Metamath Proof Explorer


Theorem ringass

Description: Associative law for multiplication in a ring. (Contributed by NM, 27-Aug-2011) (Revised by Mario Carneiro, 6-Jan-2015)

Ref Expression
Hypotheses ringcl.b 𝐵 = ( Base ‘ 𝑅 )
ringcl.t · = ( .r𝑅 )
Assertion ringass ( ( 𝑅 ∈ Ring ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( ( 𝑋 · 𝑌 ) · 𝑍 ) = ( 𝑋 · ( 𝑌 · 𝑍 ) ) )

Proof

Step Hyp Ref Expression
1 ringcl.b 𝐵 = ( Base ‘ 𝑅 )
2 ringcl.t · = ( .r𝑅 )
3 eqid ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 )
4 3 ringmgp ( 𝑅 ∈ Ring → ( mulGrp ‘ 𝑅 ) ∈ Mnd )
5 3 1 mgpbas 𝐵 = ( Base ‘ ( mulGrp ‘ 𝑅 ) )
6 3 2 mgpplusg · = ( +g ‘ ( mulGrp ‘ 𝑅 ) )
7 5 6 mndass ( ( ( mulGrp ‘ 𝑅 ) ∈ Mnd ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( ( 𝑋 · 𝑌 ) · 𝑍 ) = ( 𝑋 · ( 𝑌 · 𝑍 ) ) )
8 4 7 sylan ( ( 𝑅 ∈ Ring ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( ( 𝑋 · 𝑌 ) · 𝑍 ) = ( 𝑋 · ( 𝑌 · 𝑍 ) ) )