Theorem clmvsneg

Description: Multiplication of a vector by a negated scalar. ( lmodvsneg analog.) (Contributed by Mario Carneiro, 16-Oct-2015)

	Ref	Expression
Hypotheses	clmvsneg.b	⊢ 𝐵 = ( Base ‘ 𝑊 )
	clmvsneg.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
	clmvsneg.s	⊢ · = ( ·𝑠 ‘ 𝑊 )
	clmvsneg.n	⊢ 𝑁 = ( invg ‘ 𝑊 )
	clmvsneg.k	⊢ 𝐾 = ( Base ‘ 𝐹 )
	clmvsneg.w	⊢ ( 𝜑 → 𝑊 ∈ ℂMod )
	clmvsneg.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	clmvsneg.r	⊢ ( 𝜑 → 𝑅 ∈ 𝐾 )
Assertion	clmvsneg	⊢ ( 𝜑 → ( 𝑁 ‘ ( 𝑅 · 𝑋 ) ) = ( - 𝑅 · 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		clmvsneg.b	⊢ 𝐵 = ( Base ‘ 𝑊 )
2		clmvsneg.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
3		clmvsneg.s	⊢ · = ( ·𝑠 ‘ 𝑊 )
4		clmvsneg.n	⊢ 𝑁 = ( invg ‘ 𝑊 )
5		clmvsneg.k	⊢ 𝐾 = ( Base ‘ 𝐹 )
6		clmvsneg.w	⊢ ( 𝜑 → 𝑊 ∈ ℂMod )
7		clmvsneg.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
8		clmvsneg.r	⊢ ( 𝜑 → 𝑅 ∈ 𝐾 )
9		eqid	⊢ ( invg ‘ 𝐹 ) = ( invg ‘ 𝐹 )
10		clmlmod	⊢ ( 𝑊 ∈ ℂMod → 𝑊 ∈ LMod )
11	6 10	syl	⊢ ( 𝜑 → 𝑊 ∈ LMod )
12	1 2 3 4 5 9 11 7 8	lmodvsneg	⊢ ( 𝜑 → ( 𝑁 ‘ ( 𝑅 · 𝑋 ) ) = ( ( ( invg ‘ 𝐹 ) ‘ 𝑅 ) · 𝑋 ) )
13	2 5	clmneg	⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝑅 ∈ 𝐾 ) → - 𝑅 = ( ( invg ‘ 𝐹 ) ‘ 𝑅 ) )
14	6 8 13	syl2anc	⊢ ( 𝜑 → - 𝑅 = ( ( invg ‘ 𝐹 ) ‘ 𝑅 ) )
15	14	oveq1d	⊢ ( 𝜑 → ( - 𝑅 · 𝑋 ) = ( ( ( invg ‘ 𝐹 ) ‘ 𝑅 ) · 𝑋 ) )
16	12 15	eqtr4d	⊢ ( 𝜑 → ( 𝑁 ‘ ( 𝑅 · 𝑋 ) ) = ( - 𝑅 · 𝑋 ) )