This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem dvavsca

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

Ref Expression
Hypotheses dvafvsca.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dvafvsca.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
dvafvsca.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
dvafvsca.u ⊢ 𝑈 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 )
dvafvsca.s ⊢ · = ( ·𝑠 ‘ 𝑈 )
Assertion dvavsca ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) ) → ( 𝑅 · 𝐹 ) = ( 𝑅 ‘ 𝐹 ) )

Proof

Step Hyp Ref Expression
1 dvafvsca.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 dvafvsca.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3 dvafvsca.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
4 dvafvsca.u ⊢ 𝑈 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 )
5 dvafvsca.s ⊢ · = ( ·𝑠 ‘ 𝑈 )
6 1 2 3 4 5 dvafvsca ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → · = ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) )
7 6 oveqd ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( 𝑅 · 𝐹 ) = ( 𝑅 ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) 𝐹 ) )
8 fveq1 ⊢ ( 𝑠 = 𝑅 → ( 𝑠 ‘ 𝑓 ) = ( 𝑅 ‘ 𝑓 ) )
9 fveq2 ⊢ ( 𝑓 = 𝐹 → ( 𝑅 ‘ 𝑓 ) = ( 𝑅 ‘ 𝐹 ) )
10 eqid ⊢ ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) = ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) )
11 fvex ⊢ ( 𝑅 ‘ 𝐹 ) ∈ V
12 8 9 10 11 ovmpo ⊢ ( ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) 𝐹 ) = ( 𝑅 ‘ 𝐹 ) )
13 7 12 sylan9eq ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) ) → ( 𝑅 · 𝐹 ) = ( 𝑅 ‘ 𝐹 ) )