Metamath Proof Explorer


Theorem assaass

Description: Left-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014)

Ref Expression
Hypotheses isassa.v 𝑉 = ( Base ‘ 𝑊 )
isassa.f 𝐹 = ( Scalar ‘ 𝑊 )
isassa.b 𝐵 = ( Base ‘ 𝐹 )
isassa.s · = ( ·𝑠𝑊 )
isassa.t × = ( .r𝑊 )
Assertion assaass ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴𝐵𝑋𝑉𝑌𝑉 ) ) → ( ( 𝐴 · 𝑋 ) × 𝑌 ) = ( 𝐴 · ( 𝑋 × 𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 isassa.v 𝑉 = ( Base ‘ 𝑊 )
2 isassa.f 𝐹 = ( Scalar ‘ 𝑊 )
3 isassa.b 𝐵 = ( Base ‘ 𝐹 )
4 isassa.s · = ( ·𝑠𝑊 )
5 isassa.t × = ( .r𝑊 )
6 1 2 3 4 5 assalem ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴𝐵𝑋𝑉𝑌𝑉 ) ) → ( ( ( 𝐴 · 𝑋 ) × 𝑌 ) = ( 𝐴 · ( 𝑋 × 𝑌 ) ) ∧ ( 𝑋 × ( 𝐴 · 𝑌 ) ) = ( 𝐴 · ( 𝑋 × 𝑌 ) ) ) )
7 6 simpld ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴𝐵𝑋𝑉𝑌𝑉 ) ) → ( ( 𝐴 · 𝑋 ) × 𝑌 ) = ( 𝐴 · ( 𝑋 × 𝑌 ) ) )