Theorem clmvsass

Description: Scalar multiplication is a semigroup action. Analogue of lmodvsass . (Contributed by Mario Carneiro, 16-Oct-2015)

	Ref	Expression
Hypotheses	clmvscl.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	clmvscl.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
	clmvscl.s	⊢ · = ( ·𝑠 ‘ 𝑊 )
	clmvscl.k	⊢ 𝐾 = ( Base ‘ 𝐹 )
Assertion	clmvsass	⊢ ( ( 𝑊 ∈ ℂMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( 𝑄 · 𝑅 ) · 𝑋 ) = ( 𝑄 · ( 𝑅 · 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	2	clmmul	⊢ ( 𝑊 ∈ ℂMod → · = ( .r ‘ 𝐹 ) )
6	5	adantr	⊢ ( ( 𝑊 ∈ ℂMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) ) → · = ( .r ‘ 𝐹 ) )
7	6	oveqd	⊢ ( ( 𝑊 ∈ ℂMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) ) → ( 𝑄 · 𝑅 ) = ( 𝑄 ( .r ‘ 𝐹 ) 𝑅 ) )
8	7	oveq1d	⊢ ( ( 𝑊 ∈ ℂMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( 𝑄 · 𝑅 ) · 𝑋 ) = ( ( 𝑄 ( .r ‘ 𝐹 ) 𝑅 ) · 𝑋 ) )
9		clmlmod	⊢ ( 𝑊 ∈ ℂMod → 𝑊 ∈ LMod )
10		eqid	⊢ ( .r ‘ 𝐹 ) = ( .r ‘ 𝐹 )
11	1 2 3 4 10	lmodvsass	⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( 𝑄 ( .r ‘ 𝐹 ) 𝑅 ) · 𝑋 ) = ( 𝑄 · ( 𝑅 · 𝑋 ) ) )
12	9 11	sylan	⊢ ( ( 𝑊 ∈ ℂMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( 𝑄 ( .r ‘ 𝐹 ) 𝑅 ) · 𝑋 ) = ( 𝑄 · ( 𝑅 · 𝑋 ) ) )
13	8 12	eqtrd	⊢ ( ( 𝑊 ∈ ℂMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( 𝑄 · 𝑅 ) · 𝑋 ) = ( 𝑄 · ( 𝑅 · 𝑋 ) ) )