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Metamath Proof Explorer

Theorem wlklenvm1

Description: The number of edges of a walk is the number of its vertices minus 1. (Contributed by Alexander van der Vekens, 1-Jul-2018) (Revised by AV, 2-Jan-2021)

Ref Expression
Assertion wlklenvm1 ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ♯ ‘ 𝐹 ) = ( ( ♯ ‘ 𝑃 ) − 1 ) )

Proof

Step Hyp Ref Expression
1 wlklenvp1 ⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ♯ ‘ 𝑃 ) = ( ( ♯ ‘ 𝐹 ) + 1 ) )
2 oveq1 ⊢ ( ( ♯ ‘ 𝑃 ) = ( ( ♯ ‘ 𝐹 ) + 1 ) → ( ( ♯ ‘ 𝑃 ) − 1 ) = ( ( ( ♯ ‘ 𝐹 ) + 1 ) − 1 ) )
3 wlkcl ⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ♯ ‘ 𝐹 ) ∈ ℕ0 )
4 3 nn0cnd ⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ♯ ‘ 𝐹 ) ∈ ℂ )
5 pncan1 ⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℂ → ( ( ( ♯ ‘ 𝐹 ) + 1 ) − 1 ) = ( ♯ ‘ 𝐹 ) )
6 4 5 syl ⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ( ( ♯ ‘ 𝐹 ) + 1 ) − 1 ) = ( ♯ ‘ 𝐹 ) )
7 2 6 sylan9eqr ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( ♯ ‘ 𝑃 ) = ( ( ♯ ‘ 𝐹 ) + 1 ) ) → ( ( ♯ ‘ 𝑃 ) − 1 ) = ( ♯ ‘ 𝐹 ) )
8 7 eqcomd ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( ♯ ‘ 𝑃 ) = ( ( ♯ ‘ 𝐹 ) + 1 ) ) → ( ♯ ‘ 𝐹 ) = ( ( ♯ ‘ 𝑃 ) − 1 ) )
9 1 8 mpdan ⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ♯ ‘ 𝐹 ) = ( ( ♯ ‘ 𝑃 ) − 1 ) )