Metamath Proof Explorer


Theorem drnginvrn0

Description: The multiplicative inverse in a division ring is nonzero. ( recne0 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 drnginvrn0 R DivRing X B X 0 ˙ I X 0 ˙

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 drngring R DivRing R Ring
5 eqid Unit R = Unit R
6 5 3 unitinvcl R Ring X Unit R I X Unit R
7 6 ex R Ring X Unit R I X Unit R
8 4 7 syl R DivRing X Unit R I X Unit R
9 1 5 2 drngunit R DivRing X Unit R X B X 0 ˙
10 1 5 2 drngunit R DivRing I X Unit R I X B I X 0 ˙
11 8 9 10 3imtr3d R DivRing X B X 0 ˙ I X B I X 0 ˙
12 11 3impib R DivRing X B X 0 ˙ I X B I X 0 ˙
13 12 simprd R DivRing X B X 0 ˙ I X 0 ˙