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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > opoc1 | Structured version Visualization version GIF version |
Description: Orthocomplement of orthoposet unity. (Contributed by NM, 24-Jan-2012.) |
Ref | Expression |
---|---|
opoc1.z | β’ 0 = (0.βπΎ) |
opoc1.u | β’ 1 = (1.βπΎ) |
opoc1.o | β’ β₯ = (ocβπΎ) |
Ref | Expression |
---|---|
opoc1 | β’ (πΎ β OP β ( β₯ β 1 ) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . . . . 6 β’ (BaseβπΎ) = (BaseβπΎ) | |
2 | opoc1.z | . . . . . 6 β’ 0 = (0.βπΎ) | |
3 | 1, 2 | op0cl 38656 | . . . . 5 β’ (πΎ β OP β 0 β (BaseβπΎ)) |
4 | opoc1.o | . . . . . 6 β’ β₯ = (ocβπΎ) | |
5 | 1, 4 | opoccl 38666 | . . . . 5 β’ ((πΎ β OP β§ 0 β (BaseβπΎ)) β ( β₯ β 0 ) β (BaseβπΎ)) |
6 | 3, 5 | mpdan 686 | . . . 4 β’ (πΎ β OP β ( β₯ β 0 ) β (BaseβπΎ)) |
7 | eqid 2728 | . . . . 5 β’ (leβπΎ) = (leβπΎ) | |
8 | opoc1.u | . . . . 5 β’ 1 = (1.βπΎ) | |
9 | 1, 7, 8 | ople1 38663 | . . . 4 β’ ((πΎ β OP β§ ( β₯ β 0 ) β (BaseβπΎ)) β ( β₯ β 0 )(leβπΎ) 1 ) |
10 | 6, 9 | mpdan 686 | . . 3 β’ (πΎ β OP β ( β₯ β 0 )(leβπΎ) 1 ) |
11 | 1, 8 | op1cl 38657 | . . . 4 β’ (πΎ β OP β 1 β (BaseβπΎ)) |
12 | 1, 7, 4 | oplecon1b 38673 | . . . 4 β’ ((πΎ β OP β§ 1 β (BaseβπΎ) β§ 0 β (BaseβπΎ)) β (( β₯ β 1 )(leβπΎ) 0 β ( β₯ β 0 )(leβπΎ) 1 )) |
13 | 11, 3, 12 | mpd3an23 1460 | . . 3 β’ (πΎ β OP β (( β₯ β 1 )(leβπΎ) 0 β ( β₯ β 0 )(leβπΎ) 1 )) |
14 | 10, 13 | mpbird 257 | . 2 β’ (πΎ β OP β ( β₯ β 1 )(leβπΎ) 0 ) |
15 | 1, 4 | opoccl 38666 | . . . 4 β’ ((πΎ β OP β§ 1 β (BaseβπΎ)) β ( β₯ β 1 ) β (BaseβπΎ)) |
16 | 11, 15 | mpdan 686 | . . 3 β’ (πΎ β OP β ( β₯ β 1 ) β (BaseβπΎ)) |
17 | 1, 7, 2 | ople0 38659 | . . 3 β’ ((πΎ β OP β§ ( β₯ β 1 ) β (BaseβπΎ)) β (( β₯ β 1 )(leβπΎ) 0 β ( β₯ β 1 ) = 0 )) |
18 | 16, 17 | mpdan 686 | . 2 β’ (πΎ β OP β (( β₯ β 1 )(leβπΎ) 0 β ( β₯ β 1 ) = 0 )) |
19 | 14, 18 | mpbid 231 | 1 β’ (πΎ β OP β ( β₯ β 1 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1534 β wcel 2099 class class class wbr 5148 βcfv 6548 Basecbs 17179 lecple 17239 occoc 17240 0.cp0 18414 1.cp1 18415 OPcops 38644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-proset 18286 df-poset 18304 df-lub 18337 df-glb 18338 df-p0 18416 df-p1 18417 df-oposet 38648 |
This theorem is referenced by: opoc0 38675 olm11 38699 1cvrco 38945 1cvrjat 38948 pol1N 39383 doch1 40832 |
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