Metamath Proof Explorer


Theorem irredlmul

Description: The product of a unit and an irreducible element is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014)

Ref Expression
Hypotheses irredn0.i 𝐼 = ( Irred ‘ 𝑅 )
irredrmul.u 𝑈 = ( Unit ‘ 𝑅 )
irredrmul.t · = ( .r𝑅 )
Assertion irredlmul ( ( 𝑅 ∈ Ring ∧ 𝑋𝑈𝑌𝐼 ) → ( 𝑋 · 𝑌 ) ∈ 𝐼 )

Proof

Step Hyp Ref Expression
1 irredn0.i 𝐼 = ( Irred ‘ 𝑅 )
2 irredrmul.u 𝑈 = ( Unit ‘ 𝑅 )
3 irredrmul.t · = ( .r𝑅 )
4 eqid ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
5 eqid ( oppr𝑅 ) = ( oppr𝑅 )
6 eqid ( .r ‘ ( oppr𝑅 ) ) = ( .r ‘ ( oppr𝑅 ) )
7 4 3 5 6 opprmul ( 𝑌 ( .r ‘ ( oppr𝑅 ) ) 𝑋 ) = ( 𝑋 · 𝑌 )
8 5 opprring ( 𝑅 ∈ Ring → ( oppr𝑅 ) ∈ Ring )
9 5 1 opprirred 𝐼 = ( Irred ‘ ( oppr𝑅 ) )
10 2 5 opprunit 𝑈 = ( Unit ‘ ( oppr𝑅 ) )
11 9 10 6 irredrmul ( ( ( oppr𝑅 ) ∈ Ring ∧ 𝑌𝐼𝑋𝑈 ) → ( 𝑌 ( .r ‘ ( oppr𝑅 ) ) 𝑋 ) ∈ 𝐼 )
12 8 11 syl3an1 ( ( 𝑅 ∈ Ring ∧ 𝑌𝐼𝑋𝑈 ) → ( 𝑌 ( .r ‘ ( oppr𝑅 ) ) 𝑋 ) ∈ 𝐼 )
13 12 3com23 ( ( 𝑅 ∈ Ring ∧ 𝑋𝑈𝑌𝐼 ) → ( 𝑌 ( .r ‘ ( oppr𝑅 ) ) 𝑋 ) ∈ 𝐼 )
14 7 13 eqeltrrid ( ( 𝑅 ∈ Ring ∧ 𝑋𝑈𝑌𝐼 ) → ( 𝑋 · 𝑌 ) ∈ 𝐼 )