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 = ( 0g𝑅 )
drnginvrld.t · = ( .r𝑅 )
drnginvrld.u 1 = ( 1r𝑅 )
drnginvrld.i 𝐼 = ( invr𝑅 )
drnginvrld.r ( 𝜑𝑅 ∈ DivRing )
drnginvrld.x ( 𝜑𝑋𝐵 )
drnginvrld.1 ( 𝜑𝑋0 )
Assertion drnginvrld ( 𝜑 → ( ( 𝐼𝑋 ) · 𝑋 ) = 1 )

Proof

Step Hyp Ref Expression
1 drnginvrld.b 𝐵 = ( Base ‘ 𝑅 )
2 drnginvrld.0 0 = ( 0g𝑅 )
3 drnginvrld.t · = ( .r𝑅 )
4 drnginvrld.u 1 = ( 1r𝑅 )
5 drnginvrld.i 𝐼 = ( invr𝑅 )
6 drnginvrld.r ( 𝜑𝑅 ∈ DivRing )
7 drnginvrld.x ( 𝜑𝑋𝐵 )
8 drnginvrld.1 ( 𝜑𝑋0 )
9 1 2 3 4 5 drnginvrl ( ( 𝑅 ∈ DivRing ∧ 𝑋𝐵𝑋0 ) → ( ( 𝐼𝑋 ) · 𝑋 ) = 1 )
10 6 7 8 9 syl3anc ( 𝜑 → ( ( 𝐼𝑋 ) · 𝑋 ) = 1 )