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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 37853 | . . . . 5 β’ (πΎ β HL β πΎ β OP) | |
2 | 1 | adantr 482 | . . . 4 β’ ((πΎ β HL β§ π β π΅) β πΎ β OP) |
3 | simpr 486 | . . . 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 37676 | . . . . 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 37768 | . . . 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 1372 | . . 3 β’ ((πΎ β HL β§ π β π΅) β (ππΆ 1 β ( β₯ β 1 )πΆ( β₯ βπ))) |
12 | eqid 2737 | . . . . . 6 β’ (0.βπΎ) = (0.βπΎ) | |
13 | 12, 5, 8 | opoc1 37693 | . . . . 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 5120 | . . 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 37685 | . . . . 5 β’ ((πΎ β OP β§ π β π΅) β ( β₯ βπ) β π΅) |
17 | 1, 16 | sylan 581 | . . . 4 β’ ((πΎ β HL β§ π β π΅) β ( β₯ βπ) β π΅) |
18 | 17 | biantrurd 534 | . . 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 37777 | . . 3 β’ (πΎ β HL β (( β₯ βπ) β π΄ β (( β₯ βπ) β π΅ β§ (0.βπΎ)πΆ( β₯ βπ)))) |
22 | 21 | adantr 482 | . 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 397 = wceq 1542 β wcel 2107 class class class wbr 5110 βcfv 6501 Basecbs 17090 occoc 17148 0.cp0 18319 1.cp1 18320 OPcops 37663 β ccvr 37753 Atomscatm 37754 HLchlt 37841 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
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-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-proset 18191 df-poset 18209 df-plt 18226 df-lub 18242 df-glb 18243 df-p0 18321 df-p1 18322 df-oposet 37667 df-ol 37669 df-oml 37670 df-covers 37757 df-ats 37758 df-hlat 37842 |
This theorem is referenced by: 1cvratex 37965 lhpoc 38506 |
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