dvhset

Description: The constructed full vector space H for a lattice K . (Contributed by NM, 17-Oct-2013) (Revised by Mario Carneiro, 22-Jun-2014)

	Ref	Expression
Hypotheses	dvhset.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dvhset.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	dvhset.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
	dvhset.d	⊢ 𝐷 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
	dvhset.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
Assertion	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 ) , 𝐷 ⟩ } ∪ { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) )

Proof

Step	Hyp	Ref	Expression
1		dvhset.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		dvhset.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3		dvhset.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
4		dvhset.d	⊢ 𝐷 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
5		dvhset.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
6	1	dvhfset	⊢ ( 𝐾 ∈ 𝑋 → ( DVecH ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ ( { ⟨ ( Base ‘ ndx ) , ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ⟩ , ⟨ ( +_g ‘ ndx ) , ( 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) , 𝑔 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑤 ) ⟩ } ∪ { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) ) )
7	6	fveq1d	⊢ ( 𝐾 ∈ 𝑋 → ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( 𝑤 ∈ 𝐻 ↦ ( { ⟨ ( Base ‘ ndx ) , ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ⟩ , ⟨ ( +_g ‘ ndx ) , ( 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) , 𝑔 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑤 ) ⟩ } ∪ { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) ) ‘ 𝑊 ) )
8	5 7	syl5eq	⊢ ( 𝐾 ∈ 𝑋 → 𝑈 = ( ( 𝑤 ∈ 𝐻 ↦ ( { ⟨ ( Base ‘ ndx ) , ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ⟩ , ⟨ ( +_g ‘ ndx ) , ( 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) , 𝑔 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑤 ) ⟩ } ∪ { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) ) ‘ 𝑊 ) )
9		fveq2	⊢ ( 𝑤 = 𝑊 → ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) )
10	9 2	eqtr4di	⊢ ( 𝑤 = 𝑊 → ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) = 𝑇 )
11		fveq2	⊢ ( 𝑤 = 𝑊 → ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) )
12	11 3	eqtr4di	⊢ ( 𝑤 = 𝑊 → ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) = 𝐸 )
13	10 12	xpeq12d	⊢ ( 𝑤 = 𝑊 → ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) = ( 𝑇 × 𝐸 ) )
14	13	opeq2d	⊢ ( 𝑤 = 𝑊 → ⟨ ( Base ‘ ndx ) , ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ⟩ = ⟨ ( Base ‘ ndx ) , ( 𝑇 × 𝐸 ) ⟩ )
15	10	mpteq1d	⊢ ( 𝑤 = 𝑊 → ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) = ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) )
16	15	opeq2d	⊢ ( 𝑤 = 𝑊 → ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ = ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ )
17	13 13 16	mpoeq123dv	⊢ ( 𝑤 = 𝑊 → ( 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) , 𝑔 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) = ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) )
18	17	opeq2d	⊢ ( 𝑤 = 𝑊 → ⟨ ( +_g ‘ ndx ) , ( 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) , 𝑔 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ = ⟨ ( +_g ‘ ndx ) , ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ )
19		fveq2	⊢ ( 𝑤 = 𝑊 → ( ( EDRing ‘ 𝐾 ) ‘ 𝑤 ) = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) )
20	19 4	eqtr4di	⊢ ( 𝑤 = 𝑊 → ( ( EDRing ‘ 𝐾 ) ‘ 𝑤 ) = 𝐷 )
21	20	opeq2d	⊢ ( 𝑤 = 𝑊 → ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑤 ) ⟩ = ⟨ ( Scalar ‘ ndx ) , 𝐷 ⟩ )
22	14 18 21	tpeq123d	⊢ ( 𝑤 = 𝑊 → { ⟨ ( Base ‘ ndx ) , ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ⟩ , ⟨ ( +_g ‘ ndx ) , ( 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) , 𝑔 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑤 ) ⟩ } = { ⟨ ( 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 ) , 𝐷 ⟩ } )
23		eqidd	⊢ ( 𝑤 = 𝑊 → ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ = ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ )
24	12 13 23	mpoeq123dv	⊢ ( 𝑤 = 𝑊 → ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) = ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) )
25	24	opeq2d	⊢ ( 𝑤 = 𝑊 → ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ = ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ )
26	25	sneqd	⊢ ( 𝑤 = 𝑊 → { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } = { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } )
27	22 26	uneq12d	⊢ ( 𝑤 = 𝑊 → ( { ⟨ ( Base ‘ ndx ) , ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ⟩ , ⟨ ( +_g ‘ ndx ) , ( 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) , 𝑔 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑤 ) ⟩ } ∪ { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( 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 ) , 𝐷 ⟩ } ∪ { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) )
28		eqid	⊢ ( 𝑤 ∈ 𝐻 ↦ ( { ⟨ ( Base ‘ ndx ) , ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ⟩ , ⟨ ( +_g ‘ ndx ) , ( 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) , 𝑔 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑤 ) ⟩ } ∪ { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) ) = ( 𝑤 ∈ 𝐻 ↦ ( { ⟨ ( Base ‘ ndx ) , ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ⟩ , ⟨ ( +_g ‘ ndx ) , ( 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) , 𝑔 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑤 ) ⟩ } ∪ { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) )
29		tpex	⊢ { ⟨ ( 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 ) , 𝐷 ⟩ } ∈ V
30		snex	⊢ { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ∈ V
31	29 30	unex	⊢ ( { ⟨ ( 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 ) , 𝐷 ⟩ } ∪ { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) ∈ V
32	27 28 31	fvmpt	⊢ ( 𝑊 ∈ 𝐻 → ( ( 𝑤 ∈ 𝐻 ↦ ( { ⟨ ( Base ‘ ndx ) , ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ⟩ , ⟨ ( +_g ‘ ndx ) , ( 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) , 𝑔 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑤 ) ⟩ } ∪ { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) ) ‘ 𝑊 ) = ( { ⟨ ( 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 ) , 𝐷 ⟩ } ∪ { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) )
33	8 32	sylan9eq	⊢ ( ( 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 ) , 𝐷 ⟩ } ∪ { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) )

Metamath Proof Explorer

Theorem dvhset

Proof