Step |
Hyp |
Ref |
Expression |
1 |
|
dib0.z |
⊢ 0 = ( 0. ‘ 𝐾 ) |
2 |
|
dib0.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
3 |
|
dib0.i |
⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
dib0.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
dib0.o |
⊢ 𝑂 = ( 0g ‘ 𝑈 ) |
6 |
|
fvex |
⊢ ( Base ‘ 𝐾 ) ∈ V |
7 |
|
resiexg |
⊢ ( ( Base ‘ 𝐾 ) ∈ V → ( I ↾ ( Base ‘ 𝐾 ) ) ∈ V ) |
8 |
6 7
|
ax-mp |
⊢ ( I ↾ ( Base ‘ 𝐾 ) ) ∈ V |
9 |
|
fvex |
⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∈ V |
10 |
9
|
mptex |
⊢ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) ∈ V |
11 |
8 10
|
xpsn |
⊢ ( { ( I ↾ ( Base ‘ 𝐾 ) ) } × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) } ) = { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) 〉 } |
12 |
|
id |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
13 |
|
hlop |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP ) |
14 |
13
|
adantr |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐾 ∈ OP ) |
15 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
16 |
15 1
|
op0cl |
⊢ ( 𝐾 ∈ OP → 0 ∈ ( Base ‘ 𝐾 ) ) |
17 |
14 16
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 0 ∈ ( Base ‘ 𝐾 ) ) |
18 |
15 2
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
19 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
20 |
15 19 1
|
op0le |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → 0 ( le ‘ 𝐾 ) 𝑊 ) |
21 |
13 18 20
|
syl2an |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 0 ( le ‘ 𝐾 ) 𝑊 ) |
22 |
|
eqid |
⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
23 |
|
eqid |
⊢ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) = ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) |
24 |
|
eqid |
⊢ ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) |
25 |
15 19 2 22 23 24 3
|
dibval2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 0 ∈ ( Base ‘ 𝐾 ) ∧ 0 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐼 ‘ 0 ) = ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 0 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) } ) ) |
26 |
12 17 21 25
|
syl12anc |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐼 ‘ 0 ) = ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 0 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) } ) ) |
27 |
15 1 2 24
|
dia0 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 0 ) = { ( I ↾ ( Base ‘ 𝐾 ) ) } ) |
28 |
27
|
xpeq1d |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 0 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) } ) = ( { ( I ↾ ( Base ‘ 𝐾 ) ) } × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) } ) ) |
29 |
26 28
|
eqtrd |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐼 ‘ 0 ) = ( { ( I ↾ ( Base ‘ 𝐾 ) ) } × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) } ) ) |
30 |
15 2 22 4 5 23
|
dvh0g |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑂 = 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) 〉 ) |
31 |
30
|
sneqd |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → { 𝑂 } = { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) 〉 } ) |
32 |
11 29 31
|
3eqtr4a |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐼 ‘ 0 ) = { 𝑂 } ) |