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
|
erngset |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐷 = { 〈 ( Base ‘ ndx ) , 𝐸 〉 , 〈 ( +g ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑠 ∘ 𝑡 ) ) 〉 } ) |
7 |
6
|
fveq2d |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( .r ‘ 𝐷 ) = ( .r ‘ { 〈 ( Base ‘ ndx ) , 𝐸 〉 , 〈 ( +g ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑠 ∘ 𝑡 ) ) 〉 } ) ) |
8 |
3
|
fvexi |
⊢ 𝐸 ∈ V |
9 |
8 8
|
mpoex |
⊢ ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑠 ∘ 𝑡 ) ) ∈ V |
10 |
|
eqid |
⊢ { 〈 ( Base ‘ ndx ) , 𝐸 〉 , 〈 ( +g ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑠 ∘ 𝑡 ) ) 〉 } = { 〈 ( Base ‘ ndx ) , 𝐸 〉 , 〈 ( +g ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑠 ∘ 𝑡 ) ) 〉 } |
11 |
10
|
rngmulr |
⊢ ( ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑠 ∘ 𝑡 ) ) ∈ V → ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑠 ∘ 𝑡 ) ) = ( .r ‘ { 〈 ( Base ‘ ndx ) , 𝐸 〉 , 〈 ( +g ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑠 ∘ 𝑡 ) ) 〉 } ) ) |
12 |
9 11
|
ax-mp |
⊢ ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑠 ∘ 𝑡 ) ) = ( .r ‘ { 〈 ( Base ‘ ndx ) , 𝐸 〉 , 〈 ( +g ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑠 ∘ 𝑡 ) ) 〉 } ) |
13 |
7 5 12
|
3eqtr4g |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → · = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑠 ∘ 𝑡 ) ) ) |