Theorem lfl1sc

Description: The (right vector space) scalar product of a functional with one is the functional. (Contributed by NM, 21-Oct-2014)

	Ref	Expression
Hypotheses	lfl1sc.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	lfl1sc.d	⊢ 𝐷 = ( Scalar ‘ 𝑊 )
	lfl1sc.f	⊢ 𝐹 = ( LFnl ‘ 𝑊 )
	lfl1sc.k	⊢ 𝐾 = ( Base ‘ 𝐷 )
	lfl1sc.t	⊢ · = ( .r ‘ 𝐷 )
	lfl1sc.i	⊢ 1 = ( 1r ‘ 𝐷 )
	lfl1sc.w	⊢ ( 𝜑 → 𝑊 ∈ LMod )
	lfl1sc.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
Assertion	lfl1sc	⊢ ( 𝜑 → ( 𝐺 ∘f · ( 𝑉 × { 1 } ) ) = 𝐺 )

Proof

Step	Hyp	Ref	Expression
1		lfl1sc.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		lfl1sc.d	⊢ 𝐷 = ( Scalar ‘ 𝑊 )
3		lfl1sc.f	⊢ 𝐹 = ( LFnl ‘ 𝑊 )
4		lfl1sc.k	⊢ 𝐾 = ( Base ‘ 𝐷 )
5		lfl1sc.t	⊢ · = ( .r ‘ 𝐷 )
6		lfl1sc.i	⊢ 1 = ( 1r ‘ 𝐷 )
7		lfl1sc.w	⊢ ( 𝜑 → 𝑊 ∈ LMod )
8		lfl1sc.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
9	1	fvexi	⊢ 𝑉 ∈ V
10	9	a1i	⊢ ( 𝜑 → 𝑉 ∈ V )
11	2 4 1 3	lflf	⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ) → 𝐺 : 𝑉 ⟶ 𝐾 )
12	7 8 11	syl2anc	⊢ ( 𝜑 → 𝐺 : 𝑉 ⟶ 𝐾 )
13	6	fvexi	⊢ 1 ∈ V
14	13	a1i	⊢ ( 𝜑 → 1 ∈ V )
15	2	lmodring	⊢ ( 𝑊 ∈ LMod → 𝐷 ∈ Ring )
16	7 15	syl	⊢ ( 𝜑 → 𝐷 ∈ Ring )
17	4 5 6	ringridm	⊢ ( ( 𝐷 ∈ Ring ∧ 𝑘 ∈ 𝐾 ) → ( 𝑘 · 1 ) = 𝑘 )
18	16 17	sylan	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐾 ) → ( 𝑘 · 1 ) = 𝑘 )
19	10 12 14 18	caofid0r	⊢ ( 𝜑 → ( 𝐺 ∘f · ( 𝑉 × { 1 } ) ) = 𝐺 )