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Theorem dfgcd2 10870

Description: Alternate definition of the gcd operator, see definition in [ApostolNT] p. 15. (Contributed by AV, 8-Aug-2021.)

**Assertion**
Ref	Expression
dfgcd2	⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐷 = (𝑀 gcd 𝑁) ↔ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷))))

Distinct variable groups: 𝐷,𝑒 𝑒,𝑀 𝑒,𝑁

Proof of Theorem dfgcd2 Dummy variables 𝑓 𝑔 𝑛 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gcdcl 10825	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℕ₀)
2	1	nn0ge0d 8655	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 0 ≤ (𝑀 gcd 𝑁))
3		gcddvds 10822	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁))
4		3anass 926	. . . . . . . 8 ⊢ ((𝑒 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ↔ (𝑒 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)))
5		ancom 262	. . . . . . . 8 ⊢ ((𝑒 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑒 ∈ ℤ))
6	4, 5	bitri 182	. . . . . . 7 ⊢ ((𝑒 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑒 ∈ ℤ))
7		dvdsgcd 10868	. . . . . . 7 ⊢ ((𝑒 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ (𝑀 gcd 𝑁)))
8	6, 7	sylbir 133	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑒 ∈ ℤ) → ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ (𝑀 gcd 𝑁)))
9	8	ralrimiva 2442	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ (𝑀 gcd 𝑁)))
10	2, 3, 9	3jca 1121	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ≤ (𝑀 gcd 𝑁) ∧ ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ (𝑀 gcd 𝑁))))
11	10	adantr 270	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐷 = (𝑀 gcd 𝑁)) → (0 ≤ (𝑀 gcd 𝑁) ∧ ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ (𝑀 gcd 𝑁))))
12		breq2 3824	. . . . 5 ⊢ (𝐷 = (𝑀 gcd 𝑁) → (0 ≤ 𝐷 ↔ 0 ≤ (𝑀 gcd 𝑁)))
13		breq1 3823	. . . . . 6 ⊢ (𝐷 = (𝑀 gcd 𝑁) → (𝐷 ∥ 𝑀 ↔ (𝑀 gcd 𝑁) ∥ 𝑀))
14		breq1 3823	. . . . . 6 ⊢ (𝐷 = (𝑀 gcd 𝑁) → (𝐷 ∥ 𝑁 ↔ (𝑀 gcd 𝑁) ∥ 𝑁))
15	13, 14	anbi12d 457	. . . . 5 ⊢ (𝐷 = (𝑀 gcd 𝑁) → ((𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ↔ ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁)))
16		breq2 3824	. . . . . . 7 ⊢ (𝐷 = (𝑀 gcd 𝑁) → (𝑒 ∥ 𝐷 ↔ 𝑒 ∥ (𝑀 gcd 𝑁)))
17	16	imbi2d 228	. . . . . 6 ⊢ (𝐷 = (𝑀 gcd 𝑁) → (((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷) ↔ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ (𝑀 gcd 𝑁))))
18	17	ralbidv 2376	. . . . 5 ⊢ (𝐷 = (𝑀 gcd 𝑁) → (∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷) ↔ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ (𝑀 gcd 𝑁))))
19	12, 15, 18	3anbi123d 1246	. . . 4 ⊢ (𝐷 = (𝑀 gcd 𝑁) → ((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)) ↔ (0 ≤ (𝑀 gcd 𝑁) ∧ ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ (𝑀 gcd 𝑁)))))
20	19	adantl 271	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐷 = (𝑀 gcd 𝑁)) → ((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)) ↔ (0 ≤ (𝑀 gcd 𝑁) ∧ ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ (𝑀 gcd 𝑁)))))
21	11, 20	mpbird 165	. 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐷 = (𝑀 gcd 𝑁)) → (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)))
22		gcdval 10818	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < )))
23	22	adantr 270	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷))) → (𝑀 gcd 𝑁) = if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < )))
24		iftrue 3384	. . . . . . 7 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < )) = 0)
25	24	adantr 270	. . . . . 6 ⊢ (((𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)))) → if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < )) = 0)
26		breq2 3824	. . . . . . . . . . 11 ⊢ (𝑀 = 0 → (𝐷 ∥ 𝑀 ↔ 𝐷 ∥ 0))
27		breq2 3824	. . . . . . . . . . 11 ⊢ (𝑁 = 0 → (𝐷 ∥ 𝑁 ↔ 𝐷 ∥ 0))
28	26, 27	bi2anan9 571	. . . . . . . . . 10 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → ((𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ↔ (𝐷 ∥ 0 ∧ 𝐷 ∥ 0)))
29		breq2 3824	. . . . . . . . . . . . 13 ⊢ (𝑀 = 0 → (𝑒 ∥ 𝑀 ↔ 𝑒 ∥ 0))
30		breq2 3824	. . . . . . . . . . . . 13 ⊢ (𝑁 = 0 → (𝑒 ∥ 𝑁 ↔ 𝑒 ∥ 0))
31	29, 30	bi2anan9 571	. . . . . . . . . . . 12 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) ↔ (𝑒 ∥ 0 ∧ 𝑒 ∥ 0)))
32	31	imbi1d 229	. . . . . . . . . . 11 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷) ↔ ((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → 𝑒 ∥ 𝐷)))
33	32	ralbidv 2376	. . . . . . . . . 10 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷) ↔ ∀𝑒 ∈ ℤ ((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → 𝑒 ∥ 𝐷)))
34	28, 33	3anbi23d 1249	. . . . . . . . 9 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → ((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)) ↔ (0 ≤ 𝐷 ∧ (𝐷 ∥ 0 ∧ 𝐷 ∥ 0) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → 𝑒 ∥ 𝐷))))
35		dvdszrcl 10668	. . . . . . . . . . . . 13 ⊢ (𝐷 ∥ 0 → (𝐷 ∈ ℤ ∧ 0 ∈ ℤ))
36		dvds0 10678	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑒 ∈ ℤ → 𝑒 ∥ 0)
37	36, 36	jca 300	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 ∈ ℤ → (𝑒 ∥ 0 ∧ 𝑒 ∥ 0))
38	37	adantl 271	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐷 ∈ ℤ ∧ 0 ≤ 𝐷) ∧ 𝑒 ∈ ℤ) → (𝑒 ∥ 0 ∧ 𝑒 ∥ 0))
39		pm5.5 240	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → (((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → 𝑒 ∥ 𝐷) ↔ 𝑒 ∥ 𝐷))
40	38, 39	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐷 ∈ ℤ ∧ 0 ≤ 𝐷) ∧ 𝑒 ∈ ℤ) → (((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → 𝑒 ∥ 𝐷) ↔ 𝑒 ∥ 𝐷))
41	40	ralbidva 2372	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ ℤ ∧ 0 ≤ 𝐷) → (∀𝑒 ∈ ℤ ((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → 𝑒 ∥ 𝐷) ↔ ∀𝑒 ∈ ℤ 𝑒 ∥ 𝐷))
42		0z 8687	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℤ
43		breq1 3823	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑒 = 0 → (𝑒 ∥ 𝐷 ↔ 0 ∥ 𝐷))
44	43	rspcv 2711	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 ∈ ℤ → (∀𝑒 ∈ ℤ 𝑒 ∥ 𝐷 → 0 ∥ 𝐷))
45	42, 44	ax-mp 7	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑒 ∈ ℤ 𝑒 ∥ 𝐷 → 0 ∥ 𝐷)
46		0dvds 10683	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐷 ∈ ℤ → (0 ∥ 𝐷 ↔ 𝐷 = 0))
47	46	biimpd 142	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐷 ∈ ℤ → (0 ∥ 𝐷 → 𝐷 = 0))
48		eqcom 2087	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 = 𝐷 ↔ 𝐷 = 0)
49	47, 48	syl6ibr 160	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐷 ∈ ℤ → (0 ∥ 𝐷 → 0 = 𝐷))
50	45, 49	syl5 32	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∈ ℤ → (∀𝑒 ∈ ℤ 𝑒 ∥ 𝐷 → 0 = 𝐷))
51	50	adantr 270	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ ℤ ∧ 0 ≤ 𝐷) → (∀𝑒 ∈ ℤ 𝑒 ∥ 𝐷 → 0 = 𝐷))
52	41, 51	sylbid 148	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ ℤ ∧ 0 ≤ 𝐷) → (∀𝑒 ∈ ℤ ((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → 𝑒 ∥ 𝐷) → 0 = 𝐷))
53	52	ex 113	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ ℤ → (0 ≤ 𝐷 → (∀𝑒 ∈ ℤ ((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → 𝑒 ∥ 𝐷) → 0 = 𝐷)))
54	53	adantr 270	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ ℤ ∧ 0 ∈ ℤ) → (0 ≤ 𝐷 → (∀𝑒 ∈ ℤ ((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → 𝑒 ∥ 𝐷) → 0 = 𝐷)))
55	35, 54	syl 14	. . . . . . . . . . . 12 ⊢ (𝐷 ∥ 0 → (0 ≤ 𝐷 → (∀𝑒 ∈ ℤ ((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → 𝑒 ∥ 𝐷) → 0 = 𝐷)))
56	55	adantr 270	. . . . . . . . . . 11 ⊢ ((𝐷 ∥ 0 ∧ 𝐷 ∥ 0) → (0 ≤ 𝐷 → (∀𝑒 ∈ ℤ ((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → 𝑒 ∥ 𝐷) → 0 = 𝐷)))
57	56	com12 30	. . . . . . . . . 10 ⊢ (0 ≤ 𝐷 → ((𝐷 ∥ 0 ∧ 𝐷 ∥ 0) → (∀𝑒 ∈ ℤ ((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → 𝑒 ∥ 𝐷) → 0 = 𝐷)))
58	57	3imp 1135	. . . . . . . . 9 ⊢ ((0 ≤ 𝐷 ∧ (𝐷 ∥ 0 ∧ 𝐷 ∥ 0) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → 𝑒 ∥ 𝐷)) → 0 = 𝐷)
59	34, 58	syl6bi 161	. . . . . . . 8 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → ((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)) → 0 = 𝐷))
60	59	adantld 272	. . . . . . 7 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷))) → 0 = 𝐷))
61	60	imp 122	. . . . . 6 ⊢ (((𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)))) → 0 = 𝐷)
62	25, 61	eqtrd 2117	. . . . 5 ⊢ (((𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)))) → if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < )) = 𝐷)
63	62	ancoms 264	. . . 4 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷))) ∧ (𝑀 = 0 ∧ 𝑁 = 0)) → if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < )) = 𝐷)
64		iffalse 3387	. . . . . . 7 ⊢ (¬ (𝑀 = 0 ∧ 𝑁 = 0) → if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < )) = sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ))
65	64	adantr 270	. . . . . 6 ⊢ ((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)))) → if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < )) = sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ))
66		lttri3 7502	. . . . . . . 8 ⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
67	66	adantl 271	. . . . . . 7 ⊢ (((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)))) ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
68		dvdszrcl 10668	. . . . . . . . . . . 12 ⊢ (𝐷 ∥ 𝑀 → (𝐷 ∈ ℤ ∧ 𝑀 ∈ ℤ))
69	68	simpld 110	. . . . . . . . . . 11 ⊢ (𝐷 ∥ 𝑀 → 𝐷 ∈ ℤ)
70	69	zred 8794	. . . . . . . . . 10 ⊢ (𝐷 ∥ 𝑀 → 𝐷 ∈ ℝ)
71	70	adantr 270	. . . . . . . . 9 ⊢ ((𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) → 𝐷 ∈ ℝ)
72	71	3ad2ant2 963	. . . . . . . 8 ⊢ ((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)) → 𝐷 ∈ ℝ)
73	72	ad2antll 475	. . . . . . 7 ⊢ ((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)))) → 𝐷 ∈ ℝ)
74		breq1 3823	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑦 → (𝑛 ∥ 𝑀 ↔ 𝑦 ∥ 𝑀))
75		breq1 3823	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑦 → (𝑛 ∥ 𝑁 ↔ 𝑦 ∥ 𝑁))
76	74, 75	anbi12d 457	. . . . . . . . . 10 ⊢ (𝑛 = 𝑦 → ((𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁) ↔ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)))
77	76	elrab 2762	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)} ↔ (𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)))
78		breq1 3823	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑒 = 𝑦 → (𝑒 ∥ 𝑀 ↔ 𝑦 ∥ 𝑀))
79		breq1 3823	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑒 = 𝑦 → (𝑒 ∥ 𝑁 ↔ 𝑦 ∥ 𝑁))
80	78, 79	anbi12d 457	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑒 = 𝑦 → ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) ↔ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)))
81		breq1 3823	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑒 = 𝑦 → (𝑒 ∥ 𝐷 ↔ 𝑦 ∥ 𝐷))
82	80, 81	imbi12d 232	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = 𝑦 → (((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷) ↔ ((𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁) → 𝑦 ∥ 𝐷)))
83	82	rspcv 2711	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℤ → (∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷) → ((𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁) → 𝑦 ∥ 𝐷)))
84	83	com23 77	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℤ → ((𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁) → (∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷) → 𝑦 ∥ 𝐷)))
85	84	imp 122	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) → (∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷) → 𝑦 ∥ 𝐷))
86	85	ad2antrr 472	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷) → 𝑦 ∥ 𝐷))
87		elnn0z 8689	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐷 ∈ ℕ₀ ↔ (𝐷 ∈ ℤ ∧ 0 ≤ 𝐷))
88	87	simplbi2 377	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐷 ∈ ℤ → (0 ≤ 𝐷 → 𝐷 ∈ ℕ₀))
89	88	adantr 270	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐷 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (0 ≤ 𝐷 → 𝐷 ∈ ℕ₀))
90	68, 89	syl 14	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐷 ∥ 𝑀 → (0 ≤ 𝐷 → 𝐷 ∈ ℕ₀))
91	90	adantr 270	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) → (0 ≤ 𝐷 → 𝐷 ∈ ℕ₀))
92	91	impcom 123	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)) → 𝐷 ∈ ℕ₀)
93		zre 8680	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℝ)
94	93	ad2antrr 472	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → 𝑦 ∈ ℝ)
95	94	ad2antrl 474	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)) ∧ (𝑦 ∥ 𝐷 ∧ 𝐷 ∈ ℕ₀)) ∧ (((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))) → 𝑦 ∈ ℝ)
96	71	ad3antlr 477	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)) ∧ (𝑦 ∥ 𝐷 ∧ 𝐷 ∈ ℕ₀)) ∧ (((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))) → 𝐷 ∈ ℝ)
97		simp-5l 510	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) ∧ 𝐷 ∈ ℕ₀) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁))) → 𝑦 ∈ ℤ)
98		elnn0 8601	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐷 ∈ ℕ₀ ↔ (𝐷 ∈ ℕ ∨ 𝐷 = 0))
99		2a1 25	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐷 ∈ ℕ → ((((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)) → 𝐷 ∈ ℕ)))
100		breq1 3823	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝐷 = 0 → (𝐷 ∥ 𝑀 ↔ 0 ∥ 𝑀))
101		breq1 3823	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝐷 = 0 → (𝐷 ∥ 𝑁 ↔ 0 ∥ 𝑁))
102	100, 101	anbi12d 457	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝐷 = 0 → ((𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ↔ (0 ∥ 𝑀 ∧ 0 ∥ 𝑁)))
103	102	anbi2d 452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝐷 = 0 → ((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)) ↔ (0 ≤ 𝐷 ∧ (0 ∥ 𝑀 ∧ 0 ∥ 𝑁))))
104	103	adantr 270	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐷 = 0 ∧ (((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))) → ((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)) ↔ (0 ≤ 𝐷 ∧ (0 ∥ 𝑀 ∧ 0 ∥ 𝑁))))
105		simplr 497	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ¬ (𝑀 = 0 ∧ 𝑁 = 0))
106		zdceq 8748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑀 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑀 = 0)
107	42, 106	mpan2 416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑀 ∈ ℤ → DECID 𝑀 = 0)
108		ianordc 835	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (DECID 𝑀 = 0 → (¬ (𝑀 = 0 ∧ 𝑁 = 0) ↔ (¬ 𝑀 = 0 ∨ ¬ 𝑁 = 0)))
109	107, 108	syl 14	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑀 ∈ ℤ → (¬ (𝑀 = 0 ∧ 𝑁 = 0) ↔ (¬ 𝑀 = 0 ∨ ¬ 𝑁 = 0)))
110	109	ad2antrl 474	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (¬ (𝑀 = 0 ∧ 𝑁 = 0) ↔ (¬ 𝑀 = 0 ∨ ¬ 𝑁 = 0)))
111	105, 110	mpbid 145	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (¬ 𝑀 = 0 ∨ ¬ 𝑁 = 0))
112		dvdszrcl 10668	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (0 ∥ 𝑀 → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ))
113		0dvds 10683	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑀 ∈ ℤ → (0 ∥ 𝑀 ↔ 𝑀 = 0))
114		pm2.24 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑀 = 0 → (¬ 𝑀 = 0 → 𝐷 ∈ ℕ))
115	113, 114	syl6bi 161	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑀 ∈ ℤ → (0 ∥ 𝑀 → (¬ 𝑀 = 0 → 𝐷 ∈ ℕ)))
116	115	adantl 271	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (0 ∥ 𝑀 → (¬ 𝑀 = 0 → 𝐷 ∈ ℕ)))
117	112, 116	mpcom 36	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (0 ∥ 𝑀 → (¬ 𝑀 = 0 → 𝐷 ∈ ℕ))
118	117	adantr 270	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((0 ∥ 𝑀 ∧ 0 ∥ 𝑁) → (¬ 𝑀 = 0 → 𝐷 ∈ ℕ))
119	118	com12 30	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (¬ 𝑀 = 0 → ((0 ∥ 𝑀 ∧ 0 ∥ 𝑁) → 𝐷 ∈ ℕ))
120		dvdszrcl 10668	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (0 ∥ 𝑁 → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ))
121		0dvds 10683	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ 𝑁 = 0))
122		pm2.24 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑁 = 0 → (¬ 𝑁 = 0 → 𝐷 ∈ ℕ))
123	121, 122	syl6bi 161	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 → (¬ 𝑁 = 0 → 𝐷 ∈ ℕ)))
124	123	adantl 271	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ 𝑁 → (¬ 𝑁 = 0 → 𝐷 ∈ ℕ)))
125	120, 124	mpcom 36	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (0 ∥ 𝑁 → (¬ 𝑁 = 0 → 𝐷 ∈ ℕ))
126	125	adantl 271	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((0 ∥ 𝑀 ∧ 0 ∥ 𝑁) → (¬ 𝑁 = 0 → 𝐷 ∈ ℕ))
127	126	com12 30	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (¬ 𝑁 = 0 → ((0 ∥ 𝑀 ∧ 0 ∥ 𝑁) → 𝐷 ∈ ℕ))
128	119, 127	jaoi 669	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((¬ 𝑀 = 0 ∨ ¬ 𝑁 = 0) → ((0 ∥ 𝑀 ∧ 0 ∥ 𝑁) → 𝐷 ∈ ℕ))
129	111, 128	syl 14	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((0 ∥ 𝑀 ∧ 0 ∥ 𝑁) → 𝐷 ∈ ℕ))
130	129	adantld 272	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((0 ≤ 𝐷 ∧ (0 ∥ 𝑀 ∧ 0 ∥ 𝑁)) → 𝐷 ∈ ℕ))
131	130	adantl 271	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐷 = 0 ∧ (((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))) → ((0 ≤ 𝐷 ∧ (0 ∥ 𝑀 ∧ 0 ∥ 𝑁)) → 𝐷 ∈ ℕ))
132	104, 131	sylbid 148	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐷 = 0 ∧ (((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))) → ((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)) → 𝐷 ∈ ℕ))
133	132	ex 113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐷 = 0 → ((((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)) → 𝐷 ∈ ℕ)))
134	99, 133	jaoi 669	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐷 ∈ ℕ ∨ 𝐷 = 0) → ((((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)) → 𝐷 ∈ ℕ)))
135	98, 134	sylbi 119	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐷 ∈ ℕ₀ → ((((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)) → 𝐷 ∈ ℕ)))
136	135	impcom 123	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) ∧ 𝐷 ∈ ℕ₀) → ((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)) → 𝐷 ∈ ℕ))
137	136	imp 122	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) ∧ 𝐷 ∈ ℕ₀) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁))) → 𝐷 ∈ ℕ)
138		dvdsle 10712	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∈ ℤ ∧ 𝐷 ∈ ℕ) → (𝑦 ∥ 𝐷 → 𝑦 ≤ 𝐷))
139	97, 137, 138	syl2anc 403	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) ∧ 𝐷 ∈ ℕ₀) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁))) → (𝑦 ∥ 𝐷 → 𝑦 ≤ 𝐷))
140	139	exp31 356	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝐷 ∈ ℕ₀ → ((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)) → (𝑦 ∥ 𝐷 → 𝑦 ≤ 𝐷))))
141	140	com14 87	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∥ 𝐷 → (𝐷 ∈ ℕ₀ → ((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)) → ((((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑦 ≤ 𝐷))))
142	141	imp 122	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∥ 𝐷 ∧ 𝐷 ∈ ℕ₀) → ((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)) → ((((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑦 ≤ 𝐷)))
143	142	impcom 123	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)) ∧ (𝑦 ∥ 𝐷 ∧ 𝐷 ∈ ℕ₀)) → ((((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑦 ≤ 𝐷))
144	143	imp 122	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)) ∧ (𝑦 ∥ 𝐷 ∧ 𝐷 ∈ ℕ₀)) ∧ (((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))) → 𝑦 ≤ 𝐷)
145	95, 96, 144	lensymd 7542	. . . . . . . . . . . . . . . . . 18 ⊢ ((((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)) ∧ (𝑦 ∥ 𝐷 ∧ 𝐷 ∈ ℕ₀)) ∧ (((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))) → ¬ 𝐷 < 𝑦)
146	145	exp31 356	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)) → ((𝑦 ∥ 𝐷 ∧ 𝐷 ∈ ℕ₀) → ((((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ¬ 𝐷 < 𝑦)))
147	92, 146	mpan2d 419	. . . . . . . . . . . . . . . 16 ⊢ ((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)) → (𝑦 ∥ 𝐷 → ((((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ¬ 𝐷 < 𝑦)))
148	147	com13 79	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑦 ∥ 𝐷 → ((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)) → ¬ 𝐷 < 𝑦)))
149	86, 148	syld 44	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷) → ((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)) → ¬ 𝐷 < 𝑦)))
150	149	com13 79	. . . . . . . . . . . . 13 ⊢ ((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)) → (∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷) → ((((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ¬ 𝐷 < 𝑦)))
151	150	3impia 1138	. . . . . . . . . . . 12 ⊢ ((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)) → ((((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ¬ 𝐷 < 𝑦))
152	151	com12 30	. . . . . . . . . . 11 ⊢ ((((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)) → ¬ 𝐷 < 𝑦))
153	152	expimpd 355	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷))) → ¬ 𝐷 < 𝑦))
154	153	expimpd 355	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℤ ∧ (𝑦 ∥ 𝑀 ∧ 𝑦 ∥ 𝑁)) → ((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)))) → ¬ 𝐷 < 𝑦))
155	77, 154	sylbi 119	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)} → ((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)))) → ¬ 𝐷 < 𝑦))
156	155	impcom 123	. . . . . . 7 ⊢ (((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)))) ∧ 𝑦 ∈ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}) → ¬ 𝐷 < 𝑦)
157	69	adantr 270	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) → 𝐷 ∈ ℤ)
158	157	ancri 317	. . . . . . . . . . . 12 ⊢ ((𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) → (𝐷 ∈ ℤ ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)))
159	158	3ad2ant2 963	. . . . . . . . . . 11 ⊢ ((0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)) → (𝐷 ∈ ℤ ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)))
160	159	ad2antll 475	. . . . . . . . . 10 ⊢ ((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)))) → (𝐷 ∈ ℤ ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)))
161	160	adantr 270	. . . . . . . . 9 ⊢ (((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)))) ∧ (𝑦 ∈ ℝ ∧ 𝑦 < 𝐷)) → (𝐷 ∈ ℤ ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)))
162		breq1 3823	. . . . . . . . . . 11 ⊢ (𝑛 = 𝐷 → (𝑛 ∥ 𝑀 ↔ 𝐷 ∥ 𝑀))
163		breq1 3823	. . . . . . . . . . 11 ⊢ (𝑛 = 𝐷 → (𝑛 ∥ 𝑁 ↔ 𝐷 ∥ 𝑁))
164	162, 163	anbi12d 457	. . . . . . . . . 10 ⊢ (𝑛 = 𝐷 → ((𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁) ↔ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)))
165	164	elrab 2762	. . . . . . . . 9 ⊢ (𝐷 ∈ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)} ↔ (𝐷 ∈ ℤ ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁)))
166	161, 165	sylibr 132	. . . . . . . 8 ⊢ (((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)))) ∧ (𝑦 ∈ ℝ ∧ 𝑦 < 𝐷)) → 𝐷 ∈ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)})
167		breq2 3824	. . . . . . . . 9 ⊢ (𝑧 = 𝐷 → (𝑦 < 𝑧 ↔ 𝑦 < 𝐷))
168	167	adantl 271	. . . . . . . 8 ⊢ ((((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)))) ∧ (𝑦 ∈ ℝ ∧ 𝑦 < 𝐷)) ∧ 𝑧 = 𝐷) → (𝑦 < 𝑧 ↔ 𝑦 < 𝐷))
169		simprr 499	. . . . . . . 8 ⊢ (((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)))) ∧ (𝑦 ∈ ℝ ∧ 𝑦 < 𝐷)) → 𝑦 < 𝐷)
170	166, 168, 169	rspcedvd 2721	. . . . . . 7 ⊢ (((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)))) ∧ (𝑦 ∈ ℝ ∧ 𝑦 < 𝐷)) → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}𝑦 < 𝑧)
171	67, 73, 156, 170	eqsuptid 6629	. . . . . 6 ⊢ ((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)))) → sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ) = 𝐷)
172	65, 171	eqtrd 2117	. . . . 5 ⊢ ((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)))) → if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < )) = 𝐷)
173	172	ancoms 264	. . . 4 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷))) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < )) = 𝐷)
174		gcdmndc 10807	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID (𝑀 = 0 ∧ 𝑁 = 0))
175	174	adantr 270	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷))) → DECID (𝑀 = 0 ∧ 𝑁 = 0))
176		exmiddc 780	. . . . 5 ⊢ (DECID (𝑀 = 0 ∧ 𝑁 = 0) → ((𝑀 = 0 ∧ 𝑁 = 0) ∨ ¬ (𝑀 = 0 ∧ 𝑁 = 0)))
177	175, 176	syl 14	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷))) → ((𝑀 = 0 ∧ 𝑁 = 0) ∨ ¬ (𝑀 = 0 ∧ 𝑁 = 0)))
178	63, 173, 177	mpjaodan 745	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷))) → if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < )) = 𝐷)
179	23, 178	eqtr2d 2118	. 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷))) → 𝐷 = (𝑀 gcd 𝑁))
180	21, 179	impbida 561	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐷 = (𝑀 gcd 𝑁) ↔ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷))))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 ∨ wo 662 DECID wdc 778 ∧ w3a 922 = wceq 1287 ∈ wcel 1436 ∀wral 2355 {crab 2359 ifcif 3379 class class class wbr 3820 (class class class)co 5607 supcsup 6614 ℝcr 7286 0cc0 7287 < clt 7459 ≤ cle 7460 ℕcn 8350 ℕ₀cn0 8599 ℤcz 8676 ∥ cdvds 10663 gcd cgcd 10805

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-coll 3928 ax-sep 3931 ax-nul 3939 ax-pow 3983 ax-pr 4009 ax-un 4233 ax-setind 4325 ax-iinf 4375 ax-cnex 7373 ax-resscn 7374 ax-1cn 7375 ax-1re 7376 ax-icn 7377 ax-addcl 7378 ax-addrcl 7379 ax-mulcl 7380 ax-mulrcl 7381 ax-addcom 7382 ax-mulcom 7383 ax-addass 7384 ax-mulass 7385 ax-distr 7386 ax-i2m1 7387 ax-0lt1 7388 ax-1rid 7389 ax-0id 7390 ax-rnegex 7391 ax-precex 7392 ax-cnre 7393 ax-pre-ltirr 7394 ax-pre-ltwlin 7395 ax-pre-lttrn 7396 ax-pre-apti 7397 ax-pre-ltadd 7398 ax-pre-mulgt0 7399 ax-pre-mulext 7400 ax-arch 7401 ax-caucvg 7402

This theorem depends on definitions: df-bi 115 df-dc 779 df-3or 923 df-3an 924 df-tru 1290 df-fal 1293 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ne 2252 df-nel 2347 df-ral 2360 df-rex 2361 df-reu 2362 df-rmo 2363 df-rab 2364 df-v 2617 df-sbc 2830 df-csb 2923 df-dif 2990 df-un 2992 df-in 2994 df-ss 3001 df-nul 3276 df-if 3380 df-pw 3417 df-sn 3437 df-pr 3438 df-op 3440 df-uni 3637 df-int 3672 df-iun 3715 df-br 3821 df-opab 3875 df-mpt 3876 df-tr 3911 df-id 4093 df-po 4096 df-iso 4097 df-iord 4166 df-on 4168 df-ilim 4169 df-suc 4171 df-iom 4378 df-xp 4416 df-rel 4417 df-cnv 4418 df-co 4419 df-dm 4420 df-rn 4421 df-res 4422 df-ima 4423 df-iota 4943 df-fun 4980 df-fn 4981 df-f 4982 df-f1 4983 df-fo 4984 df-f1o 4985 df-fv 4986 df-riota 5563 df-ov 5610 df-oprab 5611 df-mpt2 5612 df-1st 5862 df-2nd 5863 df-recs 6018 df-frec 6104 df-sup 6616 df-pnf 7461 df-mnf 7462 df-xr 7463 df-ltxr 7464 df-le 7465 df-sub 7592 df-neg 7593 df-reap 7986 df-ap 7993 df-div 8072 df-inn 8351 df-2 8409 df-3 8410 df-4 8411 df-n0 8600 df-z 8677 df-uz 8945 df-q 9030 df-rp 9060 df-fz 9350 df-fzo 9475 df-fl 9598 df-mod 9651 df-iseq 9773 df-iexp 9846 df-cj 10164 df-re 10165 df-im 10166 df-rsqrt 10319 df-abs 10320 df-dvds 10664 df-gcd 10806

This theorem is referenced by: (None)

W3C validator