Step |
Hyp |
Ref |
Expression |
1 |
|
erngset.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
erngset.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
erngset.e |
⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
erngset.d |
⊢ 𝐷 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
erng.m |
⊢ · = ( .r ‘ 𝐷 ) |
6 |
1 2 3 4 5
|
erngfmul |
⊢ ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) → · = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑠 ∘ 𝑡 ) ) ) |
7 |
6
|
oveqd |
⊢ ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) → ( 𝑈 · 𝑉 ) = ( 𝑈 ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑠 ∘ 𝑡 ) ) 𝑉 ) ) |
8 |
|
coexg |
⊢ ( ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) → ( 𝑈 ∘ 𝑉 ) ∈ V ) |
9 |
|
coeq1 |
⊢ ( 𝑠 = 𝑈 → ( 𝑠 ∘ 𝑡 ) = ( 𝑈 ∘ 𝑡 ) ) |
10 |
|
coeq2 |
⊢ ( 𝑡 = 𝑉 → ( 𝑈 ∘ 𝑡 ) = ( 𝑈 ∘ 𝑉 ) ) |
11 |
|
eqid |
⊢ ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑠 ∘ 𝑡 ) ) = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑠 ∘ 𝑡 ) ) |
12 |
9 10 11
|
ovmpog |
⊢ ( ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ∘ 𝑉 ) ∈ V ) → ( 𝑈 ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑠 ∘ 𝑡 ) ) 𝑉 ) = ( 𝑈 ∘ 𝑉 ) ) |
13 |
8 12
|
mpd3an3 |
⊢ ( ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) → ( 𝑈 ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑠 ∘ 𝑡 ) ) 𝑉 ) = ( 𝑈 ∘ 𝑉 ) ) |
14 |
7 13
|
sylan9eq |
⊢ ( ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ) → ( 𝑈 · 𝑉 ) = ( 𝑈 ∘ 𝑉 ) ) |