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 | ⊢ ( 𝜑 → 𝐴 ∈ ℕ ) | |
flt4lem1.b | ⊢ ( 𝜑 → 𝐵 ∈ ℕ ) | ||
flt4lem1.c | ⊢ ( 𝜑 → 𝐶 ∈ 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 ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flt4lem1.a | ⊢ ( 𝜑 → 𝐴 ∈ ℕ ) | |
2 | flt4lem1.b | ⊢ ( 𝜑 → 𝐵 ∈ ℕ ) | |
3 | flt4lem1.c | ⊢ ( 𝜑 → 𝐶 ∈ 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 ) ) ) |