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Theorem divgcdcoprmex 16699

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 16543	. . . 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 1131	. 2 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → ∃𝑎 ∈ ℤ 𝐴 = (𝑎 · (𝐴 gcd 𝐵)))
6		zeqzmulgcd 16543	. . . . 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 1131	. 2 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → ∃𝑏 ∈ ℤ 𝐵 = (𝑏 · (𝐵 gcd 𝐴)))
10		reeanv 3226	. . 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 12615	. . . . . . . . . . . 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 16539	. . . . . . . . . . . . . . 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 12586	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝐴 gcd 𝐵) ∈ ℂ)
16	15	3adant3 1131	. . . . . . . . . . . 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 11279	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑎 ∈ ℤ) → (𝑎 · (𝐴 gcd 𝐵)) = ((𝐴 gcd 𝐵) · 𝑎))
19		simp3 1137	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → 𝑀 = (𝐴 gcd 𝐵))
20	19	eqcomd 2740	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 gcd 𝐵) = 𝑀)
21	20	oveq1d 7445	. . . . . . . . . . 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 2774	. . . . . . . . 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 2738	. . . . . . . . . 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 16546	. . . . . . . . . . . . . 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 1131	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐵 gcd 𝐴) = (𝐴 gcd 𝐵))
34	33	oveq2d 7446	. . . . . . . . . . 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 12615	. . . . . . . . . . . 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 1131	. . . . . . . . . . . . 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 12586	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → (𝐴 gcd 𝐵) ∈ ℂ)
41	37, 40	mulcomd 11279	. . . . . . . . . 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 7445	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) ∧ 𝑏 ∈ ℤ) → ((𝐴 gcd 𝐵) · 𝑏) = (𝑀 · 𝑏))
44	35, 41, 43	3eqtrd 2778	. . . . . . . . 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 2796	. . . . . . 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 12615	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
48	47	3ad2ant1 1132	. . . . . . . . . . . . 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 1135	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → 𝐴 ∈ ℤ)
52	1	3ad2ant2 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → 𝐵 ∈ ℤ)
53	51, 52	gcdcld 16541	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 gcd 𝐵) ∈ ℕ₀)
54	53	nn0cnd 12586	. . . . . . . . . . . . 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 16550	. . . . . . . . . . . . . . . . 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 2958	. . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . . 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 12074	. . . . . . . . . . 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 12615	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℂ)
66	65	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) → 𝐵 ∈ ℂ)
67	66	3ad2ant2 1133	. . . . . . . . . . . . 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 12074	. . . . . . . . . . 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 1131	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ))
72		gcdcom 16546	. . . . . . . . . . . . . . 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 7446	. . . . . . . . . . . 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 2745	. . . . . . . . . . 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 1131	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0))
82		divgcdcoprm0 16698	. . . . . . . . . . . 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 7439	. . . . . . . . . . . 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 2736	. . . . . . . . . . 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 1127	. . . . . 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 3165	. . . 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 3165	. . 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 1536 ∈ wcel 2105 ≠ wne 2937 ∃wrex 3067 (class class class)co 7430 ℂcc 11150 0cc0 11152 1c1 11153 · cmul 11157 / cdiv 11917 ℕ₀cn0 12523 ℤcz 12610 gcd cgcd 16527

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-dvds 16287 df-gcd 16528

This theorem is referenced by: cncongr1 16700

W3C validator