Metamath Proof Explorer

Theorem dvhvbase

Description: The vectors (vector base set) of the constructed full vector space H are all translations (for a fiducial co-atom W ). (Contributed by NM, 2-Nov-2013) (Revised by Mario Carneiro, 22-Jun-2014)

		Ref	Expression
	Hypotheses	dvhvbase.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
		dvhvbase.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
		dvhvbase.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
		dvhvbase.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
		dvhvbase.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	Assertion	dvhvbase	⊢ ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) → 𝑉 = ( 𝑇 × 𝐸 ) )

Proof

Step	Hyp	Ref	Expression
1		dvhvbase.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		dvhvbase.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3		dvhvbase.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
4		dvhvbase.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
5		dvhvbase.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
6		eqid	⊢ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
7	1 2 3 6 4	dvhset	⊢ ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) → 𝑈 = ( { ⟨ ( Base ‘ ndx ) , ( 𝑇 × 𝐸 ) ⟩ , ⟨ ( +_g ‘ ndx ) , ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) )
8	7	fveq2d	⊢ ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) → ( Base ‘ 𝑈 ) = ( Base ‘ ( { ⟨ ( Base ‘ ndx ) , ( 𝑇 × 𝐸 ) ⟩ , ⟨ ( +_g ‘ ndx ) , ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) ) )
9	2	fvexi	⊢ 𝑇 ∈ V
10	3	fvexi	⊢ 𝐸 ∈ V
11	9 10	xpex	⊢ ( 𝑇 × 𝐸 ) ∈ V
12		eqid	⊢ ( { ⟨ ( Base ‘ ndx ) , ( 𝑇 × 𝐸 ) ⟩ , ⟨ ( +_g ‘ ndx ) , ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) = ( { ⟨ ( Base ‘ ndx ) , ( 𝑇 × 𝐸 ) ⟩ , ⟨ ( +_g ‘ ndx ) , ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } )
13	12	lmodbase	⊢ ( ( 𝑇 × 𝐸 ) ∈ V → ( 𝑇 × 𝐸 ) = ( Base ‘ ( { ⟨ ( Base ‘ ndx ) , ( 𝑇 × 𝐸 ) ⟩ , ⟨ ( +_g ‘ ndx ) , ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) ) )
14	11 13	ax-mp	⊢ ( 𝑇 × 𝐸 ) = ( Base ‘ ( { ⟨ ( Base ‘ ndx ) , ( 𝑇 × 𝐸 ) ⟩ , ⟨ ( +_g ‘ ndx ) , ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) )
15	8 5 14	3eqtr4g	⊢ ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) → 𝑉 = ( 𝑇 × 𝐸 ) )