Nearby theorems
|
|
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
| 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 30696 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 38072 | . . . . . . 7 β’ (πΎ β OL β πΎ β OP) | |
| 2 | 1 | adantr 481 | . . . . . 6 β’ ((πΎ β OL β§ π β π΅) β πΎ β OP) |
| 3 | eqid 2732 | . . . . . . 7 β’ (0.βπΎ) = (0.βπΎ) | |
| 4 | olm1.u | . . . . . . 7 β’ 1 = (1.βπΎ) | |
| 5 | eqid 2732 | . . . . . . 7 β’ (ocβπΎ) = (ocβπΎ) | |
| 6 | 3, 4, 5 | opoc1 38060 | . . . . . 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 7421 | . . . 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 38052 | . . . . . 6 β’ ((πΎ β OP β§ π β π΅) β ((ocβπΎ)βπ) β π΅) |
| 11 | 1, 10 | sylan 580 | . . . . 5 β’ ((πΎ β OL β§ π β π΅) β ((ocβπΎ)βπ) β π΅) |
| 12 | eqid 2732 | . . . . . 6 β’ (joinβπΎ) = (joinβπΎ) | |
| 13 | 9, 12, 3 | olj01 38083 | . . . . 5 β’ ((πΎ β OL β§ ((ocβπΎ)βπ) β π΅) β (((ocβπΎ)βπ)(joinβπΎ)(0.βπΎ)) = ((ocβπΎ)βπ)) |
| 14 | 11, 13 | syldan 591 | . . . 4 β’ ((πΎ β OL β§ π β π΅) β (((ocβπΎ)βπ)(joinβπΎ)(0.βπΎ)) = ((ocβπΎ)βπ)) |
| 15 | 8, 14 | eqtrd 2772 | . . 3 β’ ((πΎ β OL β§ π β π΅) β (((ocβπΎ)βπ)(joinβπΎ)((ocβπΎ)β 1 )) = ((ocβπΎ)βπ)) |
| 16 | 15 | fveq2d 6892 | . 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 38043 | . . . 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 38082 | . . 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 1462 | . 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 38053 | . . 3 β’ ((πΎ β OP β§ π β π΅) β ((ocβπΎ)β((ocβπΎ)βπ)) = π) |
| 23 | 1, 22 | sylan 580 | . 2 β’ ((πΎ β OL β§ π β π΅) β ((ocβπΎ)β((ocβπΎ)βπ)) = π) |
| 24 | 16, 21, 23 | 3eqtr3d 2780 | 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 396 = wceq 1541 β wcel 2106 βcfv 6540 (class class class)co 7405 Basecbs 17140 occoc 17201 joincjn 18260 meetcmee 18261 0.cp0 18372 1.cp1 18373 OPcops 38030 OLcol 38032 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-proset 18244 df-poset 18262 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-p0 18374 df-p1 18375 df-lat 18381 df-oposet 38034 df-ol 38036 |
| This theorem is referenced by: olm12 38086 lhpmcvr3 38884 trljat1 39025 trljat2 39026 cdlemc1 39050 cdlemc6 39055 cdleme0cp 39073 cdleme0cq 39074 cdleme1 39086 cdleme4 39097 cdleme5 39099 cdleme8 39109 cdleme9 39112 cdleme10 39113 cdleme20c 39170 cdleme20j 39177 cdleme22e 39203 cdleme22eALTN 39204 cdleme30a 39237 cdleme35b 39309 cdleme35e 39312 cdleme42a 39330 trlcoabs2N 39581 trlcolem 39585 cdlemi1 39677 cdlemk4 39693 dia2dimlem1 39923 cdlemn10 40065 dihglbcpreN 40159 |
| Copyright terms: Public domain | W3C validator |