Step | Hyp | Ref
| Expression |
1 | | fveq2 6843 |
. . . 4
β’ (π = πΎ β (Atomsβπ) = (AtomsβπΎ)) |
2 | | iscvlat.a |
. . . 4
β’ π΄ = (AtomsβπΎ) |
3 | 1, 2 | eqtr4di 2791 |
. . 3
β’ (π = πΎ β (Atomsβπ) = π΄) |
4 | | fveq2 6843 |
. . . . . 6
β’ (π = πΎ β (Baseβπ) = (BaseβπΎ)) |
5 | | iscvlat.b |
. . . . . 6
β’ π΅ = (BaseβπΎ) |
6 | 4, 5 | eqtr4di 2791 |
. . . . 5
β’ (π = πΎ β (Baseβπ) = π΅) |
7 | | fveq2 6843 |
. . . . . . . . . 10
β’ (π = πΎ β (leβπ) = (leβπΎ)) |
8 | | iscvlat.l |
. . . . . . . . . 10
β’ β€ =
(leβπΎ) |
9 | 7, 8 | eqtr4di 2791 |
. . . . . . . . 9
β’ (π = πΎ β (leβπ) = β€ ) |
10 | 9 | breqd 5117 |
. . . . . . . 8
β’ (π = πΎ β (π(leβπ)π₯ β π β€ π₯)) |
11 | 10 | notbid 318 |
. . . . . . 7
β’ (π = πΎ β (Β¬ π(leβπ)π₯ β Β¬ π β€ π₯)) |
12 | | eqidd 2734 |
. . . . . . . 8
β’ (π = πΎ β π = π) |
13 | | fveq2 6843 |
. . . . . . . . . 10
β’ (π = πΎ β (joinβπ) = (joinβπΎ)) |
14 | | iscvlat.j |
. . . . . . . . . 10
β’ β¨ =
(joinβπΎ) |
15 | 13, 14 | eqtr4di 2791 |
. . . . . . . . 9
β’ (π = πΎ β (joinβπ) = β¨ ) |
16 | 15 | oveqd 7375 |
. . . . . . . 8
β’ (π = πΎ β (π₯(joinβπ)π) = (π₯ β¨ π)) |
17 | 12, 9, 16 | breq123d 5120 |
. . . . . . 7
β’ (π = πΎ β (π(leβπ)(π₯(joinβπ)π) β π β€ (π₯ β¨ π))) |
18 | 11, 17 | anbi12d 632 |
. . . . . 6
β’ (π = πΎ β ((Β¬ π(leβπ)π₯ β§ π(leβπ)(π₯(joinβπ)π)) β (Β¬ π β€ π₯ β§ π β€ (π₯ β¨ π)))) |
19 | | eqidd 2734 |
. . . . . . 7
β’ (π = πΎ β π = π) |
20 | 15 | oveqd 7375 |
. . . . . . 7
β’ (π = πΎ β (π₯(joinβπ)π) = (π₯ β¨ π)) |
21 | 19, 9, 20 | breq123d 5120 |
. . . . . 6
β’ (π = πΎ β (π(leβπ)(π₯(joinβπ)π) β π β€ (π₯ β¨ π))) |
22 | 18, 21 | imbi12d 345 |
. . . . 5
β’ (π = πΎ β (((Β¬ π(leβπ)π₯ β§ π(leβπ)(π₯(joinβπ)π)) β π(leβπ)(π₯(joinβπ)π)) β ((Β¬ π β€ π₯ β§ π β€ (π₯ β¨ π)) β π β€ (π₯ β¨ π)))) |
23 | 6, 22 | raleqbidv 3318 |
. . . 4
β’ (π = πΎ β (βπ₯ β (Baseβπ)((Β¬ π(leβπ)π₯ β§ π(leβπ)(π₯(joinβπ)π)) β π(leβπ)(π₯(joinβπ)π)) β βπ₯ β π΅ ((Β¬ π β€ π₯ β§ π β€ (π₯ β¨ π)) β π β€ (π₯ β¨ π)))) |
24 | 3, 23 | raleqbidv 3318 |
. . 3
β’ (π = πΎ β (βπ β (Atomsβπ)βπ₯ β (Baseβπ)((Β¬ π(leβπ)π₯ β§ π(leβπ)(π₯(joinβπ)π)) β π(leβπ)(π₯(joinβπ)π)) β βπ β π΄ βπ₯ β π΅ ((Β¬ π β€ π₯ β§ π β€ (π₯ β¨ π)) β π β€ (π₯ β¨ π)))) |
25 | 3, 24 | raleqbidv 3318 |
. 2
β’ (π = πΎ β (βπ β (Atomsβπ)βπ β (Atomsβπ)βπ₯ β (Baseβπ)((Β¬ π(leβπ)π₯ β§ π(leβπ)(π₯(joinβπ)π)) β π(leβπ)(π₯(joinβπ)π)) β βπ β π΄ βπ β π΄ βπ₯ β π΅ ((Β¬ π β€ π₯ β§ π β€ (π₯ β¨ π)) β π β€ (π₯ β¨ π)))) |
26 | | df-cvlat 37830 |
. 2
β’ CvLat =
{π β AtLat β£
βπ β
(Atomsβπ)βπ β (Atomsβπ)βπ₯ β (Baseβπ)((Β¬ π(leβπ)π₯ β§ π(leβπ)(π₯(joinβπ)π)) β π(leβπ)(π₯(joinβπ)π))} |
27 | 25, 26 | elrab2 3649 |
1
β’ (πΎ β CvLat β (πΎ β AtLat β§
βπ β π΄ βπ β π΄ βπ₯ β π΅ ((Β¬ π β€ π₯ β§ π β€ (π₯ β¨ π)) β π β€ (π₯ β¨ π)))) |