gcd 𝐵) ∈ ℂ)
41	37, 40	mulcomd 11139	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → (𝑏 · (𝐴 gcd 𝐵)) = ((𝐴 gcd 𝐵) · 𝑏))
42	20	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → (𝐴 gcd 𝐵) = 𝑀)
43	42	oveq1d 7367	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → ((𝐴 gcd 𝐵) · 𝑏) = (𝑀 · 𝑏))
44	35, 41, 43	3eqtrd 2770	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → (𝑏 · (𝐵 gcd 𝐴)) = (𝑀 · 𝑏))
45	44	adantlr 715	. . . . . . . 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 2788	. . . . . . 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 12479	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
48	47	3ad2ant1 1133	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → 𝐴 ∈ ℂ)
49	48	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → 𝐴 ∈ ℂ)
50	12	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → 𝑎 ∈ ℂ)
51		simp1 1136	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → 𝐴 ∈ ℤ)
52	1	3ad2ant2 1134	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → 𝐵 ∈ ℤ)
53	51, 52	gcdcld 16425	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 gcd 𝐵) ∈ ℕ₀)
54	53	nn0cnd 12450	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 gcd 𝐵) ∈ ℂ)
55	54	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (𝐴 gcd 𝐵) ∈ ℂ)
56		gcdeq0 16434	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
57		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → 𝐵 = 0)
58	56, 57	biimtrdi 253	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) = 0 → 𝐵 = 0))
59	58	necon3d 2949	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 ≠ 0 → (𝐴 gcd 𝐵) ≠ 0))
60	59	impr 454	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝐴 gcd 𝐵) ≠ 0)
61	60	3adant3 1132	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 gcd 𝐵) ≠ 0)
62	61	ad2antrr 726	. . . . . . . . . . . 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 11937	. . . . . . . . . . 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 223	. . . . . . . . . 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 12479	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℂ)
66	65	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) → 𝐵 ∈ ℂ)
67	66	3ad2ant2 1134	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → 𝐵 ∈ ℂ)
68	67	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → 𝐵 ∈ ℂ)
69	36	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → 𝑏 ∈ ℂ)
70	68, 69, 55, 62	divmul3d 11937	. . . . . . . . . . 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 1132	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ))
72		gcdcom 16430	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴))
73	71, 72	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴))
74	73	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴))
75	74	oveq2d 7368	. . . . . . . . . . . 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 2742	. . . . . . . . . . 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 280	. . . . . . . . . 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 632	. . . . . . . . 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 1094	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ↔ (𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)))
80	79	biimpri 228	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0))
81	80	3adant3 1132	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0))
82		divgcdcoprm0 16582	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) → ((𝐴 / (𝐴 gcd 𝐵)) gcd (𝐵 / (𝐴 gcd 𝐵))) = 1)
83	81, 82	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → ((𝐴 / (𝐴 gcd 𝐵)) gcd (𝐵 / (𝐴 gcd 𝐵))) = 1)
84		oveq12 7361	. . . . . . . . . . . 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 2733	. . . . . . . . . . 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 245	. . . . . . . . . 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 726	. . . . . . . . 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 240	. . . . . . . 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 406	. . . . . . 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 1128	. . . . . 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 412	. . . . 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 3145	. . . 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 3145	. . 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 243	. 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 699	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

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∃wrex 3056 (class class class)co 7352 ℂcc 11010 0cc0 11012 1c1 11013 · cmul 11017 / cdiv 11780 ℕ₀cn0 12387 ℤcz 12474 gcd cgcd 16411

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11068 ax-resscn 11069 ax-1cn 11070 ax-icn 11071 ax-addcl 11072 ax-addrcl 11073 ax-mulcl 11074 ax-mulrcl 11075 ax-mulcom 11076 ax-addass 11077 ax-mulass 11078 ax-distr 11079 ax-i2m1 11080 ax-1ne0 11081 ax-1rid 11082 ax-rnegex 11083 ax-rrecex 11084 ax-cnre 11085 ax-pre-lttri 11086 ax-pre-lttrn 11087 ax-pre-ltadd 11088 ax-pre-mulgt0 11089 ax-pre-sup 11090

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-er 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-sup 9332 df-inf 9333 df-pnf 11154 df-mnf 11155 df-xr 11156 df-ltxr 11157 df-le 11158 df-sub 11352 df-neg 11353 df-div 11781 df-nn 12132 df-2 12194 df-3 12195 df-n0 12388 df-z 12475 df-uz 12739 df-rp 12897 df-fl 13702 df-mod 13780 df-seq 13915 df-exp 13975 df-cj 15012 df-re 15013 df-im 15014 df-sqrt 15148 df-abs 15149 df-dvds 16170 df-gcd 16412

This theorem is referenced by: cncongr1 16584

W3C validator