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Metamath Proof Explorer

Theorem dvaplusg

Description: Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013)

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 dvaplusg ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ) ) → ( 𝑅 + 𝑆 ) = ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑅 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 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 1 2 3 4 5 6 dvafplusg ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → + = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑔 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑔 ) ∘ ( 𝑡 ‘ 𝑔 ) ) ) ) )
8 7 oveqd ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( 𝑅 + 𝑆 ) = ( 𝑅 ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑔 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑔 ) ∘ ( 𝑡 ‘ 𝑔 ) ) ) ) 𝑆 ) )
9 eqid ⊢ ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑔 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑔 ) ∘ ( 𝑡 ‘ 𝑔 ) ) ) ) = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑔 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑔 ) ∘ ( 𝑡 ‘ 𝑔 ) ) ) )
10 9 2 tendopl ⊢ ( ( 𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ) → ( 𝑅 ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑔 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑔 ) ∘ ( 𝑡 ‘ 𝑔 ) ) ) ) 𝑆 ) = ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑅 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 𝑓 ) ) ) )
11 8 10 sylan9eq ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ) ) → ( 𝑅 + 𝑆 ) = ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑅 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 𝑓 ) ) ) )