Theorem crctisclwlk

Description: A circuit is a closed walk. (Contributed by AV, 17-Feb-2021) (Proof shortened by AV, 30-Oct-2021)

	Ref	Expression
Assertion	crctisclwlk	$$⊢F\mathrm{Circuits}\left(G\right)P\to F\mathrm{ClWalks}\left(G\right)P$$

Proof

Step	Hyp	Ref	Expression
1		crctprop	$$⊢F\mathrm{Circuits}\left(G\right)P\to \left(F\mathrm{Trails}\left(G\right)P\wedge P\left(0\right)=P\left(\left|F\right|\right)\right)$$
2		trliswlk	$$⊢F\mathrm{Trails}\left(G\right)P\to F\mathrm{Walks}\left(G\right)P$$
3		isclwlk	$$⊢F\mathrm{ClWalks}\left(G\right)P\leftrightarrow \left(F\mathrm{Walks}\left(G\right)P\wedge P\left(0\right)=P\left(\left|F\right|\right)\right)$$
4	3	biimpri	$$⊢\left(F\mathrm{Walks}\left(G\right)P\wedge P\left(0\right)=P\left(\left|F\right|\right)\right)\to F\mathrm{ClWalks}\left(G\right)P$$
5	2 4	sylan	$$⊢\left(F\mathrm{Trails}\left(G\right)P\wedge P\left(0\right)=P\left(\left|F\right|\right)\right)\to F\mathrm{ClWalks}\left(G\right)P$$
6	1 5	syl	$$⊢F\mathrm{Circuits}\left(G\right)P\to F\mathrm{ClWalks}\left(G\right)P$$