Step |
Hyp |
Ref |
Expression |
1 |
|
dvhfvsca.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
dvhfvsca.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
dvhfvsca.e |
⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
dvhfvsca.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
dvhfvsca.s |
⊢ · = ( ·𝑠 ‘ 𝑈 ) |
6 |
|
eqid |
⊢ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
1 2 3 6 4
|
dvhset |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝑈 = ( { 〈 ( Base ‘ ndx ) , ( 𝑇 × 𝐸 ) 〉 , 〈 ( +g ‘ ndx ) , ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ 〈 ( ( 1st ‘ 𝑓 ) ∘ ( 1st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) ) ) 〉 ) 〉 , 〈 ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) 〉 } ∪ { 〈 ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ 〈 ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) 〉 ) 〉 } ) ) |
8 |
7
|
fveq2d |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( ·𝑠 ‘ 𝑈 ) = ( ·𝑠 ‘ ( { 〈 ( Base ‘ ndx ) , ( 𝑇 × 𝐸 ) 〉 , 〈 ( +g ‘ ndx ) , ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ 〈 ( ( 1st ‘ 𝑓 ) ∘ ( 1st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) ) ) 〉 ) 〉 , 〈 ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) 〉 } ∪ { 〈 ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ 〈 ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) 〉 ) 〉 } ) ) ) |
9 |
3
|
fvexi |
⊢ 𝐸 ∈ V |
10 |
2
|
fvexi |
⊢ 𝑇 ∈ V |
11 |
10 9
|
xpex |
⊢ ( 𝑇 × 𝐸 ) ∈ V |
12 |
9 11
|
mpoex |
⊢ ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ 〈 ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) 〉 ) ∈ V |
13 |
|
eqid |
⊢ ( { 〈 ( Base ‘ ndx ) , ( 𝑇 × 𝐸 ) 〉 , 〈 ( +g ‘ ndx ) , ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ 〈 ( ( 1st ‘ 𝑓 ) ∘ ( 1st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) ) ) 〉 ) 〉 , 〈 ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) 〉 } ∪ { 〈 ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ 〈 ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) 〉 ) 〉 } ) = ( { 〈 ( Base ‘ ndx ) , ( 𝑇 × 𝐸 ) 〉 , 〈 ( +g ‘ ndx ) , ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ 〈 ( ( 1st ‘ 𝑓 ) ∘ ( 1st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) ) ) 〉 ) 〉 , 〈 ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) 〉 } ∪ { 〈 ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ 〈 ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) 〉 ) 〉 } ) |
14 |
13
|
lmodvsca |
⊢ ( ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ 〈 ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) 〉 ) ∈ V → ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ 〈 ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) 〉 ) = ( ·𝑠 ‘ ( { 〈 ( Base ‘ ndx ) , ( 𝑇 × 𝐸 ) 〉 , 〈 ( +g ‘ ndx ) , ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ 〈 ( ( 1st ‘ 𝑓 ) ∘ ( 1st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) ) ) 〉 ) 〉 , 〈 ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) 〉 } ∪ { 〈 ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ 〈 ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) 〉 ) 〉 } ) ) ) |
15 |
12 14
|
ax-mp |
⊢ ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ 〈 ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) 〉 ) = ( ·𝑠 ‘ ( { 〈 ( Base ‘ ndx ) , ( 𝑇 × 𝐸 ) 〉 , 〈 ( +g ‘ ndx ) , ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ 〈 ( ( 1st ‘ 𝑓 ) ∘ ( 1st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) ) ) 〉 ) 〉 , 〈 ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) 〉 } ∪ { 〈 ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ 〈 ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) 〉 ) 〉 } ) ) |
16 |
8 5 15
|
3eqtr4g |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → · = ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ 〈 ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) 〉 ) ) |