Theorem flt4lem5 43423

Description: In the context of the lemmas of pythagtrip 16919, 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 1220	. . . 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 1222	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℕ)
3		simp12 1223	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℕ)
4		coprmgcdb 16732	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (∀𝑖 ∈ ℕ ((𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐵) → 𝑖 = 1) ↔ (𝐴 gcd 𝐵) = 1))
5	2, 3, 4	syl2anc 596	. . . 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 781	. . . . . . . . . 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 12635	. . . . . . . . 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 16910	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝑀 ∈ ℕ)
11	10	ad2antrr 739	. . . . . . . . . . 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 14300	. . . . . . . . . 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 12635	. . . . . . . . 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 16912	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝑁 ∈ ℕ)
16	15	ad2antrr 739	. . . . . . . . . . 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 14300	. . . . . . . . . 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 12635	. . . . . . . . 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 783	. . . . . . . . . 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 12635	. . . . . . . . . . 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 12332	. . . . . . . . . . . 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 16410	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 2 ∈ ℕ) → (𝑖 ∥ 𝑀 → 𝑖 ∥ (𝑀↑2)))
24	8, 20, 22, 23	syl3anc 1398	. . . . . . . . . 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 16	. . . . . . . . 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 785	. . . . . . . . . 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 12635	. . . . . . . . . . 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 16410	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 2 ∈ ℕ) → (𝑖 ∥ 𝑁 → 𝑖 ∥ (𝑁↑2)))
29	8, 27, 22, 28	syl3anc 1398	. . . . . . . . . 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 16	. . . . . . . . 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 16376	. . . . . . . 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 16914	. . . . . . . . 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 739	. . . . . . . 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 5144	. . . . . . 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 12644	. . . . . . . . . 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 12307	. . . . . . . . . 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 12635	. . . . . . . . 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 16382	. . . . . . . . 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 16382	. . . . . . . 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 16915	. . . . . . . . 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 739	. . . . . . . 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 5144	. . . . . . 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 521	. . . . . 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 418	. . . . 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 83	. . . 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 3180	. . 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 16	. 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 16732	. . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (∀𝑖 ∈ ℕ ((𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁) → 𝑖 = 1) ↔ (𝑀 gcd 𝑁) = 1))
50	10, 15, 49	syl2anc 596	. 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)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3082   class class class wbr 5114  ‘cfv 6543  (class class class)co 7423  1c1 11119   + caddc 11121   · cmul 11123   − cmin 11459   / cdiv 11889  ℕcn 12251  2c2 12313  ℤcz 12609  ↑cexp 14117  √csqrt 15310   ∥ cdvds 16335   gcd cgcd 16577

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195  ax-pre-sup 11196

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-er 8703  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-sup 9412  df-inf 9413  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-div 11890  df-nn 12252  df-2 12321  df-3 12322  df-n0 12523  df-z 12610  df-uz 12881  df-rp 13035  df-fz 13554  df-fl 13845  df-mod 13923  df-seq 14058  df-exp 14118  df-cj 15176  df-re 15177  df-im 15178  df-sqrt 15312  df-abs 15313  df-dvds 16336  df-gcd 16578  df-prm 16755

This theorem is used by:  flt4lem5c  43427  flt4lem5d  43428  flt4lem5e  43429