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Metamath Proof Explorer

Theorem drngmcl

Description: The product of two nonzero elements of a division ring is nonzero. (Contributed by NM, 7-Sep-2011) (Proof shortened by SN, 25-Jun-2025)

Ref Expression
Hypotheses drngmcl.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
drngmcl.t ⊢ · = ( .r ‘ 𝑅 )
drngmcl.z ⊢ 0 = ( 0g ‘ 𝑅 )
Assertion drngmcl ( ( 𝑅 ∈ DivRing ∧ 𝑋 ∈ ( 𝐵 ∖ { 0 } ) ∧ 𝑌 ∈ ( 𝐵 ∖ { 0 } ) ) → ( 𝑋 · 𝑌 ) ∈ ( 𝐵 ∖ { 0 } ) )

Proof

Step Hyp Ref Expression
1 drngmcl.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 drngmcl.t ⊢ · = ( .r ‘ 𝑅 )
3 drngmcl.z ⊢ 0 = ( 0g ‘ 𝑅 )
4 drngring ⊢ ( 𝑅 ∈ DivRing → 𝑅 ∈ Ring )
5 eldifi ⊢ ( 𝑋 ∈ ( 𝐵 ∖ { 0 } ) → 𝑋 ∈ 𝐵 )
6 eldifi ⊢ ( 𝑌 ∈ ( 𝐵 ∖ { 0 } ) → 𝑌 ∈ 𝐵 )
7 1 2 ringcl ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 · 𝑌 ) ∈ 𝐵 )
8 4 5 6 7 syl3an ⊢ ( ( 𝑅 ∈ DivRing ∧ 𝑋 ∈ ( 𝐵 ∖ { 0 } ) ∧ 𝑌 ∈ ( 𝐵 ∖ { 0 } ) ) → ( 𝑋 · 𝑌 ) ∈ 𝐵 )
9 drngdomn ⊢ ( 𝑅 ∈ DivRing → 𝑅 ∈ Domn )
10 eldifsn ⊢ ( 𝑋 ∈ ( 𝐵 ∖ { 0 } ) ↔ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) )
11 10 biimpi ⊢ ( 𝑋 ∈ ( 𝐵 ∖ { 0 } ) → ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) )
12 eldifsn ⊢ ( 𝑌 ∈ ( 𝐵 ∖ { 0 } ) ↔ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≠ 0 ) )
13 12 biimpi ⊢ ( 𝑌 ∈ ( 𝐵 ∖ { 0 } ) → ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≠ 0 ) )
14 1 2 3 domnmuln0 ⊢ ( ( 𝑅 ∈ Domn ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≠ 0 ) ) → ( 𝑋 · 𝑌 ) ≠ 0 )
15 9 11 13 14 syl3an ⊢ ( ( 𝑅 ∈ DivRing ∧ 𝑋 ∈ ( 𝐵 ∖ { 0 } ) ∧ 𝑌 ∈ ( 𝐵 ∖ { 0 } ) ) → ( 𝑋 · 𝑌 ) ≠ 0 )
16 8 15 eldifsnd ⊢ ( ( 𝑅 ∈ DivRing ∧ 𝑋 ∈ ( 𝐵 ∖ { 0 } ) ∧ 𝑌 ∈ ( 𝐵 ∖ { 0 } ) ) → ( 𝑋 · 𝑌 ) ∈ ( 𝐵 ∖ { 0 } ) )