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Theorem divgcdcoprmex 16713

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 616	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ))
3		zeqzmulgcd 16556	. . . 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 16556	. . . . 5 ⊢ ((𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ) → ∃𝑏 ∈ ℤ 𝐵 = (𝑏 · (𝐵 gcd 𝐴)))
7	6	adantlr 714	. . . 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 3235	. . 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 12644	. . . . . . . . . . . 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 16552	. . . . . . . . . . . . . . 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 12615	. . . . . . . . . . . . 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 11311	. . . . . . . . . 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 2746	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 gcd 𝐵) = 𝑀)
21	20	oveq1d 7463	. . . . . . . . . . 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 2780	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) → (𝑎 · (𝐴 gcd 𝐵)) = (𝑀 · 𝑎))
24	23	ad2antrr 725	. . . . . . . 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 2744	. . . . . . . . . 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 16559	. . . . . . . . . . . . . 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 7464	. . . . . . . . . . 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 12644	. . . . . . . . . . . 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 12615	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → (𝐴 gcd 𝐵) ∈ ℂ)
41	37, 40	mulcomd 11311	. . . . . . . . . 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 7463	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → ((𝐴 gcd 𝐵) · 𝑏) = (𝑀 · 𝑏))
44	35, 41, 43	3eqtrd 2784	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → (𝑏 · (𝐵 gcd 𝐴)) = (𝑀 · 𝑏))
45	44	adantlr 714	. . . . . . . 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 2802	. . . . . . 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 12644	. . . . . . . . . . . . . 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 725	. . . . . . . . . . . 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 16554	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 gcd 𝐵) ∈ ℕ₀)
54	53	nn0cnd 12615	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 gcd 𝐵) ∈ ℂ)
55	54	ad2antrr 725	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (𝐴 gcd 𝐵) ∈ ℂ)
56		gcdeq0 16563	. . . . . . . . . . . . . . . . 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 2967	. . . . . . . . . . . . . . 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 725	. . . . . . . . . . . 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 12104	. . . . . . . . . . 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 12644	. . . . . . . . . . . . . . 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 725	. . . . . . . . . . . 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 12104	. . . . . . . . . . 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 16559	. . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴))
75	74	oveq2d 7464	. . . . . . . . . . . 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 2751	. . . . . . . . . . 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 631	. . . . . . . . 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 1095	. . . . . . . . . . . . . 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 16712	. . . . . . . . . . . 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 7457	. . . . . . . . . . . 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 2742	. . . . . . . . . . 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 725	. . . . . . . . 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 3174	. . . 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 3174	. . 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 698	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 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∃wrex 3076 (class class class)co 7448 ℂcc 11182 0cc0 11184 1c1 11185 · cmul 11189 / cdiv 11947 ℕ₀cn0 12553 ℤcz 12639 gcd cgcd 16540

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-sup 9511 df-inf 9512 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-dvds 16303 df-gcd 16541

This theorem is referenced by: cncongr1 16714

W3C validator