Step |
Hyp |
Ref |
Expression |
1 |
|
hgmapcl.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
hgmapcl.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
hgmapcl.r |
⊢ 𝑅 = ( Scalar ‘ 𝑈 ) |
4 |
|
hgmapcl.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
5 |
|
hgmapcl.g |
⊢ 𝐺 = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
hgmapcl.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
7 |
|
hgmapcl.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝐵 ) |
8 |
|
eqid |
⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 ) |
9 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑈 ) = ( ·𝑠 ‘ 𝑈 ) |
10 |
|
eqid |
⊢ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) |
11 |
|
eqid |
⊢ ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) |
12 |
|
eqid |
⊢ ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) |
13 |
1 2 8 9 3 4 10 11 12 5 6 7
|
hgmapval |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝐹 ) = ( ℩ 𝑔 ∈ 𝐵 ∀ 𝑥 ∈ ( Base ‘ 𝑈 ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐹 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) = ( 𝑔 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) ) ) |
14 |
1 2 8 9 3 4 10 11 12 6 7
|
hdmap14lem15 |
⊢ ( 𝜑 → ∃! 𝑔 ∈ 𝐵 ∀ 𝑥 ∈ ( Base ‘ 𝑈 ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐹 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) = ( 𝑔 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) ) |
15 |
|
riotacl |
⊢ ( ∃! 𝑔 ∈ 𝐵 ∀ 𝑥 ∈ ( Base ‘ 𝑈 ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐹 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) = ( 𝑔 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) → ( ℩ 𝑔 ∈ 𝐵 ∀ 𝑥 ∈ ( Base ‘ 𝑈 ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐹 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) = ( 𝑔 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) ) ∈ 𝐵 ) |
16 |
14 15
|
syl |
⊢ ( 𝜑 → ( ℩ 𝑔 ∈ 𝐵 ∀ 𝑥 ∈ ( Base ‘ 𝑈 ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐹 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) = ( 𝑔 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) ) ∈ 𝐵 ) |
17 |
13 16
|
eqeltrd |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝐹 ) ∈ 𝐵 ) |