Theorem flt4lem5 43097

Description: In the context of the lemmas of pythagtrip 16796, 𝑀 and 𝑁 are coprime. (Contributed by SN, 23-Aug-2024.)

Hypotheses

Ref	Expression
flt4lem5.1	⊢ 𝑀 = (((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2)
flt4lem5.2	⊢ 𝑁 = (((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2)

Assertion

Ref	Expression
flt4lem5	⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝑀 gcd 𝑁) = 1)

Proof of Theorem flt4lem5

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3l 1203	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐴 gcd 𝐵) = 1)
2		simp11 1205	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℕ)
3		simp12 1206	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℕ)
4		coprmgcdb 16609	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (∀𝑖 ∈ ℕ ((𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐵) → 𝑖 = 1) ↔ (𝐴 gcd 𝐵) = 1))
5	2, 3, 4	syl2anc 585	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (∀𝑖 ∈ ℕ ((𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐵) → 𝑖 = 1) ↔ (𝐴 gcd 𝐵) = 1))
6	1, 5	mpbird 257	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ∀𝑖 ∈ ℕ ((𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐵) → 𝑖 = 1))
7		simplr 769	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → 𝑖 ∈ ℕ)
8	7	nnzd 12541	. . . . . . . . 9 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → 𝑖 ∈ ℤ)
9		flt4lem5.1	. . . . . . . . . . . . 13 ⊢ 𝑀 = (((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2)
10	9	pythagtriplem11 16787	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝑀 ∈ ℕ)
11	10	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → 𝑀 ∈ ℕ)
12	11	nnsqcld 14197	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → (𝑀↑2) ∈ ℕ)
13	12	nnzd 12541	. . . . . . . . 9 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → (𝑀↑2) ∈ ℤ)
14		flt4lem5.2	. . . . . . . . . . . . 13 ⊢ 𝑁 = (((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2)
15	14	pythagtriplem13 16789	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝑁 ∈ ℕ)
16	15	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → 𝑁 ∈ ℕ)
17	16	nnsqcld 14197	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → (𝑁↑2) ∈ ℕ)
18	17	nnzd 12541	. . . . . . . . 9 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → (𝑁↑2) ∈ ℤ)
19		simprl 771	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → 𝑖 ∥ 𝑀)
20	11	nnzd 12541	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → 𝑀 ∈ ℤ)
21		2nn 12245	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ
22	21	a1i 11	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → 2 ∈ ℕ)
23		dvdsexp2im 16287	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 2 ∈ ℕ) → (𝑖 ∥ 𝑀 → 𝑖 ∥ (𝑀↑2)))
24	8, 20, 22, 23	syl3anc 1374	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → (𝑖 ∥ 𝑀 → 𝑖 ∥ (𝑀↑2)))
25	19, 24	mpd 15	. . . . . . . . 9 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → 𝑖 ∥ (𝑀↑2))
26		simprr 773	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → 𝑖 ∥ 𝑁)
27	16	nnzd 12541	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → 𝑁 ∈ ℤ)
28		dvdsexp2im 16287	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 2 ∈ ℕ) → (𝑖 ∥ 𝑁 → 𝑖 ∥ (𝑁↑2)))
29	8, 27, 22, 28	syl3anc 1374	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → (𝑖 ∥ 𝑁 → 𝑖 ∥ (𝑁↑2)))
30	26, 29	mpd 15	. . . . . . . . 9 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → 𝑖 ∥ (𝑁↑2))
31	8, 13, 18, 25, 30	dvds2subd 16253	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → 𝑖 ∥ ((𝑀↑2) − (𝑁↑2)))
32	9, 14	pythagtriplem15 16791	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 = ((𝑀↑2) − (𝑁↑2)))
33	32	ad2antrr 727	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → 𝐴 = ((𝑀↑2) − (𝑁↑2)))
34	31, 33	breqtrrd 5114	. . . . . . 7 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → 𝑖 ∥ 𝐴)
35		2z 12550	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
36	35	a1i 11	. . . . . . . . 9 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → 2 ∈ ℤ)
37	11, 16	nnmulcld 12221	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → (𝑀 · 𝑁) ∈ ℕ)
38	37	nnzd 12541	. . . . . . . . 9 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → (𝑀 · 𝑁) ∈ ℤ)
39	8, 20, 27, 26	dvdsmultr2d 16259	. . . . . . . . 9 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → 𝑖 ∥ (𝑀 · 𝑁))
40	8, 36, 38, 39	dvdsmultr2d 16259	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → 𝑖 ∥ (2 · (𝑀 · 𝑁)))
41	9, 14	pythagtriplem16 16792	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 = (2 · (𝑀 · 𝑁)))
42	41	ad2antrr 727	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → 𝐵 = (2 · (𝑀 · 𝑁)))
43	40, 42	breqtrrd 5114	. . . . . . 7 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → 𝑖 ∥ 𝐵)
44	34, 43	jca 511	. . . . . 6 ⊢ (((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) ∧ (𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁)) → (𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐵))
45	44	ex 412	. . . . 5 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) → ((𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁) → (𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐵)))
46	45	imim1d 82	. . . 4 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 𝑖 ∈ ℕ) → (((𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐵) → 𝑖 = 1) → ((𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁) → 𝑖 = 1)))
47	46	ralimdva 3150	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (∀𝑖 ∈ ℕ ((𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐵) → 𝑖 = 1) → ∀𝑖 ∈ ℕ ((𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁) → 𝑖 = 1)))
48	6, 47	mpd 15	. 2 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ∀𝑖 ∈ ℕ ((𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁) → 𝑖 = 1))
49		coprmgcdb 16609	. . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (∀𝑖 ∈ ℕ ((𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁) → 𝑖 = 1) ↔ (𝑀 gcd 𝑁) = 1))
50	10, 15, 49	syl2anc 585	. 2 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (∀𝑖 ∈ ℕ ((𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁) → 𝑖 = 1) ↔ (𝑀 gcd 𝑁) = 1))
51	48, 50	mpbid 232	1 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝑀 gcd 𝑁) = 1)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   class class class wbr 5086  ‘cfv 6492  (class class class)co 7360  1c1 11030   + caddc 11032   · cmul 11034   − cmin 11368   / cdiv 11798  ℕcn 12165  2c2 12227  ℤcz 12515  ↑cexp 14014  √csqrt 15186   ∥ cdvds 16212   gcd cgcd 16454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9348  df-inf 9349  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-n0 12429  df-z 12516  df-uz 12780  df-rp 12934  df-fz 13453  df-fl 13742  df-mod 13820  df-seq 13955  df-exp 14015  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-dvds 16213  df-gcd 16455  df-prm 16632

This theorem is referenced by:  flt4lem5c  43101  flt4lem5d  43102  flt4lem5e  43103