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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isomliN | Structured version Visualization version GIF version |
Description: Properties that determine an orthomodular lattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
isomli.0 | β’ πΎ β OL |
isomli.b | β’ π΅ = (BaseβπΎ) |
isomli.l | β’ β€ = (leβπΎ) |
isomli.j | β’ β¨ = (joinβπΎ) |
isomli.m | β’ β§ = (meetβπΎ) |
isomli.o | β’ β₯ = (ocβπΎ) |
isomli.7 | β’ ((π₯ β π΅ β§ π¦ β π΅) β (π₯ β€ π¦ β π¦ = (π₯ β¨ (π¦ β§ ( β₯ βπ₯))))) |
Ref | Expression |
---|---|
isomliN | β’ πΎ β OML |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isomli.0 | . 2 β’ πΎ β OL | |
2 | isomli.7 | . . 3 β’ ((π₯ β π΅ β§ π¦ β π΅) β (π₯ β€ π¦ β π¦ = (π₯ β¨ (π¦ β§ ( β₯ βπ₯))))) | |
3 | 2 | rgen2 3195 | . 2 β’ βπ₯ β π΅ βπ¦ β π΅ (π₯ β€ π¦ β π¦ = (π₯ β¨ (π¦ β§ ( β₯ βπ₯)))) |
4 | isomli.b | . . 3 β’ π΅ = (BaseβπΎ) | |
5 | isomli.l | . . 3 β’ β€ = (leβπΎ) | |
6 | isomli.j | . . 3 β’ β¨ = (joinβπΎ) | |
7 | isomli.m | . . 3 β’ β§ = (meetβπΎ) | |
8 | isomli.o | . . 3 β’ β₯ = (ocβπΎ) | |
9 | 4, 5, 6, 7, 8 | isoml 37703 | . 2 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₯)))))) |
10 | 1, 3, 9 | mpbir2an 710 | 1 β’ πΎ β OML |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3065 class class class wbr 5106 βcfv 6497 (class class class)co 7358 Basecbs 17084 lecple 17141 occoc 17142 joincjn 18201 meetcmee 18202 OLcol 37639 OMLcoml 37640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fv 6505 df-ov 7361 df-oml 37644 |
This theorem is referenced by: (None) |
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