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Metamath Proof Explorer

Theorem ringdi22

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

Ref Expression
Hypotheses ringdi22.1 ⊢ 𝐵 = ( Base ‘ 𝑅 )
ringdi22.2 ⊢ + = ( +g ‘ 𝑅 )
ringdi22.3 ⊢ · = ( .r ‘ 𝑅 )
ringdi22.4 ⊢ ( 𝜑 → 𝑅 ∈ Ring )
ringdi22.5 ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
ringdi22.6 ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
ringdi22.7 ⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
ringdi22.8 ⊢ ( 𝜑 → 𝑇 ∈ 𝐵 )
Assertion ringdi22 ( 𝜑 → ( ( 𝑋 + 𝑌 ) · ( 𝑍 + 𝑇 ) ) = ( ( ( 𝑋 · 𝑍 ) + ( 𝑌 · 𝑍 ) ) + ( ( 𝑋 · 𝑇 ) + ( 𝑌 · 𝑇 ) ) ) )

Proof

Step Hyp Ref Expression
1 ringdi22.1 ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 ringdi22.2 ⊢ + = ( +g ‘ 𝑅 )
3 ringdi22.3 ⊢ · = ( .r ‘ 𝑅 )
4 ringdi22.4 ⊢ ( 𝜑 → 𝑅 ∈ Ring )
5 ringdi22.5 ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6 ringdi22.6 ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7 ringdi22.7 ⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
8 ringdi22.8 ⊢ ( 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 ) ) ) )