Step |
Hyp |
Ref |
Expression |
1 |
|
lshpkrlem.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lshpkrlem.a |
⊢ + = ( +g ‘ 𝑊 ) |
3 |
|
lshpkrlem.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
4 |
|
lshpkrlem.p |
⊢ ⊕ = ( LSSum ‘ 𝑊 ) |
5 |
|
lshpkrlem.h |
⊢ 𝐻 = ( LSHyp ‘ 𝑊 ) |
6 |
|
lshpkrlem.w |
⊢ ( 𝜑 → 𝑊 ∈ LVec ) |
7 |
|
lshpkrlem.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝐻 ) |
8 |
|
lshpkrlem.z |
⊢ ( 𝜑 → 𝑍 ∈ 𝑉 ) |
9 |
|
lshpkrlem.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
10 |
|
lshpkrlem.e |
⊢ ( 𝜑 → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑍 } ) ) = 𝑉 ) |
11 |
|
lshpkrlem.d |
⊢ 𝐷 = ( Scalar ‘ 𝑊 ) |
12 |
|
lshpkrlem.k |
⊢ 𝐾 = ( Base ‘ 𝐷 ) |
13 |
|
lshpkrlem.t |
⊢ · = ( ·𝑠 ‘ 𝑊 ) |
14 |
|
lshpkrlem.o |
⊢ 0 = ( 0g ‘ 𝐷 ) |
15 |
|
lshpkrlem.g |
⊢ 𝐺 = ( 𝑥 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝐾 ∃ 𝑦 ∈ 𝑈 𝑥 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) ) ) |
16 |
|
eqeq1 |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) ↔ 𝑋 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) ) ) |
17 |
16
|
rexbidv |
⊢ ( 𝑥 = 𝑋 → ( ∃ 𝑦 ∈ 𝑈 𝑥 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) ↔ ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) ) ) |
18 |
17
|
riotabidv |
⊢ ( 𝑥 = 𝑋 → ( ℩ 𝑘 ∈ 𝐾 ∃ 𝑦 ∈ 𝑈 𝑥 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) ) = ( ℩ 𝑘 ∈ 𝐾 ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) ) ) |
19 |
|
riotaex |
⊢ ( ℩ 𝑘 ∈ 𝐾 ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) ) ∈ V |
20 |
18 15 19
|
fvmpt |
⊢ ( 𝑋 ∈ 𝑉 → ( 𝐺 ‘ 𝑋 ) = ( ℩ 𝑘 ∈ 𝐾 ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) ) ) |
21 |
9 20
|
syl |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝑋 ) = ( ℩ 𝑘 ∈ 𝐾 ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) ) ) |
22 |
1 2 3 4 5 6 7 8 9 10 11 12 13
|
lshpsmreu |
⊢ ( 𝜑 → ∃! 𝑘 ∈ 𝐾 ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) ) |
23 |
|
riotacl |
⊢ ( ∃! 𝑘 ∈ 𝐾 ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) → ( ℩ 𝑘 ∈ 𝐾 ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) ) ∈ 𝐾 ) |
24 |
22 23
|
syl |
⊢ ( 𝜑 → ( ℩ 𝑘 ∈ 𝐾 ∃ 𝑦 ∈ 𝑈 𝑋 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) ) ∈ 𝐾 ) |
25 |
21 24
|
eqeltrd |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝑋 ) ∈ 𝐾 ) |