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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > oplecon3 | Structured version Visualization version GIF version |
Description: Contraposition law for orthoposets. (Contributed by NM, 13-Sep-2011.) |
Ref | Expression |
---|---|
opcon3.b | β’ π΅ = (BaseβπΎ) |
opcon3.l | β’ β€ = (leβπΎ) |
opcon3.o | β’ β₯ = (ocβπΎ) |
Ref | Expression |
---|---|
oplecon3 | β’ ((πΎ β OP β§ π β π΅ β§ π β π΅) β (π β€ π β ( β₯ βπ) β€ ( β₯ βπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opcon3.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
2 | opcon3.l | . . . 4 β’ β€ = (leβπΎ) | |
3 | opcon3.o | . . . 4 β’ β₯ = (ocβπΎ) | |
4 | eqid 2730 | . . . 4 β’ (joinβπΎ) = (joinβπΎ) | |
5 | eqid 2730 | . . . 4 β’ (meetβπΎ) = (meetβπΎ) | |
6 | eqid 2730 | . . . 4 β’ (0.βπΎ) = (0.βπΎ) | |
7 | eqid 2730 | . . . 4 β’ (1.βπΎ) = (1.βπΎ) | |
8 | 1, 2, 3, 4, 5, 6, 7 | oposlem 38355 | . . 3 β’ ((πΎ β OP β§ π β π΅ β§ π β π΅) β ((( β₯ βπ) β π΅ β§ ( β₯ β( β₯ βπ)) = π β§ (π β€ π β ( β₯ βπ) β€ ( β₯ βπ))) β§ (π(joinβπΎ)( β₯ βπ)) = (1.βπΎ) β§ (π(meetβπΎ)( β₯ βπ)) = (0.βπΎ))) |
9 | 8 | simp1d 1140 | . 2 β’ ((πΎ β OP β§ π β π΅ β§ π β π΅) β (( β₯ βπ) β π΅ β§ ( β₯ β( β₯ βπ)) = π β§ (π β€ π β ( β₯ βπ) β€ ( β₯ βπ)))) |
10 | 9 | simp3d 1142 | 1 β’ ((πΎ β OP β§ π β π΅ β§ π β π΅) β (π β€ π β ( β₯ βπ) β€ ( β₯ βπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1085 = wceq 1539 β wcel 2104 class class class wbr 5147 βcfv 6542 (class class class)co 7411 Basecbs 17148 lecple 17208 occoc 17209 joincjn 18268 meetcmee 18269 0.cp0 18380 1.cp1 18381 OPcops 38345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-nul 5305 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-ral 3060 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-dm 5685 df-iota 6494 df-fv 6550 df-ov 7414 df-oposet 38349 |
This theorem is referenced by: oplecon3b 38373 |
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