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Metamath Proof Explorer

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 )