Step | Hyp | Ref
| Expression |
1 | | omllaw.b |
. . . . 5
β’ π΅ = (BaseβπΎ) |
2 | | omllaw.l |
. . . . 5
β’ β€ =
(leβπΎ) |
3 | | omllaw.j |
. . . . 5
β’ β¨ =
(joinβπΎ) |
4 | | omllaw.m |
. . . . 5
β’ β§ =
(meetβπΎ) |
5 | | omllaw.o |
. . . . 5
β’ β₯ =
(ocβπΎ) |
6 | 1, 2, 3, 4, 5 | isoml 37703 |
. . . 4
β’ (πΎ β OML β (πΎ β OL β§ βπ₯ β π΅ βπ¦ β π΅ (π₯ β€ π¦ β π¦ = (π₯ β¨ (π¦ β§ ( β₯ βπ₯)))))) |
7 | 6 | simprbi 498 |
. . 3
β’ (πΎ β OML β βπ₯ β π΅ βπ¦ β π΅ (π₯ β€ π¦ β π¦ = (π₯ β¨ (π¦ β§ ( β₯ βπ₯))))) |
8 | | breq1 5109 |
. . . . 5
β’ (π₯ = π β (π₯ β€ π¦ β π β€ π¦)) |
9 | | id 22 |
. . . . . . 7
β’ (π₯ = π β π₯ = π) |
10 | | fveq2 6843 |
. . . . . . . 8
β’ (π₯ = π β ( β₯ βπ₯) = ( β₯ βπ)) |
11 | 10 | oveq2d 7374 |
. . . . . . 7
β’ (π₯ = π β (π¦ β§ ( β₯ βπ₯)) = (π¦ β§ ( β₯ βπ))) |
12 | 9, 11 | oveq12d 7376 |
. . . . . 6
β’ (π₯ = π β (π₯ β¨ (π¦ β§ ( β₯ βπ₯))) = (π β¨ (π¦ β§ ( β₯ βπ)))) |
13 | 12 | eqeq2d 2748 |
. . . . 5
β’ (π₯ = π β (π¦ = (π₯ β¨ (π¦ β§ ( β₯ βπ₯))) β π¦ = (π β¨ (π¦ β§ ( β₯ βπ))))) |
14 | 8, 13 | imbi12d 345 |
. . . 4
β’ (π₯ = π β ((π₯ β€ π¦ β π¦ = (π₯ β¨ (π¦ β§ ( β₯ βπ₯)))) β (π β€ π¦ β π¦ = (π β¨ (π¦ β§ ( β₯ βπ)))))) |
15 | | breq2 5110 |
. . . . 5
β’ (π¦ = π β (π β€ π¦ β π β€ π)) |
16 | | id 22 |
. . . . . 6
β’ (π¦ = π β π¦ = π) |
17 | | oveq1 7365 |
. . . . . . 7
β’ (π¦ = π β (π¦ β§ ( β₯ βπ)) = (π β§ ( β₯ βπ))) |
18 | 17 | oveq2d 7374 |
. . . . . 6
β’ (π¦ = π β (π β¨ (π¦ β§ ( β₯ βπ))) = (π β¨ (π β§ ( β₯ βπ)))) |
19 | 16, 18 | eqeq12d 2753 |
. . . . 5
β’ (π¦ = π β (π¦ = (π β¨ (π¦ β§ ( β₯ βπ))) β π = (π β¨ (π β§ ( β₯ βπ))))) |
20 | 15, 19 | imbi12d 345 |
. . . 4
β’ (π¦ = π β ((π β€ π¦ β π¦ = (π β¨ (π¦ β§ ( β₯ βπ)))) β (π β€ π β π = (π β¨ (π β§ ( β₯ βπ)))))) |
21 | 14, 20 | rspc2v 3591 |
. . 3
β’ ((π β π΅ β§ π β π΅) β (βπ₯ β π΅ βπ¦ β π΅ (π₯ β€ π¦ β π¦ = (π₯ β¨ (π¦ β§ ( β₯ βπ₯)))) β (π β€ π β π = (π β¨ (π β§ ( β₯ βπ)))))) |
22 | 7, 21 | syl5com 31 |
. 2
β’ (πΎ β OML β ((π β π΅ β§ π β π΅) β (π β€ π β π = (π β¨ (π β§ ( β₯ βπ)))))) |
23 | 22 | 3impib 1117 |
1
β’ ((πΎ β OML β§ π β π΅ β§ π β π΅) β (π β€ π β π = (π β¨ (π β§ ( β₯ βπ))))) |