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Theorem dfgcd2 15182
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 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷))))
Distinct variable groups:   𝐷,𝑒   𝑒,𝑀   𝑒,𝑁

Proof of Theorem dfgcd2
Dummy variables 𝑛 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gcdcl 15147 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℕ0)
21nn0ge0d 11299 . . . . 5 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 0 ≤ (𝑀 gcd 𝑁))
3 gcddvds 15144 . . . . 5 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁))
4 3anass 1040 . . . . . . . 8 ((𝑒 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ↔ (𝑒 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)))
5 ancom 466 . . . . . . . 8 ((𝑒 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑒 ∈ ℤ))
64, 5bitri 264 . . . . . . 7 ((𝑒 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑒 ∈ ℤ))
7 dvdsgcd 15180 . . . . . . 7 ((𝑒 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑒𝑀𝑒𝑁) → 𝑒 ∥ (𝑀 gcd 𝑁)))
86, 7sylbir 225 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑒 ∈ ℤ) → ((𝑒𝑀𝑒𝑁) → 𝑒 ∥ (𝑀 gcd 𝑁)))
98ralrimiva 2965 . . . . 5 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒 ∥ (𝑀 gcd 𝑁)))
102, 3, 93jca 1240 . . . 4 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ≤ (𝑀 gcd 𝑁) ∧ ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒 ∥ (𝑀 gcd 𝑁))))
1110adantr 481 . . 3 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐷 = (𝑀 gcd 𝑁)) → (0 ≤ (𝑀 gcd 𝑁) ∧ ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒 ∥ (𝑀 gcd 𝑁))))
12 breq2 4622 . . . . 5 (𝐷 = (𝑀 gcd 𝑁) → (0 ≤ 𝐷 ↔ 0 ≤ (𝑀 gcd 𝑁)))
13 breq1 4621 . . . . . 6 (𝐷 = (𝑀 gcd 𝑁) → (𝐷𝑀 ↔ (𝑀 gcd 𝑁) ∥ 𝑀))
14 breq1 4621 . . . . . 6 (𝐷 = (𝑀 gcd 𝑁) → (𝐷𝑁 ↔ (𝑀 gcd 𝑁) ∥ 𝑁))
1513, 14anbi12d 746 . . . . 5 (𝐷 = (𝑀 gcd 𝑁) → ((𝐷𝑀𝐷𝑁) ↔ ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁)))
16 breq2 4622 . . . . . . 7 (𝐷 = (𝑀 gcd 𝑁) → (𝑒𝐷𝑒 ∥ (𝑀 gcd 𝑁)))
1716imbi2d 330 . . . . . 6 (𝐷 = (𝑀 gcd 𝑁) → (((𝑒𝑀𝑒𝑁) → 𝑒𝐷) ↔ ((𝑒𝑀𝑒𝑁) → 𝑒 ∥ (𝑀 gcd 𝑁))))
1817ralbidv 2985 . . . . 5 (𝐷 = (𝑀 gcd 𝑁) → (∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷) ↔ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒 ∥ (𝑀 gcd 𝑁))))
1912, 15, 183anbi123d 1396 . . . 4 (𝐷 = (𝑀 gcd 𝑁) → ((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷)) ↔ (0 ≤ (𝑀 gcd 𝑁) ∧ ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒 ∥ (𝑀 gcd 𝑁)))))
2019adantl 482 . . 3 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐷 = (𝑀 gcd 𝑁)) → ((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷)) ↔ (0 ≤ (𝑀 gcd 𝑁) ∧ ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒 ∥ (𝑀 gcd 𝑁)))))
2111, 20mpbird 247 . 2 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐷 = (𝑀 gcd 𝑁)) → (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷)))
22 gcdval 15137 . . . 4 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑁)}, ℝ, < )))
2322adantr 481 . . 3 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷))) → (𝑀 gcd 𝑁) = if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑁)}, ℝ, < )))
24 iftrue 4069 . . . . . 6 ((𝑀 = 0 ∧ 𝑁 = 0) → if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑁)}, ℝ, < )) = 0)
2524adantr 481 . . . . 5 (((𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷)))) → if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑁)}, ℝ, < )) = 0)
26 breq2 4622 . . . . . . . . . 10 (𝑀 = 0 → (𝐷𝑀𝐷 ∥ 0))
27 breq2 4622 . . . . . . . . . 10 (𝑁 = 0 → (𝐷𝑁𝐷 ∥ 0))
2826, 27bi2anan9 916 . . . . . . . . 9 ((𝑀 = 0 ∧ 𝑁 = 0) → ((𝐷𝑀𝐷𝑁) ↔ (𝐷 ∥ 0 ∧ 𝐷 ∥ 0)))
29 breq2 4622 . . . . . . . . . . . 12 (𝑀 = 0 → (𝑒𝑀𝑒 ∥ 0))
30 breq2 4622 . . . . . . . . . . . 12 (𝑁 = 0 → (𝑒𝑁𝑒 ∥ 0))
3129, 30bi2anan9 916 . . . . . . . . . . 11 ((𝑀 = 0 ∧ 𝑁 = 0) → ((𝑒𝑀𝑒𝑁) ↔ (𝑒 ∥ 0 ∧ 𝑒 ∥ 0)))
3231imbi1d 331 . . . . . . . . . 10 ((𝑀 = 0 ∧ 𝑁 = 0) → (((𝑒𝑀𝑒𝑁) → 𝑒𝐷) ↔ ((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → 𝑒𝐷)))
3332ralbidv 2985 . . . . . . . . 9 ((𝑀 = 0 ∧ 𝑁 = 0) → (∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷) ↔ ∀𝑒 ∈ ℤ ((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → 𝑒𝐷)))
3428, 333anbi23d 1399 . . . . . . . 8 ((𝑀 = 0 ∧ 𝑁 = 0) → ((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷)) ↔ (0 ≤ 𝐷 ∧ (𝐷 ∥ 0 ∧ 𝐷 ∥ 0) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → 𝑒𝐷))))
35 dvdszrcl 14907 . . . . . . . . . . . 12 (𝐷 ∥ 0 → (𝐷 ∈ ℤ ∧ 0 ∈ ℤ))
36 dvds0 14916 . . . . . . . . . . . . . . . . . . 19 (𝑒 ∈ ℤ → 𝑒 ∥ 0)
3736, 36jca 554 . . . . . . . . . . . . . . . . . 18 (𝑒 ∈ ℤ → (𝑒 ∥ 0 ∧ 𝑒 ∥ 0))
3837adantl 482 . . . . . . . . . . . . . . . . 17 (((𝐷 ∈ ℤ ∧ 0 ≤ 𝐷) ∧ 𝑒 ∈ ℤ) → (𝑒 ∥ 0 ∧ 𝑒 ∥ 0))
39 pm5.5 351 . . . . . . . . . . . . . . . . 17 ((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → (((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → 𝑒𝐷) ↔ 𝑒𝐷))
4038, 39syl 17 . . . . . . . . . . . . . . . 16 (((𝐷 ∈ ℤ ∧ 0 ≤ 𝐷) ∧ 𝑒 ∈ ℤ) → (((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → 𝑒𝐷) ↔ 𝑒𝐷))
4140ralbidva 2984 . . . . . . . . . . . . . . 15 ((𝐷 ∈ ℤ ∧ 0 ≤ 𝐷) → (∀𝑒 ∈ ℤ ((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → 𝑒𝐷) ↔ ∀𝑒 ∈ ℤ 𝑒𝐷))
42 0z 11333 . . . . . . . . . . . . . . . . . 18 0 ∈ ℤ
43 breq1 4621 . . . . . . . . . . . . . . . . . . 19 (𝑒 = 0 → (𝑒𝐷 ↔ 0 ∥ 𝐷))
4443rspcv 3296 . . . . . . . . . . . . . . . . . 18 (0 ∈ ℤ → (∀𝑒 ∈ ℤ 𝑒𝐷 → 0 ∥ 𝐷))
4542, 44ax-mp 5 . . . . . . . . . . . . . . . . 17 (∀𝑒 ∈ ℤ 𝑒𝐷 → 0 ∥ 𝐷)
46 0dvds 14921 . . . . . . . . . . . . . . . . . . 19 (𝐷 ∈ ℤ → (0 ∥ 𝐷𝐷 = 0))
4746biimpd 219 . . . . . . . . . . . . . . . . . 18 (𝐷 ∈ ℤ → (0 ∥ 𝐷𝐷 = 0))
48 eqcom 2633 . . . . . . . . . . . . . . . . . 18 (0 = 𝐷𝐷 = 0)
4947, 48syl6ibr 242 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ ℤ → (0 ∥ 𝐷 → 0 = 𝐷))
5045, 49syl5 34 . . . . . . . . . . . . . . . 16 (𝐷 ∈ ℤ → (∀𝑒 ∈ ℤ 𝑒𝐷 → 0 = 𝐷))
5150adantr 481 . . . . . . . . . . . . . . 15 ((𝐷 ∈ ℤ ∧ 0 ≤ 𝐷) → (∀𝑒 ∈ ℤ 𝑒𝐷 → 0 = 𝐷))
5241, 51sylbid 230 . . . . . . . . . . . . . 14 ((𝐷 ∈ ℤ ∧ 0 ≤ 𝐷) → (∀𝑒 ∈ ℤ ((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → 𝑒𝐷) → 0 = 𝐷))
5352ex 450 . . . . . . . . . . . . 13 (𝐷 ∈ ℤ → (0 ≤ 𝐷 → (∀𝑒 ∈ ℤ ((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → 𝑒𝐷) → 0 = 𝐷)))
5453adantr 481 . . . . . . . . . . . 12 ((𝐷 ∈ ℤ ∧ 0 ∈ ℤ) → (0 ≤ 𝐷 → (∀𝑒 ∈ ℤ ((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → 𝑒𝐷) → 0 = 𝐷)))
5535, 54syl 17 . . . . . . . . . . 11 (𝐷 ∥ 0 → (0 ≤ 𝐷 → (∀𝑒 ∈ ℤ ((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → 𝑒𝐷) → 0 = 𝐷)))
5655adantr 481 . . . . . . . . . 10 ((𝐷 ∥ 0 ∧ 𝐷 ∥ 0) → (0 ≤ 𝐷 → (∀𝑒 ∈ ℤ ((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → 𝑒𝐷) → 0 = 𝐷)))
5756com12 32 . . . . . . . . 9 (0 ≤ 𝐷 → ((𝐷 ∥ 0 ∧ 𝐷 ∥ 0) → (∀𝑒 ∈ ℤ ((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → 𝑒𝐷) → 0 = 𝐷)))
58573imp 1254 . . . . . . . 8 ((0 ≤ 𝐷 ∧ (𝐷 ∥ 0 ∧ 𝐷 ∥ 0) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 0 ∧ 𝑒 ∥ 0) → 𝑒𝐷)) → 0 = 𝐷)
5934, 58syl6bi 243 . . . . . . 7 ((𝑀 = 0 ∧ 𝑁 = 0) → ((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷)) → 0 = 𝐷))
6059adantld 483 . . . . . 6 ((𝑀 = 0 ∧ 𝑁 = 0) → (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷))) → 0 = 𝐷))
6160imp 445 . . . . 5 (((𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷)))) → 0 = 𝐷)
6225, 61eqtrd 2660 . . . 4 (((𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷)))) → if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑁)}, ℝ, < )) = 𝐷)
63 iffalse 4072 . . . . . 6 (¬ (𝑀 = 0 ∧ 𝑁 = 0) → if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑁)}, ℝ, < )) = sup({𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑁)}, ℝ, < ))
6463adantr 481 . . . . 5 ((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷)))) → if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑁)}, ℝ, < )) = sup({𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑁)}, ℝ, < ))
65 ltso 10063 . . . . . . 7 < Or ℝ
6665a1i 11 . . . . . 6 ((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷)))) → < Or ℝ)
67 dvdszrcl 14907 . . . . . . . . . . 11 (𝐷𝑀 → (𝐷 ∈ ℤ ∧ 𝑀 ∈ ℤ))
6867simpld 475 . . . . . . . . . 10 (𝐷𝑀𝐷 ∈ ℤ)
6968zred 11426 . . . . . . . . 9 (𝐷𝑀𝐷 ∈ ℝ)
7069adantr 481 . . . . . . . 8 ((𝐷𝑀𝐷𝑁) → 𝐷 ∈ ℝ)
71703ad2ant2 1081 . . . . . . 7 ((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷)) → 𝐷 ∈ ℝ)
7271ad2antll 764 . . . . . 6 ((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷)))) → 𝐷 ∈ ℝ)
73 breq1 4621 . . . . . . . . . 10 (𝑛 = 𝑦 → (𝑛𝑀𝑦𝑀))
74 breq1 4621 . . . . . . . . . 10 (𝑛 = 𝑦 → (𝑛𝑁𝑦𝑁))
7573, 74anbi12d 746 . . . . . . . . 9 (𝑛 = 𝑦 → ((𝑛𝑀𝑛𝑁) ↔ (𝑦𝑀𝑦𝑁)))
7675elrab 3351 . . . . . . . 8 (𝑦 ∈ {𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑁)} ↔ (𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)))
77 breq1 4621 . . . . . . . . . . . . . . . . . . . . 21 (𝑒 = 𝑦 → (𝑒𝑀𝑦𝑀))
78 breq1 4621 . . . . . . . . . . . . . . . . . . . . 21 (𝑒 = 𝑦 → (𝑒𝑁𝑦𝑁))
7977, 78anbi12d 746 . . . . . . . . . . . . . . . . . . . 20 (𝑒 = 𝑦 → ((𝑒𝑀𝑒𝑁) ↔ (𝑦𝑀𝑦𝑁)))
80 breq1 4621 . . . . . . . . . . . . . . . . . . . 20 (𝑒 = 𝑦 → (𝑒𝐷𝑦𝐷))
8179, 80imbi12d 334 . . . . . . . . . . . . . . . . . . 19 (𝑒 = 𝑦 → (((𝑒𝑀𝑒𝑁) → 𝑒𝐷) ↔ ((𝑦𝑀𝑦𝑁) → 𝑦𝐷)))
8281rspcv 3296 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℤ → (∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷) → ((𝑦𝑀𝑦𝑁) → 𝑦𝐷)))
8382com23 86 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℤ → ((𝑦𝑀𝑦𝑁) → (∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷) → 𝑦𝐷)))
8483imp 445 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) → (∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷) → 𝑦𝐷))
8584ad2antrr 761 . . . . . . . . . . . . . . 15 ((((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷) → 𝑦𝐷))
86 elnn0z 11335 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐷 ∈ ℕ0 ↔ (𝐷 ∈ ℤ ∧ 0 ≤ 𝐷))
8786simplbi2 654 . . . . . . . . . . . . . . . . . . . . . 22 (𝐷 ∈ ℤ → (0 ≤ 𝐷𝐷 ∈ ℕ0))
8887adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝐷 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (0 ≤ 𝐷𝐷 ∈ ℕ0))
8967, 88syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝐷𝑀 → (0 ≤ 𝐷𝐷 ∈ ℕ0))
9089adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝐷𝑀𝐷𝑁) → (0 ≤ 𝐷𝐷 ∈ ℕ0))
9190impcom 446 . . . . . . . . . . . . . . . . . 18 ((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁)) → 𝐷 ∈ ℕ0)
92 simp-4l 805 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ 𝐷 ∈ ℕ0) ∧ (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁))) → 𝑦 ∈ ℤ)
93 elnn0 11239 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐷 ∈ ℕ0 ↔ (𝐷 ∈ ℕ ∨ 𝐷 = 0))
94 2a1 28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐷 ∈ ℕ → (((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → ((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁)) → 𝐷 ∈ ℕ)))
95 breq1 4621 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝐷 = 0 → (𝐷𝑀 ↔ 0 ∥ 𝑀))
96 breq1 4621 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝐷 = 0 → (𝐷𝑁 ↔ 0 ∥ 𝑁))
9795, 96anbi12d 746 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝐷 = 0 → ((𝐷𝑀𝐷𝑁) ↔ (0 ∥ 𝑀 ∧ 0 ∥ 𝑁)))
9897anbi2d 739 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝐷 = 0 → ((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁)) ↔ (0 ≤ 𝐷 ∧ (0 ∥ 𝑀 ∧ 0 ∥ 𝑁))))
9998adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝐷 = 0 ∧ ((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0))) → ((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁)) ↔ (0 ≤ 𝐷 ∧ (0 ∥ 𝑀 ∧ 0 ∥ 𝑁))))
100 ianor 509 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (¬ (𝑀 = 0 ∧ 𝑁 = 0) ↔ (¬ 𝑀 = 0 ∨ ¬ 𝑁 = 0))
101 dvdszrcl 14907 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (0 ∥ 𝑀 → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ))
102 0dvds 14921 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑀 ∈ ℤ → (0 ∥ 𝑀𝑀 = 0))
103 pm2.24 121 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑀 = 0 → (¬ 𝑀 = 0 → 𝐷 ∈ ℕ))
104102, 103syl6bi 243 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑀 ∈ ℤ → (0 ∥ 𝑀 → (¬ 𝑀 = 0 → 𝐷 ∈ ℕ)))
105104adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (0 ∥ 𝑀 → (¬ 𝑀 = 0 → 𝐷 ∈ ℕ)))
106101, 105mpcom 38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (0 ∥ 𝑀 → (¬ 𝑀 = 0 → 𝐷 ∈ ℕ))
107106adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((0 ∥ 𝑀 ∧ 0 ∥ 𝑁) → (¬ 𝑀 = 0 → 𝐷 ∈ ℕ))
108107com12 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 𝑀 = 0 → ((0 ∥ 𝑀 ∧ 0 ∥ 𝑁) → 𝐷 ∈ ℕ))
109 dvdszrcl 14907 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (0 ∥ 𝑁 → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ))
110 0dvds 14921 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑁 ∈ ℤ → (0 ∥ 𝑁𝑁 = 0))
111 pm2.24 121 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑁 = 0 → (¬ 𝑁 = 0 → 𝐷 ∈ ℕ))
112110, 111syl6bi 243 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑁 ∈ ℤ → (0 ∥ 𝑁 → (¬ 𝑁 = 0 → 𝐷 ∈ ℕ)))
113112adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ 𝑁 → (¬ 𝑁 = 0 → 𝐷 ∈ ℕ)))
114109, 113mpcom 38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (0 ∥ 𝑁 → (¬ 𝑁 = 0 → 𝐷 ∈ ℕ))
115114adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((0 ∥ 𝑀 ∧ 0 ∥ 𝑁) → (¬ 𝑁 = 0 → 𝐷 ∈ ℕ))
116115com12 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 𝑁 = 0 → ((0 ∥ 𝑀 ∧ 0 ∥ 𝑁) → 𝐷 ∈ ℕ))
117108, 116jaoi 394 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((¬ 𝑀 = 0 ∨ ¬ 𝑁 = 0) → ((0 ∥ 𝑀 ∧ 0 ∥ 𝑁) → 𝐷 ∈ ℕ))
118100, 117sylbi 207 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (¬ (𝑀 = 0 ∧ 𝑁 = 0) → ((0 ∥ 𝑀 ∧ 0 ∥ 𝑁) → 𝐷 ∈ ℕ))
119118adantld 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (¬ (𝑀 = 0 ∧ 𝑁 = 0) → ((0 ≤ 𝐷 ∧ (0 ∥ 𝑀 ∧ 0 ∥ 𝑁)) → 𝐷 ∈ ℕ))
120119ad2antll 764 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝐷 = 0 ∧ ((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0))) → ((0 ≤ 𝐷 ∧ (0 ∥ 𝑀 ∧ 0 ∥ 𝑁)) → 𝐷 ∈ ℕ))
12199, 120sylbid 230 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝐷 = 0 ∧ ((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0))) → ((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁)) → 𝐷 ∈ ℕ))
122121ex 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐷 = 0 → (((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → ((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁)) → 𝐷 ∈ ℕ)))
12394, 122jaoi 394 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐷 ∈ ℕ ∨ 𝐷 = 0) → (((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → ((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁)) → 𝐷 ∈ ℕ)))
12493, 123sylbi 207 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐷 ∈ ℕ0 → (((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → ((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁)) → 𝐷 ∈ ℕ)))
125124impcom 446 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ 𝐷 ∈ ℕ0) → ((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁)) → 𝐷 ∈ ℕ))
126125imp 445 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ 𝐷 ∈ ℕ0) ∧ (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁))) → 𝐷 ∈ ℕ)
127 dvdsle 14951 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑦 ∈ ℤ ∧ 𝐷 ∈ ℕ) → (𝑦𝐷𝑦𝐷))
12892, 126, 127syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ 𝐷 ∈ ℕ0) ∧ (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁))) → (𝑦𝐷𝑦𝐷))
129128exp31 629 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝐷 ∈ ℕ0 → ((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁)) → (𝑦𝐷𝑦𝐷))))
130129com14 96 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦𝐷 → (𝐷 ∈ ℕ0 → ((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁)) → (((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → 𝑦𝐷))))
131130imp 445 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦𝐷𝐷 ∈ ℕ0) → ((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁)) → (((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → 𝑦𝐷)))
132131impcom 446 . . . . . . . . . . . . . . . . . . . . 21 (((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁)) ∧ (𝑦𝐷𝐷 ∈ ℕ0)) → (((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → 𝑦𝐷))
133132imp 445 . . . . . . . . . . . . . . . . . . . 20 ((((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁)) ∧ (𝑦𝐷𝐷 ∈ ℕ0)) ∧ ((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0))) → 𝑦𝐷)
134 zre 11326 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ ℤ → 𝑦 ∈ ℝ)
135134ad2antrr 761 . . . . . . . . . . . . . . . . . . . . 21 (((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → 𝑦 ∈ ℝ)
13670ad2antlr 762 . . . . . . . . . . . . . . . . . . . . 21 (((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁)) ∧ (𝑦𝐷𝐷 ∈ ℕ0)) → 𝐷 ∈ ℝ)
137 lenlt 10061 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝑦𝐷 ↔ ¬ 𝐷 < 𝑦))
138135, 136, 137syl2anr 495 . . . . . . . . . . . . . . . . . . . 20 ((((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁)) ∧ (𝑦𝐷𝐷 ∈ ℕ0)) ∧ ((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0))) → (𝑦𝐷 ↔ ¬ 𝐷 < 𝑦))
139133, 138mpbid 222 . . . . . . . . . . . . . . . . . . 19 ((((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁)) ∧ (𝑦𝐷𝐷 ∈ ℕ0)) ∧ ((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0))) → ¬ 𝐷 < 𝑦)
140139exp31 629 . . . . . . . . . . . . . . . . . 18 ((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁)) → ((𝑦𝐷𝐷 ∈ ℕ0) → (((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → ¬ 𝐷 < 𝑦)))
14191, 140mpan2d 709 . . . . . . . . . . . . . . . . 17 ((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁)) → (𝑦𝐷 → (((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → ¬ 𝐷 < 𝑦)))
142141com13 88 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑦𝐷 → ((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁)) → ¬ 𝐷 < 𝑦)))
143142adantr 481 . . . . . . . . . . . . . . 15 ((((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑦𝐷 → ((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁)) → ¬ 𝐷 < 𝑦)))
14485, 143syld 47 . . . . . . . . . . . . . 14 ((((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷) → ((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁)) → ¬ 𝐷 < 𝑦)))
145144com13 88 . . . . . . . . . . . . 13 ((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁)) → (∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷) → ((((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ¬ 𝐷 < 𝑦)))
146145ex 450 . . . . . . . . . . . 12 (0 ≤ 𝐷 → ((𝐷𝑀𝐷𝑁) → (∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷) → ((((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ¬ 𝐷 < 𝑦))))
1471463imp 1254 . . . . . . . . . . 11 ((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷)) → ((((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ¬ 𝐷 < 𝑦))
148147com12 32 . . . . . . . . . 10 ((((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷)) → ¬ 𝐷 < 𝑦))
149148expimpd 628 . . . . . . . . 9 (((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷))) → ¬ 𝐷 < 𝑦))
150149expimpd 628 . . . . . . . 8 ((𝑦 ∈ ℤ ∧ (𝑦𝑀𝑦𝑁)) → ((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷)))) → ¬ 𝐷 < 𝑦))
15176, 150sylbi 207 . . . . . . 7 (𝑦 ∈ {𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑁)} → ((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷)))) → ¬ 𝐷 < 𝑦))
152151impcom 446 . . . . . 6 (((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷)))) ∧ 𝑦 ∈ {𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑁)}) → ¬ 𝐷 < 𝑦)
15368adantr 481 . . . . . . . . . . . 12 ((𝐷𝑀𝐷𝑁) → 𝐷 ∈ ℤ)
154153ancri 574 . . . . . . . . . . 11 ((𝐷𝑀𝐷𝑁) → (𝐷 ∈ ℤ ∧ (𝐷𝑀𝐷𝑁)))
1551543ad2ant2 1081 . . . . . . . . . 10 ((0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷)) → (𝐷 ∈ ℤ ∧ (𝐷𝑀𝐷𝑁)))
156155ad2antll 764 . . . . . . . . 9 ((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷)))) → (𝐷 ∈ ℤ ∧ (𝐷𝑀𝐷𝑁)))
157156adantr 481 . . . . . . . 8 (((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷)))) ∧ (𝑦 ∈ ℝ ∧ 𝑦 < 𝐷)) → (𝐷 ∈ ℤ ∧ (𝐷𝑀𝐷𝑁)))
158 breq1 4621 . . . . . . . . . 10 (𝑛 = 𝐷 → (𝑛𝑀𝐷𝑀))
159 breq1 4621 . . . . . . . . . 10 (𝑛 = 𝐷 → (𝑛𝑁𝐷𝑁))
160158, 159anbi12d 746 . . . . . . . . 9 (𝑛 = 𝐷 → ((𝑛𝑀𝑛𝑁) ↔ (𝐷𝑀𝐷𝑁)))
161160elrab 3351 . . . . . . . 8 (𝐷 ∈ {𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑁)} ↔ (𝐷 ∈ ℤ ∧ (𝐷𝑀𝐷𝑁)))
162157, 161sylibr 224 . . . . . . 7 (((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷)))) ∧ (𝑦 ∈ ℝ ∧ 𝑦 < 𝐷)) → 𝐷 ∈ {𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑁)})
163 breq2 4622 . . . . . . . 8 (𝑧 = 𝐷 → (𝑦 < 𝑧𝑦 < 𝐷))
164163adantl 482 . . . . . . 7 ((((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷)))) ∧ (𝑦 ∈ ℝ ∧ 𝑦 < 𝐷)) ∧ 𝑧 = 𝐷) → (𝑦 < 𝑧𝑦 < 𝐷))
165 simprr 795 . . . . . . 7 (((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷)))) ∧ (𝑦 ∈ ℝ ∧ 𝑦 < 𝐷)) → 𝑦 < 𝐷)
166162, 164, 165rspcedvd 3307 . . . . . 6 (((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷)))) ∧ (𝑦 ∈ ℝ ∧ 𝑦 < 𝐷)) → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑁)}𝑦 < 𝑧)
16766, 72, 152, 166eqsupd 8308 . . . . 5 ((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷)))) → sup({𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑁)}, ℝ, < ) = 𝐷)
16864, 167eqtrd 2660 . . . 4 ((¬ (𝑀 = 0 ∧ 𝑁 = 0) ∧ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷)))) → if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑁)}, ℝ, < )) = 𝐷)
16962, 168pm2.61ian 830 . . 3 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷))) → if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛𝑀𝑛𝑁)}, ℝ, < )) = 𝐷)
17023, 169eqtr2d 2661 . 2 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷))) → 𝐷 = (𝑀 gcd 𝑁))
17121, 170impbida 876 1 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐷 = (𝑀 gcd 𝑁) ↔ (0 ≤ 𝐷 ∧ (𝐷𝑀𝐷𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒𝑀𝑒𝑁) → 𝑒𝐷))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1992  wral 2912  {crab 2916  ifcif 4063   class class class wbr 4618   Or wor 4999  (class class class)co 6605  supcsup 8291  cr 9880  0cc0 9881   < clt 10019  cle 10020  cn 10965  0cn0 11237  cz 11322  cdvds 14902   gcd cgcd 15135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-pre-sup 9959
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-sup 8293  df-inf 8294  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-div 10630  df-nn 10966  df-2 11024  df-3 11025  df-n0 11238  df-z 11323  df-uz 11632  df-rp 11777  df-fl 12530  df-mod 12606  df-seq 12739  df-exp 12798  df-cj 13768  df-re 13769  df-im 13770  df-sqrt 13904  df-abs 13905  df-dvds 14903  df-gcd 15136
This theorem is referenced by: (None)
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