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Metamath Proof Explorer

Theorem 1rinv

Description: The inverse of the ring unity is the ring unity. (Contributed by Mario Carneiro, 18-Jun-2015)

Ref Expression
Hypotheses 1rinv.1 ⊢ 𝐼 = ( invr ‘ 𝑅 )
1rinv.2 ⊢ 1 = ( 1r ‘ 𝑅 )
Assertion 1rinv ( 𝑅 ∈ Ring → ( 𝐼 ‘ 1 ) = 1 )

Proof

Step Hyp Ref Expression
1 1rinv.1 ⊢ 𝐼 = ( invr ‘ 𝑅 )
2 1rinv.2 ⊢ 1 = ( 1r ‘ 𝑅 )
3 eqid ⊢ ( Unit ‘ 𝑅 ) = ( Unit ‘ 𝑅 )
4 3 2 1unit ⊢ ( 𝑅 ∈ Ring → 1 ∈ ( Unit ‘ 𝑅 ) )
5 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
6 3 1 5 ringinvcl ⊢ ( ( 𝑅 ∈ Ring ∧ 1 ∈ ( Unit ‘ 𝑅 ) ) → ( 𝐼 ‘ 1 ) ∈ ( Base ‘ 𝑅 ) )
7 4 6 mpdan ⊢ ( 𝑅 ∈ Ring → ( 𝐼 ‘ 1 ) ∈ ( Base ‘ 𝑅 ) )
8 eqid ⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
9 5 8 2 ringlidm ⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝐼 ‘ 1 ) ∈ ( Base ‘ 𝑅 ) ) → ( 1 ( .r ‘ 𝑅 ) ( 𝐼 ‘ 1 ) ) = ( 𝐼 ‘ 1 ) )
10 7 9 mpdan ⊢ ( 𝑅 ∈ Ring → ( 1 ( .r ‘ 𝑅 ) ( 𝐼 ‘ 1 ) ) = ( 𝐼 ‘ 1 ) )
11 3 1 8 2 unitrinv ⊢ ( ( 𝑅 ∈ Ring ∧ 1 ∈ ( Unit ‘ 𝑅 ) ) → ( 1 ( .r ‘ 𝑅 ) ( 𝐼 ‘ 1 ) ) = 1 )
12 4 11 mpdan ⊢ ( 𝑅 ∈ Ring → ( 1 ( .r ‘ 𝑅 ) ( 𝐼 ‘ 1 ) ) = 1 )
13 10 12 eqtr3d ⊢ ( 𝑅 ∈ Ring → ( 𝐼 ‘ 1 ) = 1 )