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Metamath Proof Explorer

Theorem asclmul1

Description: Left multiplication by a lifted scalar is the same as the scalar operation. (Contributed by Mario Carneiro, 9-Mar-2015)

Ref Expression
Hypotheses asclmul1.a ⊢ 𝐴 = ( algSc ‘ 𝑊 )
asclmul1.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
asclmul1.k ⊢ 𝐾 = ( Base ‘ 𝐹 )
asclmul1.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
asclmul1.t ⊢ × = ( .r ‘ 𝑊 )
asclmul1.s ⊢ · = ( ·𝑠 ‘ 𝑊 )
Assertion asclmul1 ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) → ( ( 𝐴 ‘ 𝑅 ) × 𝑋 ) = ( 𝑅 · 𝑋 ) )

Proof

Step Hyp Ref Expression
1 asclmul1.a ⊢ 𝐴 = ( algSc ‘ 𝑊 )
2 asclmul1.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
3 asclmul1.k ⊢ 𝐾 = ( Base ‘ 𝐹 )
4 asclmul1.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
5 asclmul1.t ⊢ × = ( .r ‘ 𝑊 )
6 asclmul1.s ⊢ · = ( ·𝑠 ‘ 𝑊 )
7 eqid ⊢ ( 1r ‘ 𝑊 ) = ( 1r ‘ 𝑊 )
8 1 2 3 6 7 asclval ⊢ ( 𝑅 ∈ 𝐾 → ( 𝐴 ‘ 𝑅 ) = ( 𝑅 · ( 1r ‘ 𝑊 ) ) )
9 8 3ad2ant2 ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) → ( 𝐴 ‘ 𝑅 ) = ( 𝑅 · ( 1r ‘ 𝑊 ) ) )
10 9 oveq1d ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) → ( ( 𝐴 ‘ 𝑅 ) × 𝑋 ) = ( ( 𝑅 · ( 1r ‘ 𝑊 ) ) × 𝑋 ) )
11 simp1 ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) → 𝑊 ∈ AssAlg )
12 simp2 ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) → 𝑅 ∈ 𝐾 )
13 assaring ⊢ ( 𝑊 ∈ AssAlg → 𝑊 ∈ Ring )
14 13 3ad2ant1 ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) → 𝑊 ∈ Ring )
15 4 7 ringidcl ⊢ ( 𝑊 ∈ Ring → ( 1r ‘ 𝑊 ) ∈ 𝑉 )
16 14 15 syl ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) → ( 1r ‘ 𝑊 ) ∈ 𝑉 )
17 simp3 ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) → 𝑋 ∈ 𝑉 )
18 4 2 3 6 5 assaass ⊢ ( ( 𝑊 ∈ AssAlg ∧ ( 𝑅 ∈ 𝐾 ∧ ( 1r ‘ 𝑊 ) ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( 𝑅 · ( 1r ‘ 𝑊 ) ) × 𝑋 ) = ( 𝑅 · ( ( 1r ‘ 𝑊 ) × 𝑋 ) ) )
19 11 12 16 17 18 syl13anc ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) → ( ( 𝑅 · ( 1r ‘ 𝑊 ) ) × 𝑋 ) = ( 𝑅 · ( ( 1r ‘ 𝑊 ) × 𝑋 ) ) )
20 4 5 7 ringlidm ⊢ ( ( 𝑊 ∈ Ring ∧ 𝑋 ∈ 𝑉 ) → ( ( 1r ‘ 𝑊 ) × 𝑋 ) = 𝑋 )
21 14 17 20 syl2anc ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) → ( ( 1r ‘ 𝑊 ) × 𝑋 ) = 𝑋 )
22 21 oveq2d ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) → ( 𝑅 · ( ( 1r ‘ 𝑊 ) × 𝑋 ) ) = ( 𝑅 · 𝑋 ) )
23 10 19 22 3eqtrd ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) → ( ( 𝐴 ‘ 𝑅 ) × 𝑋 ) = ( 𝑅 · 𝑋 ) )