Step |
Hyp |
Ref |
Expression |
1 |
|
lclkrslem1.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
lclkrslem1.o |
⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
lclkrslem1.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
lclkrslem1.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑈 ) |
5 |
|
lclkrslem1.f |
⊢ 𝐹 = ( LFnl ‘ 𝑈 ) |
6 |
|
lclkrslem1.l |
⊢ 𝐿 = ( LKer ‘ 𝑈 ) |
7 |
|
lclkrslem1.d |
⊢ 𝐷 = ( LDual ‘ 𝑈 ) |
8 |
|
lclkrslem1.r |
⊢ 𝑅 = ( Scalar ‘ 𝑈 ) |
9 |
|
lclkrslem1.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
10 |
|
lclkrslem1.t |
⊢ · = ( ·𝑠 ‘ 𝐷 ) |
11 |
|
lclkrslem1.c |
⊢ 𝐶 = { 𝑓 ∈ 𝐹 ∣ ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑄 ) } |
12 |
|
lclkrslem1.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
13 |
|
lclkrslem1.q |
⊢ ( 𝜑 → 𝑄 ∈ 𝑆 ) |
14 |
|
lclkrslem1.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐶 ) |
15 |
|
lclkrslem1.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
16 |
|
eqid |
⊢ { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } = { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } |
17 |
11 16
|
lcfls1c |
⊢ ( 𝐺 ∈ 𝐶 ↔ ( 𝐺 ∈ { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑄 ) ) |
18 |
17
|
simplbi |
⊢ ( 𝐺 ∈ 𝐶 → 𝐺 ∈ { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ) |
19 |
14 18
|
syl |
⊢ ( 𝜑 → 𝐺 ∈ { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ) |
20 |
1 2 3 5 6 7 8 9 10 16 12 15 19
|
lclkrlem1 |
⊢ ( 𝜑 → ( 𝑋 · 𝐺 ) ∈ { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ) |
21 |
|
eqid |
⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 ) |
22 |
1 3 12
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
23 |
11
|
lcfls1lem |
⊢ ( 𝐺 ∈ 𝐶 ↔ ( 𝐺 ∈ 𝐹 ∧ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑄 ) ) |
24 |
14 23
|
sylib |
⊢ ( 𝜑 → ( 𝐺 ∈ 𝐹 ∧ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑄 ) ) |
25 |
24
|
simp1d |
⊢ ( 𝜑 → 𝐺 ∈ 𝐹 ) |
26 |
5 8 9 7 10 22 15 25
|
ldualvscl |
⊢ ( 𝜑 → ( 𝑋 · 𝐺 ) ∈ 𝐹 ) |
27 |
21 5 6 22 26
|
lkrssv |
⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝑋 · 𝐺 ) ) ⊆ ( Base ‘ 𝑈 ) ) |
28 |
1 3 12
|
dvhlvec |
⊢ ( 𝜑 → 𝑈 ∈ LVec ) |
29 |
8 9 5 6 7 10 28 25 15
|
lkrss |
⊢ ( 𝜑 → ( 𝐿 ‘ 𝐺 ) ⊆ ( 𝐿 ‘ ( 𝑋 · 𝐺 ) ) ) |
30 |
1 3 21 2
|
dochss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐿 ‘ ( 𝑋 · 𝐺 ) ) ⊆ ( Base ‘ 𝑈 ) ∧ ( 𝐿 ‘ 𝐺 ) ⊆ ( 𝐿 ‘ ( 𝑋 · 𝐺 ) ) ) → ( ⊥ ‘ ( 𝐿 ‘ ( 𝑋 · 𝐺 ) ) ) ⊆ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) |
31 |
12 27 29 30
|
syl3anc |
⊢ ( 𝜑 → ( ⊥ ‘ ( 𝐿 ‘ ( 𝑋 · 𝐺 ) ) ) ⊆ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) |
32 |
24
|
simp3d |
⊢ ( 𝜑 → ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑄 ) |
33 |
31 32
|
sstrd |
⊢ ( 𝜑 → ( ⊥ ‘ ( 𝐿 ‘ ( 𝑋 · 𝐺 ) ) ) ⊆ 𝑄 ) |
34 |
11 16
|
lcfls1c |
⊢ ( ( 𝑋 · 𝐺 ) ∈ 𝐶 ↔ ( ( 𝑋 · 𝐺 ) ∈ { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ∧ ( ⊥ ‘ ( 𝐿 ‘ ( 𝑋 · 𝐺 ) ) ) ⊆ 𝑄 ) ) |
35 |
20 33 34
|
sylanbrc |
⊢ ( 𝜑 → ( 𝑋 · 𝐺 ) ∈ 𝐶 ) |