Step |
Hyp |
Ref |
Expression |
1 |
|
dihp.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dihp.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
3 |
|
dihp.p |
⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
dihp.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
dihp.e |
⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
dihp.o |
⊢ 𝑂 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) ) |
7 |
|
dihp.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
dihp.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
9 |
|
dihp.n |
⊢ 𝑁 = ( LSpan ‘ 𝑈 ) |
10 |
|
dihp.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
11 |
|
dihp.s |
⊢ ( 𝜑 → ( 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂 ) ) |
12 |
|
eqid |
⊢ ( 0g ‘ 𝑈 ) = ( 0g ‘ 𝑈 ) |
13 |
|
eqid |
⊢ ( LSAtoms ‘ 𝑈 ) = ( LSAtoms ‘ 𝑈 ) |
14 |
2 8 10
|
dvhlvec |
⊢ ( 𝜑 → 𝑈 ∈ LVec ) |
15 |
2 3 7 8 13 10
|
dihat |
⊢ ( 𝜑 → ( 𝐼 ‘ 𝑃 ) ∈ ( LSAtoms ‘ 𝑈 ) ) |
16 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
17 |
|
eqid |
⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 ) |
18 |
16 17 2 3
|
lhpocnel2 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝑃 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ) ) |
19 |
|
eqid |
⊢ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 ) = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 ) |
20 |
1 16 17 2 4 19
|
ltrniotaidvalN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 ) = ( I ↾ 𝐵 ) ) |
21 |
10 18 20
|
syl2anc2 |
⊢ ( 𝜑 → ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 ) = ( I ↾ 𝐵 ) ) |
22 |
21
|
fveq2d |
⊢ ( 𝜑 → ( 𝑆 ‘ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 ) ) = ( 𝑆 ‘ ( I ↾ 𝐵 ) ) ) |
23 |
11
|
simpld |
⊢ ( 𝜑 → 𝑆 ∈ 𝐸 ) |
24 |
1 2 5
|
tendoid |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( 𝑆 ‘ ( I ↾ 𝐵 ) ) = ( I ↾ 𝐵 ) ) |
25 |
10 23 24
|
syl2anc |
⊢ ( 𝜑 → ( 𝑆 ‘ ( I ↾ 𝐵 ) ) = ( I ↾ 𝐵 ) ) |
26 |
22 25
|
eqtr2d |
⊢ ( 𝜑 → ( I ↾ 𝐵 ) = ( 𝑆 ‘ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 ) ) ) |
27 |
1
|
fvexi |
⊢ 𝐵 ∈ V |
28 |
|
resiexg |
⊢ ( 𝐵 ∈ V → ( I ↾ 𝐵 ) ∈ V ) |
29 |
27 28
|
mp1i |
⊢ ( 𝜑 → ( I ↾ 𝐵 ) ∈ V ) |
30 |
|
eqeq1 |
⊢ ( 𝑔 = ( I ↾ 𝐵 ) → ( 𝑔 = ( 𝑠 ‘ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 ) ) ↔ ( I ↾ 𝐵 ) = ( 𝑠 ‘ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 ) ) ) ) |
31 |
30
|
anbi1d |
⊢ ( 𝑔 = ( I ↾ 𝐵 ) → ( ( 𝑔 = ( 𝑠 ‘ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 ) ) ∧ 𝑠 ∈ 𝐸 ) ↔ ( ( I ↾ 𝐵 ) = ( 𝑠 ‘ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 ) ) ∧ 𝑠 ∈ 𝐸 ) ) ) |
32 |
|
fveq1 |
⊢ ( 𝑠 = 𝑆 → ( 𝑠 ‘ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 ) ) = ( 𝑆 ‘ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 ) ) ) |
33 |
32
|
eqeq2d |
⊢ ( 𝑠 = 𝑆 → ( ( I ↾ 𝐵 ) = ( 𝑠 ‘ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 ) ) ↔ ( I ↾ 𝐵 ) = ( 𝑆 ‘ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 ) ) ) ) |
34 |
|
eleq1 |
⊢ ( 𝑠 = 𝑆 → ( 𝑠 ∈ 𝐸 ↔ 𝑆 ∈ 𝐸 ) ) |
35 |
33 34
|
anbi12d |
⊢ ( 𝑠 = 𝑆 → ( ( ( I ↾ 𝐵 ) = ( 𝑠 ‘ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 ) ) ∧ 𝑠 ∈ 𝐸 ) ↔ ( ( I ↾ 𝐵 ) = ( 𝑆 ‘ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 ) ) ∧ 𝑆 ∈ 𝐸 ) ) ) |
36 |
31 35
|
opelopabg |
⊢ ( ( ( I ↾ 𝐵 ) ∈ V ∧ 𝑆 ∈ 𝐸 ) → ( 〈 ( I ↾ 𝐵 ) , 𝑆 〉 ∈ { 〈 𝑔 , 𝑠 〉 ∣ ( 𝑔 = ( 𝑠 ‘ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 ) ) ∧ 𝑠 ∈ 𝐸 ) } ↔ ( ( I ↾ 𝐵 ) = ( 𝑆 ‘ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 ) ) ∧ 𝑆 ∈ 𝐸 ) ) ) |
37 |
29 23 36
|
syl2anc |
⊢ ( 𝜑 → ( 〈 ( I ↾ 𝐵 ) , 𝑆 〉 ∈ { 〈 𝑔 , 𝑠 〉 ∣ ( 𝑔 = ( 𝑠 ‘ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 ) ) ∧ 𝑠 ∈ 𝐸 ) } ↔ ( ( I ↾ 𝐵 ) = ( 𝑆 ‘ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 ) ) ∧ 𝑆 ∈ 𝐸 ) ) ) |
38 |
26 23 37
|
mpbir2and |
⊢ ( 𝜑 → 〈 ( I ↾ 𝐵 ) , 𝑆 〉 ∈ { 〈 𝑔 , 𝑠 〉 ∣ ( 𝑔 = ( 𝑠 ‘ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 ) ) ∧ 𝑠 ∈ 𝐸 ) } ) |
39 |
|
eqid |
⊢ ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) |
40 |
16 17 2 39 7
|
dihvalcqat |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐼 ‘ 𝑃 ) = ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑃 ) ) |
41 |
10 18 40
|
syl2anc2 |
⊢ ( 𝜑 → ( 𝐼 ‘ 𝑃 ) = ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑃 ) ) |
42 |
16 17 2 3 4 5 39
|
dicval |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑃 ) = { 〈 𝑔 , 𝑠 〉 ∣ ( 𝑔 = ( 𝑠 ‘ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 ) ) ∧ 𝑠 ∈ 𝐸 ) } ) |
43 |
10 18 42
|
syl2anc2 |
⊢ ( 𝜑 → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑃 ) = { 〈 𝑔 , 𝑠 〉 ∣ ( 𝑔 = ( 𝑠 ‘ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 ) ) ∧ 𝑠 ∈ 𝐸 ) } ) |
44 |
41 43
|
eqtr2d |
⊢ ( 𝜑 → { 〈 𝑔 , 𝑠 〉 ∣ ( 𝑔 = ( 𝑠 ‘ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 ) ) ∧ 𝑠 ∈ 𝐸 ) } = ( 𝐼 ‘ 𝑃 ) ) |
45 |
38 44
|
eleqtrd |
⊢ ( 𝜑 → 〈 ( I ↾ 𝐵 ) , 𝑆 〉 ∈ ( 𝐼 ‘ 𝑃 ) ) |
46 |
11
|
simprd |
⊢ ( 𝜑 → 𝑆 ≠ 𝑂 ) |
47 |
1 2 4 8 12 6
|
dvh0g |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 0g ‘ 𝑈 ) = 〈 ( I ↾ 𝐵 ) , 𝑂 〉 ) |
48 |
10 47
|
syl |
⊢ ( 𝜑 → ( 0g ‘ 𝑈 ) = 〈 ( I ↾ 𝐵 ) , 𝑂 〉 ) |
49 |
48
|
eqeq2d |
⊢ ( 𝜑 → ( 〈 ( I ↾ 𝐵 ) , 𝑆 〉 = ( 0g ‘ 𝑈 ) ↔ 〈 ( I ↾ 𝐵 ) , 𝑆 〉 = 〈 ( I ↾ 𝐵 ) , 𝑂 〉 ) ) |
50 |
27 28
|
ax-mp |
⊢ ( I ↾ 𝐵 ) ∈ V |
51 |
4
|
fvexi |
⊢ 𝑇 ∈ V |
52 |
51
|
mptex |
⊢ ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) ) ∈ V |
53 |
6 52
|
eqeltri |
⊢ 𝑂 ∈ V |
54 |
50 53
|
opth2 |
⊢ ( 〈 ( I ↾ 𝐵 ) , 𝑆 〉 = 〈 ( I ↾ 𝐵 ) , 𝑂 〉 ↔ ( ( I ↾ 𝐵 ) = ( I ↾ 𝐵 ) ∧ 𝑆 = 𝑂 ) ) |
55 |
54
|
simprbi |
⊢ ( 〈 ( I ↾ 𝐵 ) , 𝑆 〉 = 〈 ( I ↾ 𝐵 ) , 𝑂 〉 → 𝑆 = 𝑂 ) |
56 |
49 55
|
syl6bi |
⊢ ( 𝜑 → ( 〈 ( I ↾ 𝐵 ) , 𝑆 〉 = ( 0g ‘ 𝑈 ) → 𝑆 = 𝑂 ) ) |
57 |
56
|
necon3d |
⊢ ( 𝜑 → ( 𝑆 ≠ 𝑂 → 〈 ( I ↾ 𝐵 ) , 𝑆 〉 ≠ ( 0g ‘ 𝑈 ) ) ) |
58 |
46 57
|
mpd |
⊢ ( 𝜑 → 〈 ( I ↾ 𝐵 ) , 𝑆 〉 ≠ ( 0g ‘ 𝑈 ) ) |
59 |
12 9 13 14 15 45 58
|
lsatel |
⊢ ( 𝜑 → ( 𝐼 ‘ 𝑃 ) = ( 𝑁 ‘ { 〈 ( I ↾ 𝐵 ) , 𝑆 〉 } ) ) |