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Metamath Proof Explorer

Theorem dvamulr

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

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 dvamulr ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ) ) → ( 𝑅 · 𝑆 ) = ( 𝑅 ∘ 𝑆 ) )

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 1 2 3 4 5 6 dvafmulr ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → · = ( 𝑟 ∈ 𝐸 , 𝑠 ∈ 𝐸 ↦ ( 𝑟 ∘ 𝑠 ) ) )
8 7 oveqd ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( 𝑅 · 𝑆 ) = ( 𝑅 ( 𝑟 ∈ 𝐸 , 𝑠 ∈ 𝐸 ↦ ( 𝑟 ∘ 𝑠 ) ) 𝑆 ) )
9 coexg ⊢ ( ( 𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ) → ( 𝑅 ∘ 𝑆 ) ∈ V )
10 coeq1 ⊢ ( 𝑟 = 𝑅 → ( 𝑟 ∘ 𝑠 ) = ( 𝑅 ∘ 𝑠 ) )
11 coeq2 ⊢ ( 𝑠 = 𝑆 → ( 𝑅 ∘ 𝑠 ) = ( 𝑅 ∘ 𝑆 ) )
12 eqid ⊢ ( 𝑟 ∈ 𝐸 , 𝑠 ∈ 𝐸 ↦ ( 𝑟 ∘ 𝑠 ) ) = ( 𝑟 ∈ 𝐸 , 𝑠 ∈ 𝐸 ↦ ( 𝑟 ∘ 𝑠 ) )
13 10 11 12 ovmpog ⊢ ( ( 𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ∧ ( 𝑅 ∘ 𝑆 ) ∈ V ) → ( 𝑅 ( 𝑟 ∈ 𝐸 , 𝑠 ∈ 𝐸 ↦ ( 𝑟 ∘ 𝑠 ) ) 𝑆 ) = ( 𝑅 ∘ 𝑆 ) )
14 9 13 mpd3an3 ⊢ ( ( 𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ) → ( 𝑅 ( 𝑟 ∈ 𝐸 , 𝑠 ∈ 𝐸 ↦ ( 𝑟 ∘ 𝑠 ) ) 𝑆 ) = ( 𝑅 ∘ 𝑆 ) )
15 8 14 sylan9eq ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ) ) → ( 𝑅 · 𝑆 ) = ( 𝑅 ∘ 𝑆 ) )