Step |
Hyp |
Ref |
Expression |
1 |
|
erngset.h-r |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
erngset.t-r |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
erngset.e-r |
⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
erngset.d-r |
⊢ 𝐷 = ( ( EDRingR ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
erng.p-r |
⊢ + = ( +g ‘ 𝐷 ) |
6 |
1 2 3 4 5
|
erngfplus-rN |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → + = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑔 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑔 ) ∘ ( 𝑡 ‘ 𝑔 ) ) ) ) ) |
7 |
6
|
oveqd |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝑈 + 𝑉 ) = ( 𝑈 ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑔 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑔 ) ∘ ( 𝑡 ‘ 𝑔 ) ) ) ) 𝑉 ) ) |
8 |
|
eqid |
⊢ ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑔 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑔 ) ∘ ( 𝑡 ‘ 𝑔 ) ) ) ) = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑔 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑔 ) ∘ ( 𝑡 ‘ 𝑔 ) ) ) ) |
9 |
8 2
|
tendopl |
⊢ ( ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) → ( 𝑈 ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑔 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑔 ) ∘ ( 𝑡 ‘ 𝑔 ) ) ) ) 𝑉 ) = ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑈 ‘ 𝑓 ) ∘ ( 𝑉 ‘ 𝑓 ) ) ) ) |
10 |
7 9
|
sylan9eq |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ) → ( 𝑈 + 𝑉 ) = ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑈 ‘ 𝑓 ) ∘ ( 𝑉 ‘ 𝑓 ) ) ) ) |