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Metamath Proof Explorer

Theorem drnginvrcld

Description: Closure of the multiplicative inverse in a division ring. ( reccld analog). (Contributed by SN, 14-Aug-2024)

Ref Expression
Hypotheses drnginvrcl.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
drnginvrcl.z ⊢ 0 = ( 0g ‘ 𝑅 )
drnginvrcl.i ⊢ 𝐼 = ( invr ‘ 𝑅 )
drnginvrcld.r ⊢ ( 𝜑 → 𝑅 ∈ DivRing )
drnginvrcld.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
drnginvrcld.1 ⊢ ( 𝜑 → 𝑋 ≠ 0 )
Assertion drnginvrcld ( 𝜑 → ( 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 drnginvrcld.r ⊢ ( 𝜑 → 𝑅 ∈ DivRing )
5 drnginvrcld.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6 drnginvrcld.1 ⊢ ( 𝜑 → 𝑋 ≠ 0 )
7 1 2 3 drnginvrcl ⊢ ( ( 𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ( 𝐼 ‘ 𝑋 ) ∈ 𝐵 )
8 4 5 6 7 syl3anc ⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) ∈ 𝐵 )