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Metamath Proof Explorer

Theorem grplinvd

Description: The left inverse of a group element. Deduction associated with grplinv . (Contributed by SN, 29-Jan-2025)

Ref Expression
Hypotheses grplinvd.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
grplinvd.p ⊢ + = ( +g ‘ 𝐺 )
grplinvd.u ⊢ 0 = ( 0g ‘ 𝐺 )
grplinvd.n ⊢ 𝑁 = ( invg ‘ 𝐺 )
grplinvd.g ⊢ ( 𝜑 → 𝐺 ∈ Grp )
grplinvd.1 ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion grplinvd ( 𝜑 → ( ( 𝑁 ‘ 𝑋 ) + 𝑋 ) = 0 )

Proof

Step Hyp Ref Expression
1 grplinvd.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 grplinvd.p ⊢ + = ( +g ‘ 𝐺 )
3 grplinvd.u ⊢ 0 = ( 0g ‘ 𝐺 )
4 grplinvd.n ⊢ 𝑁 = ( invg ‘ 𝐺 )
5 grplinvd.g ⊢ ( 𝜑 → 𝐺 ∈ Grp )
6 grplinvd.1 ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
7 1 2 3 4 grplinv ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑁 ‘ 𝑋 ) + 𝑋 ) = 0 )
8 5 6 7 syl2anc ⊢ ( 𝜑 → ( ( 𝑁 ‘ 𝑋 ) + 𝑋 ) = 0 )