Theorem pi1grp

Description: The fundamental group is a group. Proposition 1.3 of Hatcher p. 26. (Contributed by Jeff Madsen, 19-Jun-2010) (Proof shortened by Mario Carneiro, 8-Jun-2014) (Revised by Mario Carneiro, 10-Aug-2015)

	Ref	Expression
Hypothesis	pi1grp.2	⊢ 𝐺 = ( 𝐽 π1 𝑌 )
Assertion	pi1grp	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑌 ∈ 𝑋 ) → 𝐺 ∈ Grp )

Proof

Step	Hyp	Ref	Expression
1		pi1grp.2	⊢ 𝐺 = ( 𝐽 π1 𝑌 )
2		eqid	⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
3		simpl	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑌 ∈ 𝑋 ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
4		simpr	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑌 ∈ 𝑋 ) → 𝑌 ∈ 𝑋 )
5		eqid	⊢ ( ( 0 [,] 1 ) × { 𝑌 } ) = ( ( 0 [,] 1 ) × { 𝑌 } )
6	1 2 3 4 5	pi1grplem	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑌 ∈ 𝑋 ) → ( 𝐺 ∈ Grp ∧ [ ( ( 0 [,] 1 ) × { 𝑌 } ) ] ( ≃ph ‘ 𝐽 ) = ( 0g ‘ 𝐺 ) ) )
7	6	simpld	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑌 ∈ 𝑋 ) → 𝐺 ∈ Grp )