Metamath Proof Explorer


Theorem flt4lem1

Description: Satisfy the antecedent used in several pythagtrip lemmas, with A , C coprime rather than A , B . (Contributed by SN, 21-Aug-2024)

Ref Expression
Hypotheses flt4lem1.a ( 𝜑𝐴 ∈ ℕ )
flt4lem1.b ( 𝜑𝐵 ∈ ℕ )
flt4lem1.c ( 𝜑𝐶 ∈ ℕ )
flt4lem1.1 ( 𝜑 → ¬ 2 ∥ 𝐴 )
flt4lem1.2 ( 𝜑 → ( 𝐴 gcd 𝐶 ) = 1 )
flt4lem1.3 ( 𝜑 → ( ( 𝐴 ↑ 2 ) + ( 𝐵 ↑ 2 ) ) = ( 𝐶 ↑ 2 ) )
Assertion flt4lem1 ( 𝜑 → ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ ) ∧ ( ( 𝐴 ↑ 2 ) + ( 𝐵 ↑ 2 ) ) = ( 𝐶 ↑ 2 ) ∧ ( ( 𝐴 gcd 𝐵 ) = 1 ∧ ¬ 2 ∥ 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 flt4lem1.a ( 𝜑𝐴 ∈ ℕ )
2 flt4lem1.b ( 𝜑𝐵 ∈ ℕ )
3 flt4lem1.c ( 𝜑𝐶 ∈ ℕ )
4 flt4lem1.1 ( 𝜑 → ¬ 2 ∥ 𝐴 )
5 flt4lem1.2 ( 𝜑 → ( 𝐴 gcd 𝐶 ) = 1 )
6 flt4lem1.3 ( 𝜑 → ( ( 𝐴 ↑ 2 ) + ( 𝐵 ↑ 2 ) ) = ( 𝐶 ↑ 2 ) )
7 1 2 3 3jca ( 𝜑 → ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ ) )
8 1 2 3 5 6 fltabcoprm ( 𝜑 → ( 𝐴 gcd 𝐵 ) = 1 )
9 8 4 jca ( 𝜑 → ( ( 𝐴 gcd 𝐵 ) = 1 ∧ ¬ 2 ∥ 𝐴 ) )
10 7 6 9 3jca ( 𝜑 → ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ ) ∧ ( ( 𝐴 ↑ 2 ) + ( 𝐵 ↑ 2 ) ) = ( 𝐶 ↑ 2 ) ∧ ( ( 𝐴 gcd 𝐵 ) = 1 ∧ ¬ 2 ∥ 𝐴 ) ) )