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Metamath Proof Explorer

Theorem grpinvval2

Description: A df-neg -like equation for inverse in terms of group subtraction. (Contributed by Mario Carneiro, 4-Oct-2015)

Ref Expression
Hypotheses grpsubcl.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
grpsubcl.m ⊢ − = ( -g ‘ 𝐺 )
grpinvsub.n ⊢ 𝑁 = ( invg ‘ 𝐺 )
grpinvval2.z ⊢ 0 = ( 0g ‘ 𝐺 )
Assertion grpinvval2 ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑋 ) = ( 0 − 𝑋 ) )

Proof

Step Hyp Ref Expression
1 grpsubcl.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 grpsubcl.m ⊢ − = ( -g ‘ 𝐺 )
3 grpinvsub.n ⊢ 𝑁 = ( invg ‘ 𝐺 )
4 grpinvval2.z ⊢ 0 = ( 0g ‘ 𝐺 )
5 1 4 grpidcl ⊢ ( 𝐺 ∈ Grp → 0 ∈ 𝐵 )
6 eqid ⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
7 1 6 3 2 grpsubval ⊢ ( ( 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 0 − 𝑋 ) = ( 0 ( +g ‘ 𝐺 ) ( 𝑁 ‘ 𝑋 ) ) )
8 5 7 sylan ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 0 − 𝑋 ) = ( 0 ( +g ‘ 𝐺 ) ( 𝑁 ‘ 𝑋 ) ) )
9 1 3 grpinvcl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑋 ) ∈ 𝐵 )
10 1 6 4 grplid ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑁 ‘ 𝑋 ) ∈ 𝐵 ) → ( 0 ( +g ‘ 𝐺 ) ( 𝑁 ‘ 𝑋 ) ) = ( 𝑁 ‘ 𝑋 ) )
11 9 10 syldan ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 0 ( +g ‘ 𝐺 ) ( 𝑁 ‘ 𝑋 ) ) = ( 𝑁 ‘ 𝑋 ) )
12 8 11 eqtr2d ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑋 ) = ( 0 − 𝑋 ) )