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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemg17jq | Structured version Visualization version GIF version |
Description: cdlemg17j 39163 with π and π swapped. (Contributed by NM, 13-May-2013.) |
Ref | Expression |
---|---|
cdlemg12.l | β’ β€ = (leβπΎ) |
cdlemg12.j | β’ β¨ = (joinβπΎ) |
cdlemg12.m | β’ β§ = (meetβπΎ) |
cdlemg12.a | β’ π΄ = (AtomsβπΎ) |
cdlemg12.h | β’ π» = (LHypβπΎ) |
cdlemg12.t | β’ π = ((LTrnβπΎ)βπ) |
cdlemg12b.r | β’ π = ((trLβπΎ)βπ) |
Ref | Expression |
---|---|
cdlemg17jq | β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (πΉ β π β§ πΊ β π β§ π β π) β§ ((πΊβπ) β π β§ (π βπΊ) β€ (π β¨ π) β§ Β¬ βπ β π΄ (Β¬ π β€ π β§ (π β¨ π) = (π β¨ π)))) β (πΊβ(πΉβπ)) = (πΉβ(πΊβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemg12.l | . . 3 β’ β€ = (leβπΎ) | |
2 | cdlemg12.j | . . 3 β’ β¨ = (joinβπΎ) | |
3 | cdlemg12.m | . . 3 β’ β§ = (meetβπΎ) | |
4 | cdlemg12.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
5 | cdlemg12.h | . . 3 β’ π» = (LHypβπΎ) | |
6 | cdlemg12.t | . . 3 β’ π = ((LTrnβπΎ)βπ) | |
7 | cdlemg12b.r | . . 3 β’ π = ((trLβπΎ)βπ) | |
8 | 1, 2, 3, 4, 5, 6, 7 | cdlemg17pq 39164 | . 2 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (πΉ β π β§ πΊ β π β§ π β π) β§ ((πΊβπ) β π β§ (π βπΊ) β€ (π β¨ π) β§ Β¬ βπ β π΄ (Β¬ π β€ π β§ (π β¨ π) = (π β¨ π)))) β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (πΉ β π β§ πΊ β π β§ π β π) β§ ((πΊβπ) β π β§ (π βπΊ) β€ (π β¨ π) β§ Β¬ βπ β π΄ (Β¬ π β€ π β§ (π β¨ π) = (π β¨ π))))) |
9 | 1, 2, 3, 4, 5, 6, 7 | cdlemg17j 39163 | . 2 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (πΉ β π β§ πΊ β π β§ π β π) β§ ((πΊβπ) β π β§ (π βπΊ) β€ (π β¨ π) β§ Β¬ βπ β π΄ (Β¬ π β€ π β§ (π β¨ π) = (π β¨ π)))) β (πΊβ(πΉβπ)) = (πΉβ(πΊβπ))) |
10 | 8, 9 | syl 17 | 1 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (πΉ β π β§ πΊ β π β§ π β π) β§ ((πΊβπ) β π β§ (π βπΊ) β€ (π β¨ π) β§ Β¬ βπ β π΄ (Β¬ π β€ π β§ (π β¨ π) = (π β¨ π)))) β (πΊβ(πΉβπ)) = (πΉβ(πΊβπ))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2944 βwrex 3074 class class class wbr 5110 βcfv 6501 (class class class)co 7362 lecple 17147 joincjn 18207 meetcmee 18208 Atomscatm 37754 HLchlt 37841 LHypclh 38476 LTrncltrn 38593 trLctrl 38650 |
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 ax-riotaBAD 37444 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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-rmo 3356 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-iin 4962 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-oprab 7366 df-mpo 7367 df-1st 7926 df-2nd 7927 df-undef 8209 df-map 8774 df-proset 18191 df-poset 18209 df-plt 18226 df-lub 18242 df-glb 18243 df-join 18244 df-meet 18245 df-p0 18321 df-p1 18322 df-lat 18328 df-clat 18395 df-oposet 37667 df-ol 37669 df-oml 37670 df-covers 37757 df-ats 37758 df-atl 37789 df-cvlat 37813 df-hlat 37842 df-llines 37990 df-lplanes 37991 df-lvols 37992 df-lines 37993 df-psubsp 37995 df-pmap 37996 df-padd 38288 df-lhyp 38480 df-laut 38481 df-ldil 38596 df-ltrn 38597 df-trl 38651 |
This theorem is referenced by: cdlemg17 39169 |
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