Step |
Hyp |
Ref |
Expression |
1 |
|
hlhilbase.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
hlhilbase.u |
⊢ 𝑈 = ( ( HLHil ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
hlhilbase.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
4 |
|
hlhilsca.e |
⊢ 𝐸 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
hlhilsca.g |
⊢ 𝐺 = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
hlhilsca.r |
⊢ 𝑅 = ( 𝐸 sSet 〈 ( *𝑟 ‘ ndx ) , 𝐺 〉 ) |
7 |
|
ovex |
⊢ ( 𝐸 sSet 〈 ( *𝑟 ‘ ndx ) , 𝐺 〉 ) ∈ V |
8 |
6 7
|
eqeltri |
⊢ 𝑅 ∈ V |
9 |
|
eqid |
⊢ ( { 〈 ( Base ‘ ndx ) , ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 〈 ( +g ‘ ndx ) , ( +g ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 〈 ( Scalar ‘ ndx ) , 𝑅 〉 } ∪ { 〈 ( ·𝑠 ‘ ndx ) , ( ·𝑠 ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 〈 ( ·𝑖 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑦 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ( ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑦 ) ‘ 𝑥 ) ) 〉 } ) = ( { 〈 ( Base ‘ ndx ) , ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 〈 ( +g ‘ ndx ) , ( +g ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 〈 ( Scalar ‘ ndx ) , 𝑅 〉 } ∪ { 〈 ( ·𝑠 ‘ ndx ) , ( ·𝑠 ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 〈 ( ·𝑖 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑦 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ( ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑦 ) ‘ 𝑥 ) ) 〉 } ) |
10 |
9
|
phlsca |
⊢ ( 𝑅 ∈ V → 𝑅 = ( Scalar ‘ ( { 〈 ( Base ‘ ndx ) , ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 〈 ( +g ‘ ndx ) , ( +g ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 〈 ( Scalar ‘ ndx ) , 𝑅 〉 } ∪ { 〈 ( ·𝑠 ‘ ndx ) , ( ·𝑠 ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 〈 ( ·𝑖 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑦 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ( ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑦 ) ‘ 𝑥 ) ) 〉 } ) ) ) |
11 |
8 10
|
ax-mp |
⊢ 𝑅 = ( Scalar ‘ ( { 〈 ( Base ‘ ndx ) , ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 〈 ( +g ‘ ndx ) , ( +g ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 〈 ( Scalar ‘ ndx ) , 𝑅 〉 } ∪ { 〈 ( ·𝑠 ‘ ndx ) , ( ·𝑠 ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 〈 ( ·𝑖 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑦 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ( ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑦 ) ‘ 𝑥 ) ) 〉 } ) ) |
12 |
|
eqid |
⊢ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
13 |
|
eqid |
⊢ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) |
14 |
|
eqid |
⊢ ( +g ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( +g ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) |
15 |
|
eqid |
⊢ ( ·𝑠 ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ·𝑠 ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) |
16 |
|
eqid |
⊢ ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) |
17 |
|
eqid |
⊢ ( 𝑥 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑦 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ( ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑦 ) ‘ 𝑥 ) ) = ( 𝑥 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑦 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ( ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑦 ) ‘ 𝑥 ) ) |
18 |
1 2 12 13 14 4 5 6 15 16 17 3
|
hlhilset |
⊢ ( 𝜑 → 𝑈 = ( { 〈 ( Base ‘ ndx ) , ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 〈 ( +g ‘ ndx ) , ( +g ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 〈 ( Scalar ‘ ndx ) , 𝑅 〉 } ∪ { 〈 ( ·𝑠 ‘ ndx ) , ( ·𝑠 ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 〈 ( ·𝑖 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑦 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ( ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑦 ) ‘ 𝑥 ) ) 〉 } ) ) |
19 |
18
|
fveq2d |
⊢ ( 𝜑 → ( Scalar ‘ 𝑈 ) = ( Scalar ‘ ( { 〈 ( Base ‘ ndx ) , ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 〈 ( +g ‘ ndx ) , ( +g ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 〈 ( Scalar ‘ ndx ) , 𝑅 〉 } ∪ { 〈 ( ·𝑠 ‘ ndx ) , ( ·𝑠 ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , 〈 ( ·𝑖 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑦 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ( ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑦 ) ‘ 𝑥 ) ) 〉 } ) ) ) |
20 |
11 19
|
eqtr4id |
⊢ ( 𝜑 → 𝑅 = ( Scalar ‘ 𝑈 ) ) |