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Metamath Proof Explorer

Theorem dvafmulr

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

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

Proof

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