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Metamath Proof Explorer

Theorem dvafplusg

Description: Ring addition operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013) (Revised by Mario Carneiro, 22-Jun-2014)

Ref Expression
Hypotheses dvafplus.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dvafplus.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
dvafplus.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
dvafplus.u ⊢ 𝑈 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 )
dvafplus.f ⊢ 𝐹 = ( Scalar ‘ 𝑈 )
dvafplus.p ⊢ + = ( +g ‘ 𝐹 )
Assertion dvafplusg ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → + = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 dvafplus.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 dvafplus.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3 dvafplus.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
4 dvafplus.u ⊢ 𝑈 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 )
5 dvafplus.f ⊢ 𝐹 = ( Scalar ‘ 𝑈 )
6 dvafplus.p ⊢ + = ( +g ‘ 𝐹 )
7 eqid ⊢ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
8 1 7 4 5 dvasca ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐹 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) )
9 8 fveq2d ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( +g ‘ 𝐹 ) = ( +g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) )
10 6 9 eqtrid ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → + = ( +g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) )
11 eqid ⊢ ( +g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) = ( +g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) )
12 1 2 3 7 11 erngfplus ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( +g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) )
13 10 12 eqtrd ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → + = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) )