Theorem flt4lem5 40014

Description: In the context of the lemmas of pythagtrip 16239, 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 1198	. . . 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 1200	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℕ)
3		simp12 1201	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℕ)
4		coprmgcdb 16058	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (∀𝑖 ∈ ℕ ((𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐵) → 𝑖 = 1) ↔ (𝐴 gcd 𝐵) = 1))
5	2, 3, 4	syl2anc 587	. . . 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 260	. . 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 768	. . . . . . . . . 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 12138	. . . . . . . . 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 16230	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝑀 ∈ ℕ)
11	10	ad2antrr 725	. . . . . . . . . . 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 13668	. . . . . . . . . 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 12138	. . . . . . . . 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 16232	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝑁 ∈ ℕ)
16	15	ad2antrr 725	. . . . . . . . . . 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 13668	. . . . . . . . . 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 12138	. . . . . . . . 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 770	. . . . . . . . . 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 12138	. . . . . . . . . . 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 11760	. . . . . . . . . . . 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 15741	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 2 ∈ ℕ) → (𝑖 ∥ 𝑀 → 𝑖 ∥ (𝑀↑2)))
24	8, 20, 22, 23	syl3anc 1368	. . . . . . . . . 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 772	. . . . . . . . . 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 12138	. . . . . . . . . . 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 15741	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 2 ∈ ℕ) → (𝑖 ∥ 𝑁 → 𝑖 ∥ (𝑁↑2)))
29	8, 27, 22, 28	syl3anc 1368	. . . . . . . . . 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 15707	. . . . . . . 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 16234	. . . . . . . . 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 725	. . . . . . . 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 5064	. . . . . . 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 12066	. . . . . . . . . 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 11740	. . . . . . . . . 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 12138	. . . . . . . . 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 15713	. . . . . . . . 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 15713	. . . . . . . 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 16235	. . . . . . . . 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 725	. . . . . . . 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 5064	. . . . . . 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 515	. . . . . 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 416	. . . . 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 3108	. . 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 16058	. . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (∀𝑖 ∈ ℕ ((𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁) → 𝑖 = 1) ↔ (𝑀 gcd 𝑁) = 1))
50	10, 15, 49	syl2anc 587	. 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 235	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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3070   class class class wbr 5036  ‘cfv 6340  (class class class)co 7156  1c1 10589   + caddc 10591   · cmul 10593   − cmin 10921   / cdiv 11348  ℕcn 11687  2c2 11742  ℤcz 12033  ↑cexp 13492  √csqrt 14653   ∥ cdvds 15668   gcd cgcd 15906

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-cnex 10644  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664  ax-pre-mulgt0 10665  ax-pre-sup 10666

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7586  df-1st 7699  df-2nd 7700  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-1o 8118  df-2o 8119  df-er 8305  df-en 8541  df-dom 8542  df-sdom 8543  df-fin 8544  df-sup 8952  df-inf 8953  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-sub 10923  df-neg 10924  df-div 11349  df-nn 11688  df-2 11750  df-3 11751  df-n0 11948  df-z 12034  df-uz 12296  df-rp 12444  df-fz 12953  df-fl 13224  df-mod 13300  df-seq 13432  df-exp 13493  df-cj 14519  df-re 14520  df-im 14521  df-sqrt 14655  df-abs 14656  df-dvds 15669  df-gcd 15907  df-prm 16081

This theorem is referenced by:  flt4lem5c  40018  flt4lem5d  40019  flt4lem5e  40020