Step |
Hyp |
Ref |
Expression |
1 |
|
lcfrvalsn.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
lcfrvalsn.o |
⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
lcfrvalsn.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
lcfrvalsn.f |
⊢ 𝐹 = ( LFnl ‘ 𝑈 ) |
5 |
|
lcfrvalsn.l |
⊢ 𝐿 = ( LKer ‘ 𝑈 ) |
6 |
|
lcfrvalsn.d |
⊢ 𝐷 = ( LDual ‘ 𝑈 ) |
7 |
|
lcfrvalsn.n |
⊢ 𝑁 = ( LSpan ‘ 𝐷 ) |
8 |
|
lcfrvalsn.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
9 |
|
lcfrvalsn.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐹 ) |
10 |
|
lcfrvalsn.q |
⊢ 𝑄 = ∪ 𝑓 ∈ 𝑅 ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) |
11 |
|
lcfrvalsn.r |
⊢ 𝑅 = ( 𝑁 ‘ { 𝐺 } ) |
12 |
|
eliun |
⊢ ( 𝑥 ∈ ∪ 𝑓 ∈ 𝑅 ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ↔ ∃ 𝑓 ∈ 𝑅 𝑥 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) |
13 |
11
|
eleq2i |
⊢ ( 𝑓 ∈ 𝑅 ↔ 𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) |
14 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
15 |
|
eqid |
⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 ) |
16 |
1 3 8
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
17 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → 𝑈 ∈ LMod ) |
18 |
6 16
|
lduallmod |
⊢ ( 𝜑 → 𝐷 ∈ LMod ) |
19 |
|
eqid |
⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 ) |
20 |
4 6 19 16 9
|
ldualelvbase |
⊢ ( 𝜑 → 𝐺 ∈ ( Base ‘ 𝐷 ) ) |
21 |
|
eqid |
⊢ ( LSubSp ‘ 𝐷 ) = ( LSubSp ‘ 𝐷 ) |
22 |
19 21 7
|
lspsncl |
⊢ ( ( 𝐷 ∈ LMod ∧ 𝐺 ∈ ( Base ‘ 𝐷 ) ) → ( 𝑁 ‘ { 𝐺 } ) ∈ ( LSubSp ‘ 𝐷 ) ) |
23 |
18 20 22
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝐺 } ) ∈ ( LSubSp ‘ 𝐷 ) ) |
24 |
19 21
|
lssel |
⊢ ( ( ( 𝑁 ‘ { 𝐺 } ) ∈ ( LSubSp ‘ 𝐷 ) ∧ 𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → 𝑓 ∈ ( Base ‘ 𝐷 ) ) |
25 |
23 24
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → 𝑓 ∈ ( Base ‘ 𝐷 ) ) |
26 |
4 6 19 16
|
ldualvbase |
⊢ ( 𝜑 → ( Base ‘ 𝐷 ) = 𝐹 ) |
27 |
26
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → ( Base ‘ 𝐷 ) = 𝐹 ) |
28 |
25 27
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → 𝑓 ∈ 𝐹 ) |
29 |
15 4 5 17 28
|
lkrssv |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → ( 𝐿 ‘ 𝑓 ) ⊆ ( Base ‘ 𝑈 ) ) |
30 |
|
eqid |
⊢ ( Scalar ‘ 𝐷 ) = ( Scalar ‘ 𝐷 ) |
31 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝐷 ) ) = ( Base ‘ ( Scalar ‘ 𝐷 ) ) |
32 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝐷 ) = ( ·𝑠 ‘ 𝐷 ) |
33 |
30 31 19 32 7
|
lspsnel |
⊢ ( ( 𝐷 ∈ LMod ∧ 𝐺 ∈ ( Base ‘ 𝐷 ) ) → ( 𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ↔ ∃ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) 𝑓 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐺 ) ) ) |
34 |
18 20 33
|
syl2anc |
⊢ ( 𝜑 → ( 𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ↔ ∃ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) 𝑓 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐺 ) ) ) |
35 |
|
eqid |
⊢ ( Scalar ‘ 𝑈 ) = ( Scalar ‘ 𝑈 ) |
36 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑈 ) ) = ( Base ‘ ( Scalar ‘ 𝑈 ) ) |
37 |
35 36 6 30 31 16
|
ldualsbase |
⊢ ( 𝜑 → ( Base ‘ ( Scalar ‘ 𝐷 ) ) = ( Base ‘ ( Scalar ‘ 𝑈 ) ) ) |
38 |
37
|
rexeqdv |
⊢ ( 𝜑 → ( ∃ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) 𝑓 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐺 ) ↔ ∃ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) 𝑓 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐺 ) ) ) |
39 |
34 38
|
bitrd |
⊢ ( 𝜑 → ( 𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ↔ ∃ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) 𝑓 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐺 ) ) ) |
40 |
39
|
biimpa |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → ∃ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) 𝑓 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐺 ) ) |
41 |
1 3 8
|
dvhlvec |
⊢ ( 𝜑 → 𝑈 ∈ LVec ) |
42 |
41
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → 𝑈 ∈ LVec ) |
43 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → 𝐺 ∈ 𝐹 ) |
44 |
35 36 4 5 6 32 42 43 28
|
lkrss2N |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → ( ( 𝐿 ‘ 𝐺 ) ⊆ ( 𝐿 ‘ 𝑓 ) ↔ ∃ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) 𝑓 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐺 ) ) ) |
45 |
40 44
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → ( 𝐿 ‘ 𝐺 ) ⊆ ( 𝐿 ‘ 𝑓 ) ) |
46 |
1 3 15 2
|
dochss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐿 ‘ 𝑓 ) ⊆ ( Base ‘ 𝑈 ) ∧ ( 𝐿 ‘ 𝐺 ) ⊆ ( 𝐿 ‘ 𝑓 ) ) → ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) |
47 |
14 29 45 46
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) |
48 |
47
|
sseld |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → ( 𝑥 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) → 𝑥 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ) |
49 |
48
|
ex |
⊢ ( 𝜑 → ( 𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) → ( 𝑥 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) → 𝑥 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ) ) |
50 |
13 49
|
syl5bi |
⊢ ( 𝜑 → ( 𝑓 ∈ 𝑅 → ( 𝑥 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) → 𝑥 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ) ) |
51 |
50
|
rexlimdv |
⊢ ( 𝜑 → ( ∃ 𝑓 ∈ 𝑅 𝑥 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) → 𝑥 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ) |
52 |
19 7
|
lspsnid |
⊢ ( ( 𝐷 ∈ LMod ∧ 𝐺 ∈ ( Base ‘ 𝐷 ) ) → 𝐺 ∈ ( 𝑁 ‘ { 𝐺 } ) ) |
53 |
18 20 52
|
syl2anc |
⊢ ( 𝜑 → 𝐺 ∈ ( 𝑁 ‘ { 𝐺 } ) ) |
54 |
53 11
|
eleqtrrdi |
⊢ ( 𝜑 → 𝐺 ∈ 𝑅 ) |
55 |
|
2fveq3 |
⊢ ( 𝑓 = 𝐺 → ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) = ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) |
56 |
55
|
eleq2d |
⊢ ( 𝑓 = 𝐺 → ( 𝑥 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ↔ 𝑥 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ) |
57 |
56
|
rspcev |
⊢ ( ( 𝐺 ∈ 𝑅 ∧ 𝑥 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) → ∃ 𝑓 ∈ 𝑅 𝑥 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) |
58 |
54 57
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) → ∃ 𝑓 ∈ 𝑅 𝑥 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) |
59 |
58
|
ex |
⊢ ( 𝜑 → ( 𝑥 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) → ∃ 𝑓 ∈ 𝑅 𝑥 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) ) |
60 |
51 59
|
impbid |
⊢ ( 𝜑 → ( ∃ 𝑓 ∈ 𝑅 𝑥 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ↔ 𝑥 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ) |
61 |
12 60
|
syl5bb |
⊢ ( 𝜑 → ( 𝑥 ∈ ∪ 𝑓 ∈ 𝑅 ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ↔ 𝑥 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ) |
62 |
61
|
eqrdv |
⊢ ( 𝜑 → ∪ 𝑓 ∈ 𝑅 ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) = ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) |
63 |
10 62
|
syl5eq |
⊢ ( 𝜑 → 𝑄 = ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) |