Theorem flt4lem5 43196

Description: In the context of the lemmas of pythagtrip 16853, 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 1214	. . . 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 1216	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℕ)
3		simp12 1217	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℕ)
4		coprmgcdb 16666	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (∀𝑖 ∈ ℕ ((𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐵) → 𝑖 = 1) ↔ (𝐴 gcd 𝐵) = 1))
5	2, 3, 4	syl2anc 593	. . . 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 259	. . 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 778	. . . . . . . . . 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 12591	. . . . . . . . 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 16844	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝑀 ∈ ℕ)
11	10	ad2antrr 736	. . . . . . . . . . 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 14254	. . . . . . . . . 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 12591	. . . . . . . . 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 16846	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝑁 ∈ ℕ)
16	15	ad2antrr 736	. . . . . . . . . . 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 14254	. . . . . . . . . 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 12591	. . . . . . . . 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 780	. . . . . . . . . 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 12591	. . . . . . . . . . 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 12288	. . . . . . . . . . . 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 16344	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 2 ∈ ℕ) → (𝑖 ∥ 𝑀 → 𝑖 ∥ (𝑀↑2)))
24	8, 20, 22, 23	syl3anc 1389	. . . . . . . . . 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 782	. . . . . . . . . 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 12591	. . . . . . . . . . 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 16344	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 2 ∈ ℕ) → (𝑖 ∥ 𝑁 → 𝑖 ∥ (𝑁↑2)))
29	8, 27, 22, 28	syl3anc 1389	. . . . . . . . . 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 16310	. . . . . . . 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 16848	. . . . . . . . 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 736	. . . . . . . 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 5127	. . . . . . 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 12600	. . . . . . . . . 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 12263	. . . . . . . . . 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 12591	. . . . . . . . 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 16316	. . . . . . . . 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 16316	. . . . . . . 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 16849	. . . . . . . . 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 736	. . . . . . . 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 5127	. . . . . . 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 519	. . . . . 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 3173	. . 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 16666	. . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (∀𝑖 ∈ ℕ ((𝑖 ∥ 𝑀 ∧ 𝑖 ∥ 𝑁) → 𝑖 = 1) ↔ (𝑀 gcd 𝑁) = 1))
50	10, 15, 49	syl2anc 593	. 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 234	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 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075   class class class wbr 5099  ‘cfv 6517  (class class class)co 7392  1c1 11071   + caddc 11073   · cmul 11075   − cmin 11411   / cdiv 11841  ℕcn 12207  2c2 12269  ℤcz 12565  ↑cexp 14071  √csqrt 15243   ∥ cdvds 16269   gcd cgcd 16511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147  ax-pre-sup 11148

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-2o 8433  df-er 8673  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-sup 9385  df-inf 9386  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-div 11842  df-nn 12208  df-2 12277  df-3 12278  df-n0 12479  df-z 12566  df-uz 12837  df-rp 12991  df-fz 13510  df-fl 13799  df-mod 13877  df-seq 14012  df-exp 14072  df-cj 15109  df-re 15110  df-im 15111  df-sqrt 15245  df-abs 15246  df-dvds 16270  df-gcd 16512  df-prm 16689

This theorem is referenced by:  flt4lem5c  43200  flt4lem5d  43201  flt4lem5e  43202