This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem domnmuln0rd

Description: In a domain, factors of a nonzero product are nonzero. (Contributed by Thierry Arnoux, 8-Jun-2025)

Ref Expression
Hypotheses domnmuln0rd.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
domnmuln0rd.t ⊢ · = ( .r ‘ 𝑅 )
domnmuln0rd.z ⊢ 0 = ( 0g ‘ 𝑅 )
domnmuln0rd.1 ⊢ ( 𝜑 → 𝑅 ∈ Domn )
domnmuln0rd.2 ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
domnmuln0rd.3 ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
domnmuln0rd.4 ⊢ ( 𝜑 → ( 𝑋 · 𝑌 ) ≠ 0 )
Assertion domnmuln0rd ( 𝜑 → ( 𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ) )

Proof

Step Hyp Ref Expression
1 domnmuln0rd.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 domnmuln0rd.t ⊢ · = ( .r ‘ 𝑅 )
3 domnmuln0rd.z ⊢ 0 = ( 0g ‘ 𝑅 )
4 domnmuln0rd.1 ⊢ ( 𝜑 → 𝑅 ∈ Domn )
5 domnmuln0rd.2 ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6 domnmuln0rd.3 ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7 domnmuln0rd.4 ⊢ ( 𝜑 → ( 𝑋 · 𝑌 ) ≠ 0 )
8 1 2 3 domneq0 ⊢ ( ( 𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 · 𝑌 ) = 0 ↔ ( 𝑋 = 0 ∨ 𝑌 = 0 ) ) )
9 4 5 6 8 syl3anc ⊢ ( 𝜑 → ( ( 𝑋 · 𝑌 ) = 0 ↔ ( 𝑋 = 0 ∨ 𝑌 = 0 ) ) )
10 9 necon3abid ⊢ ( 𝜑 → ( ( 𝑋 · 𝑌 ) ≠ 0 ↔ ¬ ( 𝑋 = 0 ∨ 𝑌 = 0 ) ) )
11 7 10 mpbid ⊢ ( 𝜑 → ¬ ( 𝑋 = 0 ∨ 𝑌 = 0 ) )
12 ioran ⊢ ( ¬ ( 𝑋 = 0 ∨ 𝑌 = 0 ) ↔ ( ¬ 𝑋 = 0 ∧ ¬ 𝑌 = 0 ) )
13 11 12 sylib ⊢ ( 𝜑 → ( ¬ 𝑋 = 0 ∧ ¬ 𝑌 = 0 ) )
14 neqne ⊢ ( ¬ 𝑋 = 0 → 𝑋 ≠ 0 )
15 neqne ⊢ ( ¬ 𝑌 = 0 → 𝑌 ≠ 0 )
16 14 15 anim12i ⊢ ( ( ¬ 𝑋 = 0 ∧ ¬ 𝑌 = 0 ) → ( 𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ) )
17 13 16 syl ⊢ ( 𝜑 → ( 𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ) )