Theorem elringlsmd

Description: Membership in a product of two subsets of a ring, one direction. (Contributed by Thierry Arnoux, 13-Apr-2024)

	Ref	Expression
Hypotheses	elringlsm.1	⊢ 𝐵 = ( Base ‘ 𝑅 )
	elringlsm.2	⊢ · = ( .r ‘ 𝑅 )
	elringlsm.3	⊢ 𝐺 = ( mulGrp ‘ 𝑅 )
	elringlsm.4	⊢ × = ( LSSum ‘ 𝐺 )
	elringlsm.6	⊢ ( 𝜑 → 𝐸 ⊆ 𝐵 )
	elringlsm.7	⊢ ( 𝜑 → 𝐹 ⊆ 𝐵 )
	elringlsmd.1	⊢ ( 𝜑 → 𝑋 ∈ 𝐸 )
	elringlsmd.2	⊢ ( 𝜑 → 𝑌 ∈ 𝐹 )
Assertion	elringlsmd	⊢ ( 𝜑 → ( 𝑋 · 𝑌 ) ∈ ( 𝐸 × 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		elringlsm.1	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		elringlsm.2	⊢ · = ( .r ‘ 𝑅 )
3		elringlsm.3	⊢ 𝐺 = ( mulGrp ‘ 𝑅 )
4		elringlsm.4	⊢ × = ( LSSum ‘ 𝐺 )
5		elringlsm.6	⊢ ( 𝜑 → 𝐸 ⊆ 𝐵 )
6		elringlsm.7	⊢ ( 𝜑 → 𝐹 ⊆ 𝐵 )
7		elringlsmd.1	⊢ ( 𝜑 → 𝑋 ∈ 𝐸 )
8		elringlsmd.2	⊢ ( 𝜑 → 𝑌 ∈ 𝐹 )
9		eqidd	⊢ ( 𝜑 → ( 𝑋 · 𝑌 ) = ( 𝑋 · 𝑌 ) )
10		rspceov	⊢ ( ( 𝑋 ∈ 𝐸 ∧ 𝑌 ∈ 𝐹 ∧ ( 𝑋 · 𝑌 ) = ( 𝑋 · 𝑌 ) ) → ∃ 𝑥 ∈ 𝐸 ∃ 𝑦 ∈ 𝐹 ( 𝑋 · 𝑌 ) = ( 𝑥 · 𝑦 ) )
11	7 8 9 10	syl3anc	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐸 ∃ 𝑦 ∈ 𝐹 ( 𝑋 · 𝑌 ) = ( 𝑥 · 𝑦 ) )
12	1 2 3 4 5 6	elringlsm	⊢ ( 𝜑 → ( ( 𝑋 · 𝑌 ) ∈ ( 𝐸 × 𝐹 ) ↔ ∃ 𝑥 ∈ 𝐸 ∃ 𝑦 ∈ 𝐹 ( 𝑋 · 𝑌 ) = ( 𝑥 · 𝑦 ) ) )
13	11 12	mpbird	⊢ ( 𝜑 → ( 𝑋 · 𝑌 ) ∈ ( 𝐸 × 𝐹 ) )