Step |
Hyp |
Ref |
Expression |
1 |
|
lshpkr.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lshpkr.a |
⊢ + = ( +g ‘ 𝑊 ) |
3 |
|
lshpkr.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
4 |
|
lshpkr.p |
⊢ ⊕ = ( LSSum ‘ 𝑊 ) |
5 |
|
lshpkr.h |
⊢ 𝐻 = ( LSHyp ‘ 𝑊 ) |
6 |
|
lshpkr.w |
⊢ ( 𝜑 → 𝑊 ∈ LVec ) |
7 |
|
lshpkr.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝐻 ) |
8 |
|
lshpkr.z |
⊢ ( 𝜑 → 𝑍 ∈ 𝑉 ) |
9 |
|
lshpkr.e |
⊢ ( 𝜑 → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑍 } ) ) = 𝑉 ) |
10 |
|
lshpkr.d |
⊢ 𝐷 = ( Scalar ‘ 𝑊 ) |
11 |
|
lshpkr.k |
⊢ 𝐾 = ( Base ‘ 𝐷 ) |
12 |
|
lshpkr.t |
⊢ · = ( ·𝑠 ‘ 𝑊 ) |
13 |
|
lshpkr.g |
⊢ 𝐺 = ( 𝑥 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝐾 ∃ 𝑦 ∈ 𝑈 𝑥 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) ) ) |
14 |
|
lshpkr.f |
⊢ 𝐹 = ( LFnl ‘ 𝑊 ) |
15 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑉 ) → 𝑊 ∈ LVec ) |
16 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑉 ) → 𝑈 ∈ 𝐻 ) |
17 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑉 ) → 𝑍 ∈ 𝑉 ) |
18 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑉 ) → 𝑎 ∈ 𝑉 ) |
19 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑉 ) → ( 𝑈 ⊕ ( 𝑁 ‘ { 𝑍 } ) ) = 𝑉 ) |
20 |
1 2 3 4 5 15 16 17 18 19 10 11 12
|
lshpsmreu |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑉 ) → ∃! 𝑘 ∈ 𝐾 ∃ 𝑦 ∈ 𝑈 𝑎 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) ) |
21 |
|
riotacl |
⊢ ( ∃! 𝑘 ∈ 𝐾 ∃ 𝑦 ∈ 𝑈 𝑎 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) → ( ℩ 𝑘 ∈ 𝐾 ∃ 𝑦 ∈ 𝑈 𝑎 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) ) ∈ 𝐾 ) |
22 |
20 21
|
syl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑉 ) → ( ℩ 𝑘 ∈ 𝐾 ∃ 𝑦 ∈ 𝑈 𝑎 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) ) ∈ 𝐾 ) |
23 |
|
eqeq1 |
⊢ ( 𝑥 = 𝑎 → ( 𝑥 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) ↔ 𝑎 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) ) ) |
24 |
23
|
rexbidv |
⊢ ( 𝑥 = 𝑎 → ( ∃ 𝑦 ∈ 𝑈 𝑥 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) ↔ ∃ 𝑦 ∈ 𝑈 𝑎 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) ) ) |
25 |
24
|
riotabidv |
⊢ ( 𝑥 = 𝑎 → ( ℩ 𝑘 ∈ 𝐾 ∃ 𝑦 ∈ 𝑈 𝑥 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) ) = ( ℩ 𝑘 ∈ 𝐾 ∃ 𝑦 ∈ 𝑈 𝑎 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) ) ) |
26 |
25
|
cbvmptv |
⊢ ( 𝑥 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝐾 ∃ 𝑦 ∈ 𝑈 𝑥 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) ) ) = ( 𝑎 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝐾 ∃ 𝑦 ∈ 𝑈 𝑎 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) ) ) |
27 |
13 26
|
eqtri |
⊢ 𝐺 = ( 𝑎 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝐾 ∃ 𝑦 ∈ 𝑈 𝑎 = ( 𝑦 + ( 𝑘 · 𝑍 ) ) ) ) |
28 |
22 27
|
fmptd |
⊢ ( 𝜑 → 𝐺 : 𝑉 ⟶ 𝐾 ) |
29 |
|
eqid |
⊢ ( 0g ‘ 𝐷 ) = ( 0g ‘ 𝐷 ) |
30 |
1 2 3 4 5 6 7 8 8 9 10 11 12 29 13
|
lshpkrlem6 |
⊢ ( ( 𝜑 ∧ ( 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉 ) ) → ( 𝐺 ‘ ( ( 𝑙 · 𝑢 ) + 𝑣 ) ) = ( ( 𝑙 ( .r ‘ 𝐷 ) ( 𝐺 ‘ 𝑢 ) ) ( +g ‘ 𝐷 ) ( 𝐺 ‘ 𝑣 ) ) ) |
31 |
30
|
ralrimivvva |
⊢ ( 𝜑 → ∀ 𝑙 ∈ 𝐾 ∀ 𝑢 ∈ 𝑉 ∀ 𝑣 ∈ 𝑉 ( 𝐺 ‘ ( ( 𝑙 · 𝑢 ) + 𝑣 ) ) = ( ( 𝑙 ( .r ‘ 𝐷 ) ( 𝐺 ‘ 𝑢 ) ) ( +g ‘ 𝐷 ) ( 𝐺 ‘ 𝑣 ) ) ) |
32 |
|
eqid |
⊢ ( +g ‘ 𝐷 ) = ( +g ‘ 𝐷 ) |
33 |
|
eqid |
⊢ ( .r ‘ 𝐷 ) = ( .r ‘ 𝐷 ) |
34 |
1 2 10 12 11 32 33 14
|
islfl |
⊢ ( 𝑊 ∈ LVec → ( 𝐺 ∈ 𝐹 ↔ ( 𝐺 : 𝑉 ⟶ 𝐾 ∧ ∀ 𝑙 ∈ 𝐾 ∀ 𝑢 ∈ 𝑉 ∀ 𝑣 ∈ 𝑉 ( 𝐺 ‘ ( ( 𝑙 · 𝑢 ) + 𝑣 ) ) = ( ( 𝑙 ( .r ‘ 𝐷 ) ( 𝐺 ‘ 𝑢 ) ) ( +g ‘ 𝐷 ) ( 𝐺 ‘ 𝑣 ) ) ) ) ) |
35 |
6 34
|
syl |
⊢ ( 𝜑 → ( 𝐺 ∈ 𝐹 ↔ ( 𝐺 : 𝑉 ⟶ 𝐾 ∧ ∀ 𝑙 ∈ 𝐾 ∀ 𝑢 ∈ 𝑉 ∀ 𝑣 ∈ 𝑉 ( 𝐺 ‘ ( ( 𝑙 · 𝑢 ) + 𝑣 ) ) = ( ( 𝑙 ( .r ‘ 𝐷 ) ( 𝐺 ‘ 𝑢 ) ) ( +g ‘ 𝐷 ) ( 𝐺 ‘ 𝑣 ) ) ) ) ) |
36 |
28 31 35
|
mpbir2and |
⊢ ( 𝜑 → 𝐺 ∈ 𝐹 ) |