Theorem flt4lem5 43083

Description: In the context of the lemmas of pythagtrip 16805, 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 16618	. . . . 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 12550	. . . . . . . . 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 16796	. . . . . . . . . . . 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 14206	. . . . . . . . . 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 12550	. . . . . . . . 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 16798	. . . . . . . . . . . 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 14206	. . . . . . . . . 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 12550	. . . . . . . . 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 12550	. . . . . . . . . . 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 12254	. . . . . . . . . . . 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 16296	. . . . . . . . . . 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 12550	. . . . . . . . . . 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 16296	. . . . . . . . . . 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 16262	. . . . . . . 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 16800	. . . . . . . . 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 5113	. . . . . . 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 12559	. . . . . . . . . 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 12230	. . . . . . . . . 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 12550	. . . . . . . . 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 16268	. . . . . . . . 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 16268	. . . . . . . 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 16801	. . . . . . . . 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 5113	. . . . . . 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 3149	. . 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 16618	. . 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 3051   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  1c1 11039   + caddc 11041   · cmul 11043   − cmin 11377   / cdiv 11807  ℕcn 12174  2c2 12236  ℤcz 12524  ↑cexp 14023  √csqrt 15195   ∥ cdvds 16221   gcd cgcd 16463

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 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-sup 9355  df-inf 9356  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-n0 12438  df-z 12525  df-uz 12789  df-rp 12943  df-fz 13462  df-fl 13751  df-mod 13829  df-seq 13964  df-exp 14024  df-cj 15061  df-re 15062  df-im 15063  df-sqrt 15197  df-abs 15198  df-dvds 16222  df-gcd 16464  df-prm 16641

This theorem is referenced by:  flt4lem5c  43087  flt4lem5d  43088  flt4lem5e  43089