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Metamath Proof Explorer

Theorem drnginvrcl

Description: Closure of the multiplicative inverse in a division ring. ( reccl analog). (Contributed by NM, 19-Apr-2014)

Ref Expression
Hypotheses drnginvrcl.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
drnginvrcl.z ⊢ 0 = ( 0g ‘ 𝑅 )
drnginvrcl.i ⊢ 𝐼 = ( invr ‘ 𝑅 )
Assertion drnginvrcl ( ( 𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ( 𝐼 ‘ 𝑋 ) ∈ 𝐵 )

Proof

Step Hyp Ref Expression
1 drnginvrcl.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 drnginvrcl.z ⊢ 0 = ( 0g ‘ 𝑅 )
3 drnginvrcl.i ⊢ 𝐼 = ( invr ‘ 𝑅 )
4 eqid ⊢ ( Unit ‘ 𝑅 ) = ( Unit ‘ 𝑅 )
5 1 4 2 drngunit ⊢ ( 𝑅 ∈ DivRing → ( 𝑋 ∈ ( Unit ‘ 𝑅 ) ↔ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ) )
6 drngring ⊢ ( 𝑅 ∈ DivRing → 𝑅 ∈ Ring )
7 4 3 1 ringinvcl ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ ( Unit ‘ 𝑅 ) ) → ( 𝐼 ‘ 𝑋 ) ∈ 𝐵 )
8 7 ex ⊢ ( 𝑅 ∈ Ring → ( 𝑋 ∈ ( Unit ‘ 𝑅 ) → ( 𝐼 ‘ 𝑋 ) ∈ 𝐵 ) )
9 6 8 syl ⊢ ( 𝑅 ∈ DivRing → ( 𝑋 ∈ ( Unit ‘ 𝑅 ) → ( 𝐼 ‘ 𝑋 ) ∈ 𝐵 ) )
10 5 9 sylbird ⊢ ( 𝑅 ∈ DivRing → ( ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ( 𝐼 ‘ 𝑋 ) ∈ 𝐵 ) )
11 10 3impib ⊢ ( ( 𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ( 𝐼 ‘ 𝑋 ) ∈ 𝐵 )