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Metamath Proof Explorer

Theorem clmvscom

Description: Commutative law for the scalar product. (Contributed by NM, 14-Feb-2008) (Revised by AV, 7-Oct-2021)

Ref Expression
Hypotheses clmvscl.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
clmvscl.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
clmvscl.s ⊢ · = ( ·𝑠 ‘ 𝑊 )
clmvscl.k ⊢ 𝐾 = ( Base ‘ 𝐹 )
Assertion clmvscom ( ( 𝑊 ∈ ℂMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) ) → ( 𝑄 · ( 𝑅 · 𝑋 ) ) = ( 𝑅 · ( 𝑄 · 𝑋 ) ) )

Proof

Step Hyp Ref Expression
1 clmvscl.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
2 clmvscl.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
3 clmvscl.s ⊢ · = ( ·𝑠 ‘ 𝑊 )
4 clmvscl.k ⊢ 𝐾 = ( Base ‘ 𝐹 )
5 ssel ⊢ ( 𝐾 ⊆ ℂ → ( 𝑄 ∈ 𝐾 → 𝑄 ∈ ℂ ) )
6 ssel ⊢ ( 𝐾 ⊆ ℂ → ( 𝑅 ∈ 𝐾 → 𝑅 ∈ ℂ ) )
7 5 6 anim12d ⊢ ( 𝐾 ⊆ ℂ → ( ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ) → ( 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ ) ) )
8 2 4 clmsscn ⊢ ( 𝑊 ∈ ℂMod → 𝐾 ⊆ ℂ )
9 7 8 syl11 ⊢ ( ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ) → ( 𝑊 ∈ ℂMod → ( 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ ) ) )
10 9 3adant3 ⊢ ( ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) → ( 𝑊 ∈ ℂMod → ( 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ ) ) )
11 10 impcom ⊢ ( ( 𝑊 ∈ ℂMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) ) → ( 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ ) )
12 mulcom ⊢ ( ( 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ ) → ( 𝑄 · 𝑅 ) = ( 𝑅 · 𝑄 ) )
13 11 12 syl ⊢ ( ( 𝑊 ∈ ℂMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) ) → ( 𝑄 · 𝑅 ) = ( 𝑅 · 𝑄 ) )
14 13 oveq1d ⊢ ( ( 𝑊 ∈ ℂMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( 𝑄 · 𝑅 ) · 𝑋 ) = ( ( 𝑅 · 𝑄 ) · 𝑋 ) )
15 1 2 3 4 clmvsass ⊢ ( ( 𝑊 ∈ ℂMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( 𝑄 · 𝑅 ) · 𝑋 ) = ( 𝑄 · ( 𝑅 · 𝑋 ) ) )
16 3ancoma ⊢ ( ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) ↔ ( 𝑅 ∈ 𝐾 ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) )
17 1 2 3 4 clmvsass ⊢ ( ( 𝑊 ∈ ℂMod ∧ ( 𝑅 ∈ 𝐾 ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( 𝑅 · 𝑄 ) · 𝑋 ) = ( 𝑅 · ( 𝑄 · 𝑋 ) ) )
18 16 17 sylan2b ⊢ ( ( 𝑊 ∈ ℂMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( 𝑅 · 𝑄 ) · 𝑋 ) = ( 𝑅 · ( 𝑄 · 𝑋 ) ) )
19 14 15 18 3eqtr3d ⊢ ( ( 𝑊 ∈ ℂMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) ) → ( 𝑄 · ( 𝑅 · 𝑋 ) ) = ( 𝑅 · ( 𝑄 · 𝑋 ) ) )