Theorem sconnpconn

Description: A simply connected space is path-connected. (Contributed by Mario Carneiro, 11-Feb-2015)

	Ref	Expression
Assertion	sconnpconn	⊢ ( 𝐽 ∈ SConn → 𝐽 ∈ PConn )

Proof

Step	Hyp	Ref	Expression
1		issconn	⊢ ( 𝐽 ∈ SConn ↔ ( 𝐽 ∈ PConn ∧ ∀ 𝑓 ∈ ( II Cn 𝐽 ) ( ( 𝑓 ‘ 0 ) = ( 𝑓 ‘ 1 ) → 𝑓 ( ≃ph ‘ 𝐽 ) ( ( 0 [,] 1 ) × { ( 𝑓 ‘ 0 ) } ) ) ) )
2	1	simplbi	⊢ ( 𝐽 ∈ SConn → 𝐽 ∈ PConn )