Metamath Proof Explorer

Theorem isdomn2

Description: A ring is a domain iff all nonzero elements are nonzero-divisors. (Contributed by Mario Carneiro, 28-Mar-2015)

Ref Expression
Hypotheses isdomn2.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
isdomn2.t ⊢ 𝐸 = ( RLReg ‘ 𝑅 )
isdomn2.z ⊢ 0 = ( 0g ‘ 𝑅 )
Assertion isdomn2 ( 𝑅 ∈ Domn ↔ ( 𝑅 ∈ NzRing ∧ ( 𝐵 ∖ { 0 } ) ⊆ 𝐸 ) )

Proof

Step Hyp Ref Expression
1 isdomn2.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 isdomn2.t ⊢ 𝐸 = ( RLReg ‘ 𝑅 )
3 isdomn2.z ⊢ 0 = ( 0g ‘ 𝑅 )
4 eqid ⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
5 1 4 3 isdomn ⊢ ( 𝑅 ∈ Domn ↔ ( 𝑅 ∈ NzRing ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → ( 𝑥 = 0 ∨ 𝑦 = 0 ) ) ) )
6 dfss3 ⊢ ( ( 𝐵 ∖ { 0 } ) ⊆ 𝐸 ↔ ∀ 𝑥 ∈ ( 𝐵 ∖ { 0 } ) 𝑥 ∈ 𝐸 )
7 2 1 4 3 isrrg ⊢ ( 𝑥 ∈ 𝐸 ↔ ( 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → 𝑦 = 0 ) ) )
8 7 baib ⊢ ( 𝑥 ∈ 𝐵 → ( 𝑥 ∈ 𝐸 ↔ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → 𝑦 = 0 ) ) )
9 8 imbi2d ⊢ ( 𝑥 ∈ 𝐵 → ( ( 𝑥 ≠ 0 → 𝑥 ∈ 𝐸 ) ↔ ( 𝑥 ≠ 0 → ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → 𝑦 = 0 ) ) ) )
10 9 ralbiia ⊢ ( ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≠ 0 → 𝑥 ∈ 𝐸 ) ↔ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≠ 0 → ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → 𝑦 = 0 ) ) )
11 eldifsn ⊢ ( 𝑥 ∈ ( 𝐵 ∖ { 0 } ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) )
12 11 imbi1i ⊢ ( ( 𝑥 ∈ ( 𝐵 ∖ { 0 } ) → 𝑥 ∈ 𝐸 ) ↔ ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) → 𝑥 ∈ 𝐸 ) )
13 impexp ⊢ ( ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) → 𝑥 ∈ 𝐸 ) ↔ ( 𝑥 ∈ 𝐵 → ( 𝑥 ≠ 0 → 𝑥 ∈ 𝐸 ) ) )
14 12 13 bitri ⊢ ( ( 𝑥 ∈ ( 𝐵 ∖ { 0 } ) → 𝑥 ∈ 𝐸 ) ↔ ( 𝑥 ∈ 𝐵 → ( 𝑥 ≠ 0 → 𝑥 ∈ 𝐸 ) ) )
15 14 ralbii2 ⊢ ( ∀ 𝑥 ∈ ( 𝐵 ∖ { 0 } ) 𝑥 ∈ 𝐸 ↔ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≠ 0 → 𝑥 ∈ 𝐸 ) )
16 con34b ⊢ ( ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → ( 𝑥 = 0 ∨ 𝑦 = 0 ) ) ↔ ( ¬ ( 𝑥 = 0 ∨ 𝑦 = 0 ) → ¬ ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 ) )
17 impexp ⊢ ( ( ( ¬ 𝑥 = 0 ∧ ¬ 𝑦 = 0 ) → ¬ ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 ) ↔ ( ¬ 𝑥 = 0 → ( ¬ 𝑦 = 0 → ¬ ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 ) ) )
18 ioran ⊢ ( ¬ ( 𝑥 = 0 ∨ 𝑦 = 0 ) ↔ ( ¬ 𝑥 = 0 ∧ ¬ 𝑦 = 0 ) )
19 18 imbi1i ⊢ ( ( ¬ ( 𝑥 = 0 ∨ 𝑦 = 0 ) → ¬ ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 ) ↔ ( ( ¬ 𝑥 = 0 ∧ ¬ 𝑦 = 0 ) → ¬ ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 ) )
20 df-ne ⊢ ( 𝑥 ≠ 0 ↔ ¬ 𝑥 = 0 )
21 con34b ⊢ ( ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → 𝑦 = 0 ) ↔ ( ¬ 𝑦 = 0 → ¬ ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 ) )
22 20 21 imbi12i ⊢ ( ( 𝑥 ≠ 0 → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → 𝑦 = 0 ) ) ↔ ( ¬ 𝑥 = 0 → ( ¬ 𝑦 = 0 → ¬ ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 ) ) )
23 17 19 22 3bitr4i ⊢ ( ( ¬ ( 𝑥 = 0 ∨ 𝑦 = 0 ) → ¬ ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 ) ↔ ( 𝑥 ≠ 0 → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → 𝑦 = 0 ) ) )
24 16 23 bitri ⊢ ( ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → ( 𝑥 = 0 ∨ 𝑦 = 0 ) ) ↔ ( 𝑥 ≠ 0 → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → 𝑦 = 0 ) ) )
25 24 ralbii ⊢ ( ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → ( 𝑥 = 0 ∨ 𝑦 = 0 ) ) ↔ ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≠ 0 → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → 𝑦 = 0 ) ) )
26 r19.21v ⊢ ( ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≠ 0 → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → 𝑦 = 0 ) ) ↔ ( 𝑥 ≠ 0 → ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → 𝑦 = 0 ) ) )
27 25 26 bitri ⊢ ( ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → ( 𝑥 = 0 ∨ 𝑦 = 0 ) ) ↔ ( 𝑥 ≠ 0 → ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → 𝑦 = 0 ) ) )
28 27 ralbii ⊢ ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → ( 𝑥 = 0 ∨ 𝑦 = 0 ) ) ↔ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≠ 0 → ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → 𝑦 = 0 ) ) )
29 10 15 28 3bitr4i ⊢ ( ∀ 𝑥 ∈ ( 𝐵 ∖ { 0 } ) 𝑥 ∈ 𝐸 ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → ( 𝑥 = 0 ∨ 𝑦 = 0 ) ) )
30 6 29 bitr2i ⊢ ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → ( 𝑥 = 0 ∨ 𝑦 = 0 ) ) ↔ ( 𝐵 ∖ { 0 } ) ⊆ 𝐸 )
31 30 anbi2i ⊢ ( ( 𝑅 ∈ NzRing ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → ( 𝑥 = 0 ∨ 𝑦 = 0 ) ) ) ↔ ( 𝑅 ∈ NzRing ∧ ( 𝐵 ∖ { 0 } ) ⊆ 𝐸 ) )
32 5 31 bitri ⊢ ( 𝑅 ∈ Domn ↔ ( 𝑅 ∈ NzRing ∧ ( 𝐵 ∖ { 0 } ) ⊆ 𝐸 ) )