Description: A number which is twice an integer is even. (Contributed by AV, 16-Jul-2021)
Ref | Expression | ||
---|---|---|---|
Assertion | 2teven | ⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 = ( 2 · 𝐴 ) ) → 2 ∥ 𝐵 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2z | ⊢ 2 ∈ ℤ | |
2 | dvdsmul1 | ⊢ ( ( 2 ∈ ℤ ∧ 𝐴 ∈ ℤ ) → 2 ∥ ( 2 · 𝐴 ) ) | |
3 | 1 2 | mpan | ⊢ ( 𝐴 ∈ ℤ → 2 ∥ ( 2 · 𝐴 ) ) |
4 | 3 | adantr | ⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 = ( 2 · 𝐴 ) ) → 2 ∥ ( 2 · 𝐴 ) ) |
5 | breq2 | ⊢ ( 𝐵 = ( 2 · 𝐴 ) → ( 2 ∥ 𝐵 ↔ 2 ∥ ( 2 · 𝐴 ) ) ) | |
6 | 5 | adantl | ⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 = ( 2 · 𝐴 ) ) → ( 2 ∥ 𝐵 ↔ 2 ∥ ( 2 · 𝐴 ) ) ) |
7 | 4 6 | mpbird | ⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 = ( 2 · 𝐴 ) ) → 2 ∥ 𝐵 ) |