Metamath Proof Explorer
Description: The properties of a circuit: A circuit is a closed trail. (Contributed by AV, 31-Jan-2021) (Proof shortened by AV, 30-Oct-2021)
|
|
Ref |
Expression |
|
Assertion |
crctprop |
⊢ ( 𝐹 ( Circuits ‘ 𝐺 ) 𝑃 → ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
iscrct |
⊢ ( 𝐹 ( Circuits ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ) |
| 2 |
1
|
biimpi |
⊢ ( 𝐹 ( Circuits ‘ 𝐺 ) 𝑃 → ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ) |