Theorem isdomn5

Description: The right conjunct in the right hand side of the equivalence of isdomn is logically equivalent to a less symmetric version where one of the variables is restricted to be nonzero. (Contributed by SN, 16-Sep-2024)

	Ref	Expression
Assertion	isdomn5	⊢ ( ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( ( 𝑎 · 𝑏 ) = 0 → ( 𝑎 = 0 ∨ 𝑏 = 0 ) ) ↔ ∀ 𝑎 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑏 ∈ 𝐵 ( ( 𝑎 · 𝑏 ) = 0 → 𝑏 = 0 ) )

Proof

Step	Hyp	Ref	Expression
1		bi2.04	⊢ ( ( ¬ 𝑎 = 0 → ( ( 𝑎 · 𝑏 ) = 0 → 𝑏 = 0 ) ) ↔ ( ( 𝑎 · 𝑏 ) = 0 → ( ¬ 𝑎 = 0 → 𝑏 = 0 ) ) )
2		df-ne	⊢ ( 𝑎 ≠ 0 ↔ ¬ 𝑎 = 0 )
3	2	imbi1i	⊢ ( ( 𝑎 ≠ 0 → ( ( 𝑎 · 𝑏 ) = 0 → 𝑏 = 0 ) ) ↔ ( ¬ 𝑎 = 0 → ( ( 𝑎 · 𝑏 ) = 0 → 𝑏 = 0 ) ) )
4		df-or	⊢ ( ( 𝑎 = 0 ∨ 𝑏 = 0 ) ↔ ( ¬ 𝑎 = 0 → 𝑏 = 0 ) )
5	4	imbi2i	⊢ ( ( ( 𝑎 · 𝑏 ) = 0 → ( 𝑎 = 0 ∨ 𝑏 = 0 ) ) ↔ ( ( 𝑎 · 𝑏 ) = 0 → ( ¬ 𝑎 = 0 → 𝑏 = 0 ) ) )
6	1 3 5	3bitr4ri	⊢ ( ( ( 𝑎 · 𝑏 ) = 0 → ( 𝑎 = 0 ∨ 𝑏 = 0 ) ) ↔ ( 𝑎 ≠ 0 → ( ( 𝑎 · 𝑏 ) = 0 → 𝑏 = 0 ) ) )
7	6	2ralbii	⊢ ( ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( ( 𝑎 · 𝑏 ) = 0 → ( 𝑎 = 0 ∨ 𝑏 = 0 ) ) ↔ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ≠ 0 → ( ( 𝑎 · 𝑏 ) = 0 → 𝑏 = 0 ) ) )
8		r19.21v	⊢ ( ∀ 𝑏 ∈ 𝐵 ( 𝑎 ≠ 0 → ( ( 𝑎 · 𝑏 ) = 0 → 𝑏 = 0 ) ) ↔ ( 𝑎 ≠ 0 → ∀ 𝑏 ∈ 𝐵 ( ( 𝑎 · 𝑏 ) = 0 → 𝑏 = 0 ) ) )
9	8	ralbii	⊢ ( ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ≠ 0 → ( ( 𝑎 · 𝑏 ) = 0 → 𝑏 = 0 ) ) ↔ ∀ 𝑎 ∈ 𝐵 ( 𝑎 ≠ 0 → ∀ 𝑏 ∈ 𝐵 ( ( 𝑎 · 𝑏 ) = 0 → 𝑏 = 0 ) ) )
10		raldifsnb	⊢ ( ∀ 𝑎 ∈ 𝐵 ( 𝑎 ≠ 0 → ∀ 𝑏 ∈ 𝐵 ( ( 𝑎 · 𝑏 ) = 0 → 𝑏 = 0 ) ) ↔ ∀ 𝑎 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑏 ∈ 𝐵 ( ( 𝑎 · 𝑏 ) = 0 → 𝑏 = 0 ) )
11	7 9 10	3bitri	⊢ ( ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( ( 𝑎 · 𝑏 ) = 0 → ( 𝑎 = 0 ∨ 𝑏 = 0 ) ) ↔ ∀ 𝑎 ∈ ( 𝐵 ∖ { 0 } ) ∀ 𝑏 ∈ 𝐵 ( ( 𝑎 · 𝑏 ) = 0 → 𝑏 = 0 ) )