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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 30401 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 37679 | . . . . . . 7 β’ (πΎ β OL β πΎ β OP) | |
2 | 1 | adantr 482 | . . . . . 6 β’ ((πΎ β OL β§ π β π΅) β πΎ β OP) |
3 | eqid 2737 | . . . . . . 7 β’ (0.βπΎ) = (0.βπΎ) | |
4 | olm1.u | . . . . . . 7 β’ 1 = (1.βπΎ) | |
5 | eqid 2737 | . . . . . . 7 β’ (ocβπΎ) = (ocβπΎ) | |
6 | 3, 4, 5 | opoc1 37667 | . . . . . 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 7374 | . . . 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 37659 | . . . . . 6 β’ ((πΎ β OP β§ π β π΅) β ((ocβπΎ)βπ) β π΅) |
11 | 1, 10 | sylan 581 | . . . . 5 β’ ((πΎ β OL β§ π β π΅) β ((ocβπΎ)βπ) β π΅) |
12 | eqid 2737 | . . . . . 6 β’ (joinβπΎ) = (joinβπΎ) | |
13 | 9, 12, 3 | olj01 37690 | . . . . 5 β’ ((πΎ β OL β§ ((ocβπΎ)βπ) β π΅) β (((ocβπΎ)βπ)(joinβπΎ)(0.βπΎ)) = ((ocβπΎ)βπ)) |
14 | 11, 13 | syldan 592 | . . . 4 β’ ((πΎ β OL β§ π β π΅) β (((ocβπΎ)βπ)(joinβπΎ)(0.βπΎ)) = ((ocβπΎ)βπ)) |
15 | 8, 14 | eqtrd 2777 | . . 3 β’ ((πΎ β OL β§ π β π΅) β (((ocβπΎ)βπ)(joinβπΎ)((ocβπΎ)β 1 )) = ((ocβπΎ)βπ)) |
16 | 15 | fveq2d 6847 | . 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 37650 | . . . 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 37689 | . . 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 1463 | . 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 37660 | . . 3 β’ ((πΎ β OP β§ π β π΅) β ((ocβπΎ)β((ocβπΎ)βπ)) = π) |
23 | 1, 22 | sylan 581 | . 2 β’ ((πΎ β OL β§ π β π΅) β ((ocβπΎ)β((ocβπΎ)βπ)) = π) |
24 | 16, 21, 23 | 3eqtr3d 2785 | 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 397 = wceq 1542 β wcel 2107 βcfv 6497 (class class class)co 7358 Basecbs 17084 occoc 17142 joincjn 18201 meetcmee 18202 0.cp0 18313 1.cp1 18314 OPcops 37637 OLcol 37639 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-proset 18185 df-poset 18203 df-lub 18236 df-glb 18237 df-join 18238 df-meet 18239 df-p0 18315 df-p1 18316 df-lat 18322 df-oposet 37641 df-ol 37643 |
This theorem is referenced by: olm12 37693 lhpmcvr3 38491 trljat1 38632 trljat2 38633 cdlemc1 38657 cdlemc6 38662 cdleme0cp 38680 cdleme0cq 38681 cdleme1 38693 cdleme4 38704 cdleme5 38706 cdleme8 38716 cdleme9 38719 cdleme10 38720 cdleme20c 38777 cdleme20j 38784 cdleme22e 38810 cdleme22eALTN 38811 cdleme30a 38844 cdleme35b 38916 cdleme35e 38919 cdleme42a 38937 trlcoabs2N 39188 trlcolem 39192 cdlemi1 39284 cdlemk4 39300 dia2dimlem1 39530 cdlemn10 39672 dihglbcpreN 39766 |
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