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Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > olm11 | Structured version Visualization version GIF version | ||
| Description: The meet of an ortholattice element with one equals itself. (chm1i 31204 analog.) (Contributed by NM, 22-May-2012.) |
| Ref | Expression |
|---|---|
| olm1.b | β’ π΅ = (BaseβπΎ) |
| olm1.m | β’ β§ = (meetβπΎ) |
| olm1.u | β’ 1 = (1.βπΎ) |
| Ref | Expression |
|---|---|
| olm11 | β’ ((πΎ β OL β§ π β π΅) β (π β§ 1 ) = π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | olop 38588 | . . . . . . 7 β’ (πΎ β OL β πΎ β OP) | |
| 2 | 1 | adantr 480 | . . . . . 6 β’ ((πΎ β OL β§ π β π΅) β πΎ β OP) |
| 3 | eqid 2724 | . . . . . . 7 β’ (0.βπΎ) = (0.βπΎ) | |
| 4 | olm1.u | . . . . . . 7 β’ 1 = (1.βπΎ) | |
| 5 | eqid 2724 | . . . . . . 7 β’ (ocβπΎ) = (ocβπΎ) | |
| 6 | 3, 4, 5 | opoc1 38576 | . . . . . 6 β’ (πΎ β OP β ((ocβπΎ)β 1 ) = (0.βπΎ)) |
| 7 | 2, 6 | syl 17 | . . . . 5 β’ ((πΎ β OL β§ π β π΅) β ((ocβπΎ)β 1 ) = (0.βπΎ)) |
| 8 | 7 | oveq2d 7418 | . . . 4 β’ ((πΎ β OL β§ π β π΅) β (((ocβπΎ)βπ)(joinβπΎ)((ocβπΎ)β 1 )) = (((ocβπΎ)βπ)(joinβπΎ)(0.βπΎ))) |
| 9 | olm1.b | . . . . . . 7 β’ π΅ = (BaseβπΎ) | |
| 10 | 9, 5 | opoccl 38568 | . . . . . 6 β’ ((πΎ β OP β§ π β π΅) β ((ocβπΎ)βπ) β π΅) |
| 11 | 1, 10 | sylan 579 | . . . . 5 β’ ((πΎ β OL β§ π β π΅) β ((ocβπΎ)βπ) β π΅) |
| 12 | eqid 2724 | . . . . . 6 β’ (joinβπΎ) = (joinβπΎ) | |
| 13 | 9, 12, 3 | olj01 38599 | . . . . 5 β’ ((πΎ β OL β§ ((ocβπΎ)βπ) β π΅) β (((ocβπΎ)βπ)(joinβπΎ)(0.βπΎ)) = ((ocβπΎ)βπ)) |
| 14 | 11, 13 | syldan 590 | . . . 4 β’ ((πΎ β OL β§ π β π΅) β (((ocβπΎ)βπ)(joinβπΎ)(0.βπΎ)) = ((ocβπΎ)βπ)) |
| 15 | 8, 14 | eqtrd 2764 | . . 3 β’ ((πΎ β OL β§ π β π΅) β (((ocβπΎ)βπ)(joinβπΎ)((ocβπΎ)β 1 )) = ((ocβπΎ)βπ)) |
| 16 | 15 | fveq2d 6886 | . 2 β’ ((πΎ β OL β§ π β π΅) β ((ocβπΎ)β(((ocβπΎ)βπ)(joinβπΎ)((ocβπΎ)β 1 ))) = ((ocβπΎ)β((ocβπΎ)βπ))) |
| 17 | 9, 4 | op1cl 38559 | . . . 4 β’ (πΎ β OP β 1 β π΅) |
| 18 | 2, 17 | syl 17 | . . 3 β’ ((πΎ β OL β§ π β π΅) β 1 β π΅) |
| 19 | olm1.m | . . . 4 β’ β§ = (meetβπΎ) | |
| 20 | 9, 12, 19, 5 | oldmj4 38598 | . . 3 β’ ((πΎ β OL β§ π β π΅ β§ 1 β π΅) β ((ocβπΎ)β(((ocβπΎ)βπ)(joinβπΎ)((ocβπΎ)β 1 ))) = (π β§ 1 )) |
| 21 | 18, 20 | mpd3an3 1458 | . 2 β’ ((πΎ β OL β§ π β π΅) β ((ocβπΎ)β(((ocβπΎ)βπ)(joinβπΎ)((ocβπΎ)β 1 ))) = (π β§ 1 )) |
| 22 | 9, 5 | opococ 38569 | . . 3 β’ ((πΎ β OP β§ π β π΅) β ((ocβπΎ)β((ocβπΎ)βπ)) = π) |
| 23 | 1, 22 | sylan 579 | . 2 β’ ((πΎ β OL β§ π β π΅) β ((ocβπΎ)β((ocβπΎ)βπ)) = π) |
| 24 | 16, 21, 23 | 3eqtr3d 2772 | 1 β’ ((πΎ β OL β§ π β π΅) β (π β§ 1 ) = π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 βcfv 6534 (class class class)co 7402 Basecbs 17149 occoc 17210 joincjn 18272 meetcmee 18273 0.cp0 18384 1.cp1 18385 OPcops 38546 OLcol 38548 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-proset 18256 df-poset 18274 df-lub 18307 df-glb 18308 df-join 18309 df-meet 18310 df-p0 18386 df-p1 18387 df-lat 18393 df-oposet 38550 df-ol 38552 |
| This theorem is referenced by: olm12 38602 lhpmcvr3 39400 trljat1 39541 trljat2 39542 cdlemc1 39566 cdlemc6 39571 cdleme0cp 39589 cdleme0cq 39590 cdleme1 39602 cdleme4 39613 cdleme5 39615 cdleme8 39625 cdleme9 39628 cdleme10 39629 cdleme20c 39686 cdleme20j 39693 cdleme22e 39719 cdleme22eALTN 39720 cdleme30a 39753 cdleme35b 39825 cdleme35e 39828 cdleme42a 39846 trlcoabs2N 40097 trlcolem 40101 cdlemi1 40193 cdlemk4 40209 dia2dimlem1 40439 cdlemn10 40581 dihglbcpreN 40675 |
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