Step |
Hyp |
Ref |
Expression |
1 |
|
dib1dim.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dib1dim.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
3 |
|
dib1dim.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
dib1dim.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
dib1dim.e |
⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
dib1dim.o |
⊢ 𝑂 = ( ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵 ) ) |
7 |
|
dib1dim.i |
⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
simpl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
9 |
1 2 3 4
|
trlcl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐹 ) ∈ 𝐵 ) |
10 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
11 |
10 2 3 4
|
trlle |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐹 ) ( le ‘ 𝐾 ) 𝑊 ) |
12 |
|
eqid |
⊢ ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) |
13 |
1 10 2 3 6 12 7
|
dibval2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑅 ‘ 𝐹 ) ∈ 𝐵 ∧ ( 𝑅 ‘ 𝐹 ) ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) = ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑅 ‘ 𝐹 ) ) × { 𝑂 } ) ) |
14 |
8 9 11 13
|
syl12anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) = ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑅 ‘ 𝐹 ) ) × { 𝑂 } ) ) |
15 |
|
relxp |
⊢ Rel ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑅 ‘ 𝐹 ) ) × { 𝑂 } ) |
16 |
|
opelxp |
⊢ ( 〈 𝑓 , 𝑡 〉 ∈ ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑅 ‘ 𝐹 ) ) × { 𝑂 } ) ↔ ( 𝑓 ∈ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑅 ‘ 𝐹 ) ) ∧ 𝑡 ∈ { 𝑂 } ) ) |
17 |
2 3 4 5 12
|
dia1dim |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑅 ‘ 𝐹 ) ) = { 𝑓 ∣ ∃ 𝑠 ∈ 𝐸 𝑓 = ( 𝑠 ‘ 𝐹 ) } ) |
18 |
17
|
abeq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑓 ∈ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑅 ‘ 𝐹 ) ) ↔ ∃ 𝑠 ∈ 𝐸 𝑓 = ( 𝑠 ‘ 𝐹 ) ) ) |
19 |
18
|
anbi1d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑓 ∈ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑅 ‘ 𝐹 ) ) ∧ 𝑡 ∈ { 𝑂 } ) ↔ ( ∃ 𝑠 ∈ 𝐸 𝑓 = ( 𝑠 ‘ 𝐹 ) ∧ 𝑡 ∈ { 𝑂 } ) ) ) |
20 |
2 3 5
|
tendocl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑠 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑠 ‘ 𝐹 ) ∈ 𝑇 ) |
21 |
20
|
3expa |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑠 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑠 ‘ 𝐹 ) ∈ 𝑇 ) |
22 |
21
|
an32s |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝑠 ∈ 𝐸 ) → ( 𝑠 ‘ 𝐹 ) ∈ 𝑇 ) |
23 |
1 2 3 5 6
|
tendo0cl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑂 ∈ 𝐸 ) |
24 |
23
|
ad2antrr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝑠 ∈ 𝐸 ) → 𝑂 ∈ 𝐸 ) |
25 |
22 24
|
jca |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝑠 ∈ 𝐸 ) → ( ( 𝑠 ‘ 𝐹 ) ∈ 𝑇 ∧ 𝑂 ∈ 𝐸 ) ) |
26 |
|
eleq1 |
⊢ ( 𝑓 = ( 𝑠 ‘ 𝐹 ) → ( 𝑓 ∈ 𝑇 ↔ ( 𝑠 ‘ 𝐹 ) ∈ 𝑇 ) ) |
27 |
|
eleq1 |
⊢ ( 𝑡 = 𝑂 → ( 𝑡 ∈ 𝐸 ↔ 𝑂 ∈ 𝐸 ) ) |
28 |
26 27
|
bi2anan9 |
⊢ ( ( 𝑓 = ( 𝑠 ‘ 𝐹 ) ∧ 𝑡 = 𝑂 ) → ( ( 𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸 ) ↔ ( ( 𝑠 ‘ 𝐹 ) ∈ 𝑇 ∧ 𝑂 ∈ 𝐸 ) ) ) |
29 |
25 28
|
syl5ibrcom |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝑠 ∈ 𝐸 ) → ( ( 𝑓 = ( 𝑠 ‘ 𝐹 ) ∧ 𝑡 = 𝑂 ) → ( 𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸 ) ) ) |
30 |
29
|
rexlimdva |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ∃ 𝑠 ∈ 𝐸 ( 𝑓 = ( 𝑠 ‘ 𝐹 ) ∧ 𝑡 = 𝑂 ) → ( 𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸 ) ) ) |
31 |
30
|
pm4.71rd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ∃ 𝑠 ∈ 𝐸 ( 𝑓 = ( 𝑠 ‘ 𝐹 ) ∧ 𝑡 = 𝑂 ) ↔ ( ( 𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸 ) ∧ ∃ 𝑠 ∈ 𝐸 ( 𝑓 = ( 𝑠 ‘ 𝐹 ) ∧ 𝑡 = 𝑂 ) ) ) ) |
32 |
|
velsn |
⊢ ( 𝑡 ∈ { 𝑂 } ↔ 𝑡 = 𝑂 ) |
33 |
32
|
anbi2i |
⊢ ( ( ∃ 𝑠 ∈ 𝐸 𝑓 = ( 𝑠 ‘ 𝐹 ) ∧ 𝑡 ∈ { 𝑂 } ) ↔ ( ∃ 𝑠 ∈ 𝐸 𝑓 = ( 𝑠 ‘ 𝐹 ) ∧ 𝑡 = 𝑂 ) ) |
34 |
|
r19.41v |
⊢ ( ∃ 𝑠 ∈ 𝐸 ( 𝑓 = ( 𝑠 ‘ 𝐹 ) ∧ 𝑡 = 𝑂 ) ↔ ( ∃ 𝑠 ∈ 𝐸 𝑓 = ( 𝑠 ‘ 𝐹 ) ∧ 𝑡 = 𝑂 ) ) |
35 |
33 34
|
bitr4i |
⊢ ( ( ∃ 𝑠 ∈ 𝐸 𝑓 = ( 𝑠 ‘ 𝐹 ) ∧ 𝑡 ∈ { 𝑂 } ) ↔ ∃ 𝑠 ∈ 𝐸 ( 𝑓 = ( 𝑠 ‘ 𝐹 ) ∧ 𝑡 = 𝑂 ) ) |
36 |
|
df-3an |
⊢ ( ( 𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸 ∧ ∃ 𝑠 ∈ 𝐸 ( 𝑓 = ( 𝑠 ‘ 𝐹 ) ∧ 𝑡 = 𝑂 ) ) ↔ ( ( 𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸 ) ∧ ∃ 𝑠 ∈ 𝐸 ( 𝑓 = ( 𝑠 ‘ 𝐹 ) ∧ 𝑡 = 𝑂 ) ) ) |
37 |
31 35 36
|
3bitr4g |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( ∃ 𝑠 ∈ 𝐸 𝑓 = ( 𝑠 ‘ 𝐹 ) ∧ 𝑡 ∈ { 𝑂 } ) ↔ ( 𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸 ∧ ∃ 𝑠 ∈ 𝐸 ( 𝑓 = ( 𝑠 ‘ 𝐹 ) ∧ 𝑡 = 𝑂 ) ) ) ) |
38 |
19 37
|
bitrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑓 ∈ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑅 ‘ 𝐹 ) ) ∧ 𝑡 ∈ { 𝑂 } ) ↔ ( 𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸 ∧ ∃ 𝑠 ∈ 𝐸 ( 𝑓 = ( 𝑠 ‘ 𝐹 ) ∧ 𝑡 = 𝑂 ) ) ) ) |
39 |
16 38
|
syl5bb |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 〈 𝑓 , 𝑡 〉 ∈ ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑅 ‘ 𝐹 ) ) × { 𝑂 } ) ↔ ( 𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸 ∧ ∃ 𝑠 ∈ 𝐸 ( 𝑓 = ( 𝑠 ‘ 𝐹 ) ∧ 𝑡 = 𝑂 ) ) ) ) |
40 |
15 39
|
opabbi2dv |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑅 ‘ 𝐹 ) ) × { 𝑂 } ) = { 〈 𝑓 , 𝑡 〉 ∣ ( 𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸 ∧ ∃ 𝑠 ∈ 𝐸 ( 𝑓 = ( 𝑠 ‘ 𝐹 ) ∧ 𝑡 = 𝑂 ) ) } ) |
41 |
14 40
|
eqtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) = { 〈 𝑓 , 𝑡 〉 ∣ ( 𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸 ∧ ∃ 𝑠 ∈ 𝐸 ( 𝑓 = ( 𝑠 ‘ 𝐹 ) ∧ 𝑡 = 𝑂 ) ) } ) |
42 |
|
eqeq1 |
⊢ ( 𝑔 = 〈 𝑓 , 𝑡 〉 → ( 𝑔 = 〈 ( 𝑠 ‘ 𝐹 ) , 𝑂 〉 ↔ 〈 𝑓 , 𝑡 〉 = 〈 ( 𝑠 ‘ 𝐹 ) , 𝑂 〉 ) ) |
43 |
|
vex |
⊢ 𝑓 ∈ V |
44 |
|
vex |
⊢ 𝑡 ∈ V |
45 |
43 44
|
opth |
⊢ ( 〈 𝑓 , 𝑡 〉 = 〈 ( 𝑠 ‘ 𝐹 ) , 𝑂 〉 ↔ ( 𝑓 = ( 𝑠 ‘ 𝐹 ) ∧ 𝑡 = 𝑂 ) ) |
46 |
42 45
|
syl6bb |
⊢ ( 𝑔 = 〈 𝑓 , 𝑡 〉 → ( 𝑔 = 〈 ( 𝑠 ‘ 𝐹 ) , 𝑂 〉 ↔ ( 𝑓 = ( 𝑠 ‘ 𝐹 ) ∧ 𝑡 = 𝑂 ) ) ) |
47 |
46
|
rexbidv |
⊢ ( 𝑔 = 〈 𝑓 , 𝑡 〉 → ( ∃ 𝑠 ∈ 𝐸 𝑔 = 〈 ( 𝑠 ‘ 𝐹 ) , 𝑂 〉 ↔ ∃ 𝑠 ∈ 𝐸 ( 𝑓 = ( 𝑠 ‘ 𝐹 ) ∧ 𝑡 = 𝑂 ) ) ) |
48 |
47
|
rabxp |
⊢ { 𝑔 ∈ ( 𝑇 × 𝐸 ) ∣ ∃ 𝑠 ∈ 𝐸 𝑔 = 〈 ( 𝑠 ‘ 𝐹 ) , 𝑂 〉 } = { 〈 𝑓 , 𝑡 〉 ∣ ( 𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸 ∧ ∃ 𝑠 ∈ 𝐸 ( 𝑓 = ( 𝑠 ‘ 𝐹 ) ∧ 𝑡 = 𝑂 ) ) } |
49 |
41 48
|
eqtr4di |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) = { 𝑔 ∈ ( 𝑇 × 𝐸 ) ∣ ∃ 𝑠 ∈ 𝐸 𝑔 = 〈 ( 𝑠 ‘ 𝐹 ) , 𝑂 〉 } ) |