Step |
Hyp |
Ref |
Expression |
1 |
|
hlhilnvl.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
hlhilnvl.u |
⊢ 𝑈 = ( ( HLHil ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
hlhilnvl.r |
⊢ 𝑅 = ( Scalar ‘ 𝑈 ) |
4 |
|
hlhilnvl.i |
⊢ ∗ = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
hlhilnvl.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
6 |
|
fvex |
⊢ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ∈ V |
7 |
4
|
fvexi |
⊢ ∗ ∈ V |
8 |
|
starvid |
⊢ *𝑟 = Slot ( *𝑟 ‘ ndx ) |
9 |
8
|
setsid |
⊢ ( ( ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ∈ V ∧ ∗ ∈ V ) → ∗ = ( *𝑟 ‘ ( ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) sSet 〈 ( *𝑟 ‘ ndx ) , ∗ 〉 ) ) ) |
10 |
6 7 9
|
mp2an |
⊢ ∗ = ( *𝑟 ‘ ( ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) sSet 〈 ( *𝑟 ‘ ndx ) , ∗ 〉 ) ) |
11 |
|
eqid |
⊢ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) |
12 |
|
eqid |
⊢ ( ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) sSet 〈 ( *𝑟 ‘ ndx ) , ∗ 〉 ) = ( ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) sSet 〈 ( *𝑟 ‘ ndx ) , ∗ 〉 ) |
13 |
1 2 5 11 4 12
|
hlhilsca |
⊢ ( 𝜑 → ( ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) sSet 〈 ( *𝑟 ‘ ndx ) , ∗ 〉 ) = ( Scalar ‘ 𝑈 ) ) |
14 |
13 3
|
eqtr4di |
⊢ ( 𝜑 → ( ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) sSet 〈 ( *𝑟 ‘ ndx ) , ∗ 〉 ) = 𝑅 ) |
15 |
14
|
fveq2d |
⊢ ( 𝜑 → ( *𝑟 ‘ ( ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) sSet 〈 ( *𝑟 ‘ ndx ) , ∗ 〉 ) ) = ( *𝑟 ‘ 𝑅 ) ) |
16 |
10 15
|
syl5eq |
⊢ ( 𝜑 → ∗ = ( *𝑟 ‘ 𝑅 ) ) |