Metamath Proof Explorer


Theorem rngoaddneg1

Description: Adding the negative in a ring gives zero. (Contributed by Jeff Madsen, 10-Jun-2010)

Ref Expression
Hypotheses ringnegcl.1 𝐺 = ( 1st𝑅 )
ringnegcl.2 𝑋 = ran 𝐺
ringnegcl.3 𝑁 = ( inv ‘ 𝐺 )
ringaddneg.4 𝑍 = ( GId ‘ 𝐺 )
Assertion rngoaddneg1 ( ( 𝑅 ∈ RingOps ∧ 𝐴𝑋 ) → ( 𝐴 𝐺 ( 𝑁𝐴 ) ) = 𝑍 )

Proof

Step Hyp Ref Expression
1 ringnegcl.1 𝐺 = ( 1st𝑅 )
2 ringnegcl.2 𝑋 = ran 𝐺
3 ringnegcl.3 𝑁 = ( inv ‘ 𝐺 )
4 ringaddneg.4 𝑍 = ( GId ‘ 𝐺 )
5 1 rngogrpo ( 𝑅 ∈ RingOps → 𝐺 ∈ GrpOp )
6 2 4 3 grporinv ( ( 𝐺 ∈ GrpOp ∧ 𝐴𝑋 ) → ( 𝐴 𝐺 ( 𝑁𝐴 ) ) = 𝑍 )
7 5 6 sylan ( ( 𝑅 ∈ RingOps ∧ 𝐴𝑋 ) → ( 𝐴 𝐺 ( 𝑁𝐴 ) ) = 𝑍 )