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Metamath Proof Explorer

Theorem mulgm1

Description: Group multiple (exponentiation) operation at negative one. (Contributed by Paul Chapman, 17-Apr-2009) (Revised by Mario Carneiro, 20-Dec-2014)

Ref Expression
Hypotheses mulgnncl.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
mulgnncl.t ⊢ · = ( .g ‘ 𝐺 )
mulgneg.i ⊢ 𝐼 = ( invg ‘ 𝐺 )
Assertion mulgm1 ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( - 1 · 𝑋 ) = ( 𝐼 ‘ 𝑋 ) )

Proof

Step Hyp Ref Expression
1 mulgnncl.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 mulgnncl.t ⊢ · = ( .g ‘ 𝐺 )
3 mulgneg.i ⊢ 𝐼 = ( invg ‘ 𝐺 )
4 1z ⊢ 1 ∈ ℤ
5 1 2 3 mulgneg ⊢ ( ( 𝐺 ∈ Grp ∧ 1 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ) → ( - 1 · 𝑋 ) = ( 𝐼 ‘ ( 1 · 𝑋 ) ) )
6 4 5 mp3an2 ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( - 1 · 𝑋 ) = ( 𝐼 ‘ ( 1 · 𝑋 ) ) )
7 1 2 mulg1 ⊢ ( 𝑋 ∈ 𝐵 → ( 1 · 𝑋 ) = 𝑋 )
8 7 adantl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 1 · 𝑋 ) = 𝑋 )
9 8 fveq2d ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝐼 ‘ ( 1 · 𝑋 ) ) = ( 𝐼 ‘ 𝑋 ) )
10 6 9 eqtrd ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( - 1 · 𝑋 ) = ( 𝐼 ‘ 𝑋 ) )