Metamath Proof Explorer

Theorem dvhopvsca

Description: Scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Feb-2014)

Ref Expression
Hypotheses dvhfvsca.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dvhfvsca.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
dvhfvsca.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
dvhfvsca.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dvhfvsca.s ⊢ · = ( ·𝑠 ‘ 𝑈 )
Assertion dvhopvsca ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸 ) ) → ( 𝑅 · ⟨ 𝐹 , 𝑋 ⟩ ) = ⟨ ( 𝑅 ‘ 𝐹 ) , ( 𝑅 ∘ 𝑋 ) ⟩ )

Proof

Step Hyp Ref Expression
1 dvhfvsca.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 dvhfvsca.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3 dvhfvsca.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
4 dvhfvsca.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
5 dvhfvsca.s ⊢ · = ( ·𝑠 ‘ 𝑈 )
6 simpl ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸 ) ) → ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) )
7 simpr1 ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸 ) ) → 𝑅 ∈ 𝐸 )
8 simpr2 ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸 ) ) → 𝐹 ∈ 𝑇 )
9 simpr3 ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸 ) ) → 𝑋 ∈ 𝐸 )
10 opelxpi ⊢ ( ( 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸 ) → ⟨ 𝐹 , 𝑋 ⟩ ∈ ( 𝑇 × 𝐸 ) )
11 8 9 10 syl2anc ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸 ) ) → ⟨ 𝐹 , 𝑋 ⟩ ∈ ( 𝑇 × 𝐸 ) )
12 1 2 3 4 5 dvhvsca ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ ⟨ 𝐹 , 𝑋 ⟩ ∈ ( 𝑇 × 𝐸 ) ) ) → ( 𝑅 · ⟨ 𝐹 , 𝑋 ⟩ ) = ⟨ ( 𝑅 ‘ ( 1st ‘ ⟨ 𝐹 , 𝑋 ⟩ ) ) , ( 𝑅 ∘ ( 2nd ‘ ⟨ 𝐹 , 𝑋 ⟩ ) ) ⟩ )
13 6 7 11 12 syl12anc ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸 ) ) → ( 𝑅 · ⟨ 𝐹 , 𝑋 ⟩ ) = ⟨ ( 𝑅 ‘ ( 1st ‘ ⟨ 𝐹 , 𝑋 ⟩ ) ) , ( 𝑅 ∘ ( 2nd ‘ ⟨ 𝐹 , 𝑋 ⟩ ) ) ⟩ )
14 op1stg ⊢ ( ( 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸 ) → ( 1st ‘ ⟨ 𝐹 , 𝑋 ⟩ ) = 𝐹 )
15 8 9 14 syl2anc ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸 ) ) → ( 1st ‘ ⟨ 𝐹 , 𝑋 ⟩ ) = 𝐹 )
16 15 fveq2d ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸 ) ) → ( 𝑅 ‘ ( 1st ‘ ⟨ 𝐹 , 𝑋 ⟩ ) ) = ( 𝑅 ‘ 𝐹 ) )
17 op2ndg ⊢ ( ( 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸 ) → ( 2nd ‘ ⟨ 𝐹 , 𝑋 ⟩ ) = 𝑋 )
18 8 9 17 syl2anc ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸 ) ) → ( 2nd ‘ ⟨ 𝐹 , 𝑋 ⟩ ) = 𝑋 )
19 18 coeq2d ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸 ) ) → ( 𝑅 ∘ ( 2nd ‘ ⟨ 𝐹 , 𝑋 ⟩ ) ) = ( 𝑅 ∘ 𝑋 ) )
20 16 19 opeq12d ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸 ) ) → ⟨ ( 𝑅 ‘ ( 1st ‘ ⟨ 𝐹 , 𝑋 ⟩ ) ) , ( 𝑅 ∘ ( 2nd ‘ ⟨ 𝐹 , 𝑋 ⟩ ) ) ⟩ = ⟨ ( 𝑅 ‘ 𝐹 ) , ( 𝑅 ∘ 𝑋 ) ⟩ )
21 13 20 eqtrd ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸 ) ) → ( 𝑅 · ⟨ 𝐹 , 𝑋 ⟩ ) = ⟨ ( 𝑅 ‘ 𝐹 ) , ( 𝑅 ∘ 𝑋 ) ⟩ )