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 invrcl.b B = Base R
invrcl.z 0 ˙ = 0 R
invrcl.i I = inv r R
Assertion drnginvrcl R DivRing X B X 0 ˙ I X B

Proof

Step Hyp Ref Expression
1 invrcl.b B = Base R
2 invrcl.z 0 ˙ = 0 R
3 invrcl.i I = inv r 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