Theorem ldualsca

Description: The ring of scalars of the dual of a vector space. (Contributed by NM, 18-Oct-2014)

	Ref	Expression
Hypotheses	ldualsca.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
	ldualsca.o	⊢ 𝑂 = ( oppr ‘ 𝐹 )
	ldualsca.d	⊢ 𝐷 = ( LDual ‘ 𝑊 )
	ldualsca.r	⊢ 𝑅 = ( Scalar ‘ 𝐷 )
	ldualsca.w	⊢ ( 𝜑 → 𝑊 ∈ 𝑋 )
Assertion	ldualsca	⊢ ( 𝜑 → 𝑅 = 𝑂 )

Proof

Step	Hyp	Ref	Expression
1		ldualsca.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2		ldualsca.o	⊢ 𝑂 = ( oppr ‘ 𝐹 )
3		ldualsca.d	⊢ 𝐷 = ( LDual ‘ 𝑊 )
4		ldualsca.r	⊢ 𝑅 = ( Scalar ‘ 𝐷 )
5		ldualsca.w	⊢ ( 𝜑 → 𝑊 ∈ 𝑋 )
6		eqid	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
7		eqid	⊢ ( +g ‘ 𝐹 ) = ( +g ‘ 𝐹 )
8		eqid	⊢ ( ∘f ( +g ‘ 𝐹 ) ↾ ( ( LFnl ‘ 𝑊 ) × ( LFnl ‘ 𝑊 ) ) ) = ( ∘f ( +g ‘ 𝐹 ) ↾ ( ( LFnl ‘ 𝑊 ) × ( LFnl ‘ 𝑊 ) ) )
9		eqid	⊢ ( LFnl ‘ 𝑊 ) = ( LFnl ‘ 𝑊 )
10		eqid	⊢ ( Base ‘ 𝐹 ) = ( Base ‘ 𝐹 )
11		eqid	⊢ ( .r ‘ 𝐹 ) = ( .r ‘ 𝐹 )
12		eqid	⊢ ( 𝑘 ∈ ( Base ‘ 𝐹 ) , 𝑓 ∈ ( LFnl ‘ 𝑊 ) ↦ ( 𝑓 ∘f ( .r ‘ 𝐹 ) ( ( Base ‘ 𝑊 ) × { 𝑘 } ) ) ) = ( 𝑘 ∈ ( Base ‘ 𝐹 ) , 𝑓 ∈ ( LFnl ‘ 𝑊 ) ↦ ( 𝑓 ∘f ( .r ‘ 𝐹 ) ( ( Base ‘ 𝑊 ) × { 𝑘 } ) ) )
13	6 7 8 9 3 1 10 11 2 12 5	ldualset	⊢ ( 𝜑 → 𝐷 = ( { ⟨ ( Base ‘ ndx ) , ( LFnl ‘ 𝑊 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( ∘f ( +g ‘ 𝐹 ) ↾ ( ( LFnl ‘ 𝑊 ) × ( LFnl ‘ 𝑊 ) ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑂 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑘 ∈ ( Base ‘ 𝐹 ) , 𝑓 ∈ ( LFnl ‘ 𝑊 ) ↦ ( 𝑓 ∘f ( .r ‘ 𝐹 ) ( ( Base ‘ 𝑊 ) × { 𝑘 } ) ) ) ⟩ } ) )
14	13	fveq2d	⊢ ( 𝜑 → ( Scalar ‘ 𝐷 ) = ( Scalar ‘ ( { ⟨ ( Base ‘ ndx ) , ( LFnl ‘ 𝑊 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( ∘f ( +g ‘ 𝐹 ) ↾ ( ( LFnl ‘ 𝑊 ) × ( LFnl ‘ 𝑊 ) ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑂 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑘 ∈ ( Base ‘ 𝐹 ) , 𝑓 ∈ ( LFnl ‘ 𝑊 ) ↦ ( 𝑓 ∘f ( .r ‘ 𝐹 ) ( ( Base ‘ 𝑊 ) × { 𝑘 } ) ) ) ⟩ } ) ) )
15	2	fvexi	⊢ 𝑂 ∈ V
16		eqid	⊢ ( { ⟨ ( Base ‘ ndx ) , ( LFnl ‘ 𝑊 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( ∘f ( +g ‘ 𝐹 ) ↾ ( ( LFnl ‘ 𝑊 ) × ( LFnl ‘ 𝑊 ) ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑂 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑘 ∈ ( Base ‘ 𝐹 ) , 𝑓 ∈ ( LFnl ‘ 𝑊 ) ↦ ( 𝑓 ∘f ( .r ‘ 𝐹 ) ( ( Base ‘ 𝑊 ) × { 𝑘 } ) ) ) ⟩ } ) = ( { ⟨ ( Base ‘ ndx ) , ( LFnl ‘ 𝑊 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( ∘f ( +g ‘ 𝐹 ) ↾ ( ( LFnl ‘ 𝑊 ) × ( LFnl ‘ 𝑊 ) ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑂 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑘 ∈ ( Base ‘ 𝐹 ) , 𝑓 ∈ ( LFnl ‘ 𝑊 ) ↦ ( 𝑓 ∘f ( .r ‘ 𝐹 ) ( ( Base ‘ 𝑊 ) × { 𝑘 } ) ) ) ⟩ } )
17	16	lmodsca	⊢ ( 𝑂 ∈ V → 𝑂 = ( Scalar ‘ ( { ⟨ ( Base ‘ ndx ) , ( LFnl ‘ 𝑊 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( ∘f ( +g ‘ 𝐹 ) ↾ ( ( LFnl ‘ 𝑊 ) × ( LFnl ‘ 𝑊 ) ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑂 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑘 ∈ ( Base ‘ 𝐹 ) , 𝑓 ∈ ( LFnl ‘ 𝑊 ) ↦ ( 𝑓 ∘f ( .r ‘ 𝐹 ) ( ( Base ‘ 𝑊 ) × { 𝑘 } ) ) ) ⟩ } ) ) )
18	15 17	ax-mp	⊢ 𝑂 = ( Scalar ‘ ( { ⟨ ( Base ‘ ndx ) , ( LFnl ‘ 𝑊 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( ∘f ( +g ‘ 𝐹 ) ↾ ( ( LFnl ‘ 𝑊 ) × ( LFnl ‘ 𝑊 ) ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑂 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑘 ∈ ( Base ‘ 𝐹 ) , 𝑓 ∈ ( LFnl ‘ 𝑊 ) ↦ ( 𝑓 ∘f ( .r ‘ 𝐹 ) ( ( Base ‘ 𝑊 ) × { 𝑘 } ) ) ) ⟩ } ) )
19	14 4 18	3eqtr4g	⊢ ( 𝜑 → 𝑅 = 𝑂 )