Theorem cycliscrct

Description: A cycle is a circuit. (Contributed by Alexander van der Vekens, 30-Oct-2017) (Revised by AV, 31-Jan-2021) (Proof shortened by AV, 30-Oct-2021)

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

Proof

Step	Hyp	Ref	Expression
1		pthistrl	$$⊢F\mathrm{Paths}\left(G\right)P\to F\mathrm{Trails}\left(G\right)P$$
2	1	anim1i	$$⊢\left(F\mathrm{Paths}\left(G\right)P\wedge P\left(0\right)=P\left(\left|F\right|\right)\right)\to \left(F\mathrm{Trails}\left(G\right)P\wedge P\left(0\right)=P\left(\left|F\right|\right)\right)$$
3		iscycl	$$⊢F\mathrm{Cycles}\left(G\right)P\leftrightarrow \left(F\mathrm{Paths}\left(G\right)P\wedge P\left(0\right)=P\left(\left|F\right|\right)\right)$$
4		iscrct	$$⊢F\mathrm{Circuits}\left(G\right)P\leftrightarrow \left(F\mathrm{Trails}\left(G\right)P\wedge P\left(0\right)=P\left(\left|F\right|\right)\right)$$
5	2 3 4	3imtr4i	$$⊢F\mathrm{Cycles}\left(G\right)P\to F\mathrm{Circuits}\left(G\right)P$$