Metamath Proof Explorer

Theorem assa2ass2

Description: Left- and right-associative property of an associative algebra. Notice that the scalars are not commuted! (Contributed by Zhi Wang, 11-Sep-2025)

Ref Expression
Hypotheses assa2ass.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
assa2ass.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
assa2ass.b ⊢ 𝐵 = ( Base ‘ 𝐹 )
assa2ass.m ⊢ ∗ = ( .r ‘ 𝐹 )
assa2ass.s ⊢ · = ( ·𝑠 ‘ 𝑊 )
assa2ass.t ⊢ × = ( .r ‘ 𝑊 )
Assertion assa2ass2 ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( ( 𝐴 · 𝑋 ) × ( 𝐶 · 𝑌 ) ) = ( ( 𝐴 ∗ 𝐶 ) · ( 𝑋 × 𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 assa2ass.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
2 assa2ass.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
3 assa2ass.b ⊢ 𝐵 = ( Base ‘ 𝐹 )
4 assa2ass.m ⊢ ∗ = ( .r ‘ 𝐹 )
5 assa2ass.s ⊢ · = ( ·𝑠 ‘ 𝑊 )
6 assa2ass.t ⊢ × = ( .r ‘ 𝑊 )
7 simp1 ⊢ ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → 𝑊 ∈ AssAlg )
8 simpl ⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) → 𝐴 ∈ 𝐵 )
9 8 3ad2ant2 ⊢ ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → 𝐴 ∈ 𝐵 )
10 simpl ⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → 𝑋 ∈ 𝑉 )
11 10 3ad2ant3 ⊢ ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → 𝑋 ∈ 𝑉 )
12 assalmod ⊢ ( 𝑊 ∈ AssAlg → 𝑊 ∈ LMod )
13 12 3ad2ant1 ⊢ ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → 𝑊 ∈ LMod )
14 simpr ⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) → 𝐶 ∈ 𝐵 )
15 14 3ad2ant2 ⊢ ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → 𝐶 ∈ 𝐵 )
16 simpr ⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → 𝑌 ∈ 𝑉 )
17 16 3ad2ant3 ⊢ ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → 𝑌 ∈ 𝑉 )
18 1 2 5 3 13 15 17 lmodvscld ⊢ ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝐶 · 𝑌 ) ∈ 𝑉 )
19 1 2 3 5 6 assaass ⊢ ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ ( 𝐶 · 𝑌 ) ∈ 𝑉 ) ) → ( ( 𝐴 · 𝑋 ) × ( 𝐶 · 𝑌 ) ) = ( 𝐴 · ( 𝑋 × ( 𝐶 · 𝑌 ) ) ) )
20 7 9 11 18 19 syl13anc ⊢ ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( ( 𝐴 · 𝑋 ) × ( 𝐶 · 𝑌 ) ) = ( 𝐴 · ( 𝑋 × ( 𝐶 · 𝑌 ) ) ) )
21 1 2 3 5 6 assaassr ⊢ ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ ( 𝐶 · 𝑌 ) ∈ 𝑉 ) ) → ( 𝑋 × ( 𝐴 · ( 𝐶 · 𝑌 ) ) ) = ( 𝐴 · ( 𝑋 × ( 𝐶 · 𝑌 ) ) ) )
22 21 eqcomd ⊢ ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ ( 𝐶 · 𝑌 ) ∈ 𝑉 ) ) → ( 𝐴 · ( 𝑋 × ( 𝐶 · 𝑌 ) ) ) = ( 𝑋 × ( 𝐴 · ( 𝐶 · 𝑌 ) ) ) )
23 7 9 11 18 22 syl13anc ⊢ ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝐴 · ( 𝑋 × ( 𝐶 · 𝑌 ) ) ) = ( 𝑋 × ( 𝐴 · ( 𝐶 · 𝑌 ) ) ) )
24 1 2 5 3 4 lmodvsass ⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉 ) ) → ( ( 𝐴 ∗ 𝐶 ) · 𝑌 ) = ( 𝐴 · ( 𝐶 · 𝑌 ) ) )
25 24 eqcomd ⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝐴 · ( 𝐶 · 𝑌 ) ) = ( ( 𝐴 ∗ 𝐶 ) · 𝑌 ) )
26 25 oveq2d ⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝑋 × ( 𝐴 · ( 𝐶 · 𝑌 ) ) ) = ( 𝑋 × ( ( 𝐴 ∗ 𝐶 ) · 𝑌 ) ) )
27 13 9 15 17 26 syl13anc ⊢ ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝑋 × ( 𝐴 · ( 𝐶 · 𝑌 ) ) ) = ( 𝑋 × ( ( 𝐴 ∗ 𝐶 ) · 𝑌 ) ) )
28 2 assasca ⊢ ( 𝑊 ∈ AssAlg → 𝐹 ∈ Ring )
29 28 adantr ⊢ ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ) → 𝐹 ∈ Ring )
30 8 adantl ⊢ ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ) → 𝐴 ∈ 𝐵 )
31 14 adantl ⊢ ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ) → 𝐶 ∈ 𝐵 )
32 3 4 29 30 31 ringcld ⊢ ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ) → ( 𝐴 ∗ 𝐶 ) ∈ 𝐵 )
33 32 3adant3 ⊢ ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝐴 ∗ 𝐶 ) ∈ 𝐵 )
34 1 2 3 5 6 assaassr ⊢ ( ( 𝑊 ∈ AssAlg ∧ ( ( 𝐴 ∗ 𝐶 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝑋 × ( ( 𝐴 ∗ 𝐶 ) · 𝑌 ) ) = ( ( 𝐴 ∗ 𝐶 ) · ( 𝑋 × 𝑌 ) ) )
35 7 33 11 17 34 syl13anc ⊢ ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝑋 × ( ( 𝐴 ∗ 𝐶 ) · 𝑌 ) ) = ( ( 𝐴 ∗ 𝐶 ) · ( 𝑋 × 𝑌 ) ) )
36 27 35 eqtrd ⊢ ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝑋 × ( 𝐴 · ( 𝐶 · 𝑌 ) ) ) = ( ( 𝐴 ∗ 𝐶 ) · ( 𝑋 × 𝑌 ) ) )
37 20 23 36 3eqtrd ⊢ ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( ( 𝐴 · 𝑋 ) × ( 𝐶 · 𝑌 ) ) = ( ( 𝐴 ∗ 𝐶 ) · ( 𝑋 × 𝑌 ) ) )