Theorem ringdi22

Description: Expand the product of two sums in a ring. (Contributed by Thierry Arnoux, 3-Jun-2025)

	Ref	Expression
Hypotheses	ringdid.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	ringdid.p	⊢ + = ( +g ‘ 𝑅 )
	ringdid.m	⊢ · = ( .r ‘ 𝑅 )
	ringdid.r	⊢ ( 𝜑 → 𝑅 ∈ Ring )
	ringdid.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	ringdid.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	ringdid.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
	ringdi22.t	⊢ ( 𝜑 → 𝑇 ∈ 𝐵 )
Assertion	ringdi22	⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) · ( 𝑍 + 𝑇 ) ) = ( ( ( 𝑋 · 𝑍 ) + ( 𝑌 · 𝑍 ) ) + ( ( 𝑋 · 𝑇 ) + ( 𝑌 · 𝑇 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ringdid.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		ringdid.p	⊢ + = ( +g ‘ 𝑅 )
3		ringdid.m	⊢ · = ( .r ‘ 𝑅 )
4		ringdid.r	⊢ ( 𝜑 → 𝑅 ∈ Ring )
5		ringdid.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6		ringdid.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7		ringdid.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
8		ringdi22.t	⊢ ( 𝜑 → 𝑇 ∈ 𝐵 )
9	4	ringgrpd	⊢ ( 𝜑 → 𝑅 ∈ Grp )
10	1 2 9 5 6	grpcld	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝐵 )
11	1 2 3 4 10 7 8	ringdid	⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) · ( 𝑍 + 𝑇 ) ) = ( ( ( 𝑋 + 𝑌 ) · 𝑍 ) + ( ( 𝑋 + 𝑌 ) · 𝑇 ) ) )
12	1 2 3 4 5 6 7	ringdird	⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) · 𝑍 ) = ( ( 𝑋 · 𝑍 ) + ( 𝑌 · 𝑍 ) ) )
13	1 2 3 4 5 6 8	ringdird	⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) · 𝑇 ) = ( ( 𝑋 · 𝑇 ) + ( 𝑌 · 𝑇 ) ) )
14	12 13	oveq12d	⊢ ( 𝜑 → ( ( ( 𝑋 + 𝑌 ) · 𝑍 ) + ( ( 𝑋 + 𝑌 ) · 𝑇 ) ) = ( ( ( 𝑋 · 𝑍 ) + ( 𝑌 · 𝑍 ) ) + ( ( 𝑋 · 𝑇 ) + ( 𝑌 · 𝑇 ) ) ) )
15	11 14	eqtrd	⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) · ( 𝑍 + 𝑇 ) ) = ( ( ( 𝑋 · 𝑍 ) + ( 𝑌 · 𝑍 ) ) + ( ( 𝑋 · 𝑇 ) + ( 𝑌 · 𝑇 ) ) ) )