Metamath Proof Explorer


Theorem isnirred

Description: The property of being a non-irreducible (reducible) element in a ring. (Contributed by Mario Carneiro, 4-Dec-2014)

Ref Expression
Hypotheses irred.1 𝐵 = ( Base ‘ 𝑅 )
irred.2 𝑈 = ( Unit ‘ 𝑅 )
irred.3 𝐼 = ( Irred ‘ 𝑅 )
irred.4 𝑁 = ( 𝐵𝑈 )
irred.5 · = ( .r𝑅 )
Assertion isnirred ( 𝑋𝐵 → ( ¬ 𝑋𝐼 ↔ ( 𝑋𝑈 ∨ ∃ 𝑥𝑁𝑦𝑁 ( 𝑥 · 𝑦 ) = 𝑋 ) ) )

Proof

Step Hyp Ref Expression
1 irred.1 𝐵 = ( Base ‘ 𝑅 )
2 irred.2 𝑈 = ( Unit ‘ 𝑅 )
3 irred.3 𝐼 = ( Irred ‘ 𝑅 )
4 irred.4 𝑁 = ( 𝐵𝑈 )
5 irred.5 · = ( .r𝑅 )
6 4 eleq2i ( 𝑋𝑁𝑋 ∈ ( 𝐵𝑈 ) )
7 eldif ( 𝑋 ∈ ( 𝐵𝑈 ) ↔ ( 𝑋𝐵 ∧ ¬ 𝑋𝑈 ) )
8 6 7 bitri ( 𝑋𝑁 ↔ ( 𝑋𝐵 ∧ ¬ 𝑋𝑈 ) )
9 8 baibr ( 𝑋𝐵 → ( ¬ 𝑋𝑈𝑋𝑁 ) )
10 df-ne ( ( 𝑥 · 𝑦 ) ≠ 𝑋 ↔ ¬ ( 𝑥 · 𝑦 ) = 𝑋 )
11 10 ralbii ( ∀ 𝑦𝑁 ( 𝑥 · 𝑦 ) ≠ 𝑋 ↔ ∀ 𝑦𝑁 ¬ ( 𝑥 · 𝑦 ) = 𝑋 )
12 ralnex ( ∀ 𝑦𝑁 ¬ ( 𝑥 · 𝑦 ) = 𝑋 ↔ ¬ ∃ 𝑦𝑁 ( 𝑥 · 𝑦 ) = 𝑋 )
13 11 12 bitri ( ∀ 𝑦𝑁 ( 𝑥 · 𝑦 ) ≠ 𝑋 ↔ ¬ ∃ 𝑦𝑁 ( 𝑥 · 𝑦 ) = 𝑋 )
14 13 ralbii ( ∀ 𝑥𝑁𝑦𝑁 ( 𝑥 · 𝑦 ) ≠ 𝑋 ↔ ∀ 𝑥𝑁 ¬ ∃ 𝑦𝑁 ( 𝑥 · 𝑦 ) = 𝑋 )
15 ralnex ( ∀ 𝑥𝑁 ¬ ∃ 𝑦𝑁 ( 𝑥 · 𝑦 ) = 𝑋 ↔ ¬ ∃ 𝑥𝑁𝑦𝑁 ( 𝑥 · 𝑦 ) = 𝑋 )
16 14 15 bitr2i ( ¬ ∃ 𝑥𝑁𝑦𝑁 ( 𝑥 · 𝑦 ) = 𝑋 ↔ ∀ 𝑥𝑁𝑦𝑁 ( 𝑥 · 𝑦 ) ≠ 𝑋 )
17 16 a1i ( 𝑋𝐵 → ( ¬ ∃ 𝑥𝑁𝑦𝑁 ( 𝑥 · 𝑦 ) = 𝑋 ↔ ∀ 𝑥𝑁𝑦𝑁 ( 𝑥 · 𝑦 ) ≠ 𝑋 ) )
18 9 17 anbi12d ( 𝑋𝐵 → ( ( ¬ 𝑋𝑈 ∧ ¬ ∃ 𝑥𝑁𝑦𝑁 ( 𝑥 · 𝑦 ) = 𝑋 ) ↔ ( 𝑋𝑁 ∧ ∀ 𝑥𝑁𝑦𝑁 ( 𝑥 · 𝑦 ) ≠ 𝑋 ) ) )
19 ioran ( ¬ ( 𝑋𝑈 ∨ ∃ 𝑥𝑁𝑦𝑁 ( 𝑥 · 𝑦 ) = 𝑋 ) ↔ ( ¬ 𝑋𝑈 ∧ ¬ ∃ 𝑥𝑁𝑦𝑁 ( 𝑥 · 𝑦 ) = 𝑋 ) )
20 1 2 3 4 5 isirred ( 𝑋𝐼 ↔ ( 𝑋𝑁 ∧ ∀ 𝑥𝑁𝑦𝑁 ( 𝑥 · 𝑦 ) ≠ 𝑋 ) )
21 18 19 20 3bitr4g ( 𝑋𝐵 → ( ¬ ( 𝑋𝑈 ∨ ∃ 𝑥𝑁𝑦𝑁 ( 𝑥 · 𝑦 ) = 𝑋 ) ↔ 𝑋𝐼 ) )
22 21 con1bid ( 𝑋𝐵 → ( ¬ 𝑋𝐼 ↔ ( 𝑋𝑈 ∨ ∃ 𝑥𝑁𝑦𝑁 ( 𝑥 · 𝑦 ) = 𝑋 ) ) )