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Metamath Proof Explorer

Theorem drnginvrld

Description: Property of the multiplicative inverse in a division ring. ( recid2d analog). (Contributed by SN, 14-Aug-2024)

Ref Expression
Hypotheses drnginvrld.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
drnginvrld.0 ⊢ 0 = ( 0g ‘ 𝑅 )
drnginvrld.t ⊢ · = ( .r ‘ 𝑅 )
drnginvrld.u ⊢ 1 = ( 1r ‘ 𝑅 )
drnginvrld.i ⊢ 𝐼 = ( invr ‘ 𝑅 )
drnginvrld.r ⊢ ( 𝜑 → 𝑅 ∈ DivRing )
drnginvrld.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
drnginvrld.1 ⊢ ( 𝜑 → 𝑋 ≠ 0 )
Assertion drnginvrld ( 𝜑 → ( ( 𝐼 ‘ 𝑋 ) · 𝑋 ) = 1 )

Proof

Step Hyp Ref Expression
1 drnginvrld.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 drnginvrld.0 ⊢ 0 = ( 0g ‘ 𝑅 )
3 drnginvrld.t ⊢ · = ( .r ‘ 𝑅 )
4 drnginvrld.u ⊢ 1 = ( 1r ‘ 𝑅 )
5 drnginvrld.i ⊢ 𝐼 = ( invr ‘ 𝑅 )
6 drnginvrld.r ⊢ ( 𝜑 → 𝑅 ∈ DivRing )
7 drnginvrld.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
8 drnginvrld.1 ⊢ ( 𝜑 → 𝑋 ≠ 0 )
9 1 2 3 4 5 drnginvrl ⊢ ( ( 𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ( ( 𝐼 ‘ 𝑋 ) · 𝑋 ) = 1 )
10 6 7 8 9 syl3anc ⊢ ( 𝜑 → ( ( 𝐼 ‘ 𝑋 ) · 𝑋 ) = 1 )