Metamath Proof Explorer


Theorem dvhfplusr

Description: Ring addition operation for the constructed full vector space H. (Contributed by NM, 29-Oct-2013) (Revised by Mario Carneiro, 22-Jun-2014)

Ref Expression
Hypotheses dvhfplusr.h 𝐻 = ( LHyp ‘ 𝐾 )
dvhfplusr.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
dvhfplusr.e 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
dvhfplusr.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dvhfplusr.f 𝐹 = ( Scalar ‘ 𝑈 )
dvhfplusr.p + = ( 𝑠𝐸 , 𝑡𝐸 ↦ ( 𝑓𝑇 ↦ ( ( 𝑠𝑓 ) ∘ ( 𝑡𝑓 ) ) ) )
dvhfplusr.s = ( +g𝐹 )
Assertion dvhfplusr ( ( 𝐾𝑉𝑊𝐻 ) → = + )

Proof

Step Hyp Ref Expression
1 dvhfplusr.h 𝐻 = ( LHyp ‘ 𝐾 )
2 dvhfplusr.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3 dvhfplusr.e 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
4 dvhfplusr.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
5 dvhfplusr.f 𝐹 = ( Scalar ‘ 𝑈 )
6 dvhfplusr.p + = ( 𝑠𝐸 , 𝑡𝐸 ↦ ( 𝑓𝑇 ↦ ( ( 𝑠𝑓 ) ∘ ( 𝑡𝑓 ) ) ) )
7 dvhfplusr.s = ( +g𝐹 )
8 eqid ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
9 1 8 4 5 dvhsca ( ( 𝐾𝑉𝑊𝐻 ) → 𝐹 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) )
10 9 fveq2d ( ( 𝐾𝑉𝑊𝐻 ) → ( +g𝐹 ) = ( +g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) )
11 eqid ( +g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) = ( +g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) )
12 1 2 3 8 11 erngfplus ( ( 𝐾𝑉𝑊𝐻 ) → ( +g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) = ( 𝑠𝐸 , 𝑡𝐸 ↦ ( 𝑓𝑇 ↦ ( ( 𝑠𝑓 ) ∘ ( 𝑡𝑓 ) ) ) ) )
13 10 12 eqtrd ( ( 𝐾𝑉𝑊𝐻 ) → ( +g𝐹 ) = ( 𝑠𝐸 , 𝑡𝐸 ↦ ( 𝑓𝑇 ↦ ( ( 𝑠𝑓 ) ∘ ( 𝑡𝑓 ) ) ) ) )
14 13 7 6 3eqtr4g ( ( 𝐾𝑉𝑊𝐻 ) → = + )