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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 1cvrco | Structured version Visualization version GIF version |
Description: The orthocomplement of an element covered by 1 is an atom. (Contributed by NM, 7-May-2012.) |
Ref | Expression |
---|---|
1cvrco.b | β’ π΅ = (BaseβπΎ) |
1cvrco.u | β’ 1 = (1.βπΎ) |
1cvrco.o | β’ β₯ = (ocβπΎ) |
1cvrco.c | β’ πΆ = ( β βπΎ) |
1cvrco.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
1cvrco | β’ ((πΎ β HL β§ π β π΅) β (ππΆ 1 β ( β₯ βπ) β π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlop 38834 | . . . . 5 β’ (πΎ β HL β πΎ β OP) | |
2 | 1 | adantr 480 | . . . 4 β’ ((πΎ β HL β§ π β π΅) β πΎ β OP) |
3 | simpr 484 | . . . 4 β’ ((πΎ β HL β§ π β π΅) β π β π΅) | |
4 | 1cvrco.b | . . . . . 6 β’ π΅ = (BaseβπΎ) | |
5 | 1cvrco.u | . . . . . 6 β’ 1 = (1.βπΎ) | |
6 | 4, 5 | op1cl 38657 | . . . . 5 β’ (πΎ β OP β 1 β π΅) |
7 | 2, 6 | syl 17 | . . . 4 β’ ((πΎ β HL β§ π β π΅) β 1 β π΅) |
8 | 1cvrco.o | . . . . 5 β’ β₯ = (ocβπΎ) | |
9 | 1cvrco.c | . . . . 5 β’ πΆ = ( β βπΎ) | |
10 | 4, 8, 9 | cvrcon3b 38749 | . . . 4 β’ ((πΎ β OP β§ π β π΅ β§ 1 β π΅) β (ππΆ 1 β ( β₯ β 1 )πΆ( β₯ βπ))) |
11 | 2, 3, 7, 10 | syl3anc 1369 | . . 3 β’ ((πΎ β HL β§ π β π΅) β (ππΆ 1 β ( β₯ β 1 )πΆ( β₯ βπ))) |
12 | eqid 2728 | . . . . . 6 β’ (0.βπΎ) = (0.βπΎ) | |
13 | 12, 5, 8 | opoc1 38674 | . . . . 5 β’ (πΎ β OP β ( β₯ β 1 ) = (0.βπΎ)) |
14 | 2, 13 | syl 17 | . . . 4 β’ ((πΎ β HL β§ π β π΅) β ( β₯ β 1 ) = (0.βπΎ)) |
15 | 14 | breq1d 5158 | . . 3 β’ ((πΎ β HL β§ π β π΅) β (( β₯ β 1 )πΆ( β₯ βπ) β (0.βπΎ)πΆ( β₯ βπ))) |
16 | 4, 8 | opoccl 38666 | . . . . 5 β’ ((πΎ β OP β§ π β π΅) β ( β₯ βπ) β π΅) |
17 | 1, 16 | sylan 579 | . . . 4 β’ ((πΎ β HL β§ π β π΅) β ( β₯ βπ) β π΅) |
18 | 17 | biantrurd 532 | . . 3 β’ ((πΎ β HL β§ π β π΅) β ((0.βπΎ)πΆ( β₯ βπ) β (( β₯ βπ) β π΅ β§ (0.βπΎ)πΆ( β₯ βπ)))) |
19 | 11, 15, 18 | 3bitrd 305 | . 2 β’ ((πΎ β HL β§ π β π΅) β (ππΆ 1 β (( β₯ βπ) β π΅ β§ (0.βπΎ)πΆ( β₯ βπ)))) |
20 | 1cvrco.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
21 | 4, 12, 9, 20 | isat 38758 | . . 3 β’ (πΎ β HL β (( β₯ βπ) β π΄ β (( β₯ βπ) β π΅ β§ (0.βπΎ)πΆ( β₯ βπ)))) |
22 | 21 | adantr 480 | . 2 β’ ((πΎ β HL β§ π β π΅) β (( β₯ βπ) β π΄ β (( β₯ βπ) β π΅ β§ (0.βπΎ)πΆ( β₯ βπ)))) |
23 | 19, 22 | bitr4d 282 | 1 β’ ((πΎ β HL β§ π β π΅) β (ππΆ 1 β ( β₯ βπ) β π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 class class class wbr 5148 βcfv 6548 Basecbs 17180 occoc 17241 0.cp0 18415 1.cp1 18416 OPcops 38644 β ccvr 38734 Atomscatm 38735 HLchlt 38822 |
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 ax-un 7740 |
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 18287 df-poset 18305 df-plt 18322 df-lub 18338 df-glb 18339 df-p0 18417 df-p1 18418 df-oposet 38648 df-ol 38650 df-oml 38651 df-covers 38738 df-ats 38739 df-hlat 38823 |
This theorem is referenced by: 1cvratex 38946 lhpoc 39487 |
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