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 |
⊢ + = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) |
7 |
|
dvhfplusr.s |
⊢ ✚ = ( +g ‘ 𝐹 ) |
8 |
|
eqid |
⊢ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) |
9 |
1 8 4 5
|
dvhsca |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐹 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) |
10 |
9
|
fveq2d |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( +g ‘ 𝐹 ) = ( +g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
11 |
|
eqid |
⊢ ( +g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) = ( +g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) |
12 |
1 2 3 8 11
|
erngfplus |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( +g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) ) |
13 |
10 12
|
eqtrd |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( +g ‘ 𝐹 ) = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) ) |
14 |
13 7 6
|
3eqtr4g |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ✚ = + ) |