Theorem grpinvex

Description: Every member of a group has a left inverse. (Contributed by NM, 16-Aug-2011) (Revised by Mario Carneiro, 6-Jan-2015)

	Ref	Expression
Hypotheses	grpcl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	grpcl.p	⊢ + = ( +g ‘ 𝐺 )
	grpinvex.p	⊢ 0 = ( 0g ‘ 𝐺 )
Assertion	grpinvex	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ∃ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑋 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		grpcl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		grpcl.p	⊢ + = ( +g ‘ 𝐺 )
3		grpinvex.p	⊢ 0 = ( 0g ‘ 𝐺 )
4	1 2 3	isgrp	⊢ ( 𝐺 ∈ Grp ↔ ( 𝐺 ∈ Mnd ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) )
5	4	simprbi	⊢ ( 𝐺 ∈ Grp → ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 )
6		oveq2	⊢ ( 𝑥 = 𝑋 → ( 𝑦 + 𝑥 ) = ( 𝑦 + 𝑋 ) )
7	6	eqeq1d	⊢ ( 𝑥 = 𝑋 → ( ( 𝑦 + 𝑥 ) = 0 ↔ ( 𝑦 + 𝑋 ) = 0 ) )
8	7	rexbidv	⊢ ( 𝑥 = 𝑋 → ( ∃ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ↔ ∃ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑋 ) = 0 ) )
9	8	rspccva	⊢ ( ( ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ∧ 𝑋 ∈ 𝐵 ) → ∃ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑋 ) = 0 )
10	5 9	sylan	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ∃ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑋 ) = 0 )