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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemkoatnle | Structured version Visualization version GIF version |
Description: Utility lemma. (Contributed by NM, 2-Jul-2013.) |
Ref | Expression |
---|---|
cdlemk1.b | β’ π΅ = (BaseβπΎ) |
cdlemk1.l | β’ β€ = (leβπΎ) |
cdlemk1.j | β’ β¨ = (joinβπΎ) |
cdlemk1.m | β’ β§ = (meetβπΎ) |
cdlemk1.a | β’ π΄ = (AtomsβπΎ) |
cdlemk1.h | β’ π» = (LHypβπΎ) |
cdlemk1.t | β’ π = ((LTrnβπΎ)βπ) |
cdlemk1.r | β’ π = ((trLβπΎ)βπ) |
cdlemk1.s | β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) |
cdlemk1.o | β’ π = (πβπ·) |
Ref | Expression |
---|---|
cdlemkoatnle | β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ (π βπ·) β (π βπΉ))) β ((πβπ) β π΄ β§ Β¬ (πβπ) β€ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp11 1204 | . 2 β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ (π βπ·) β (π βπΉ))) β (πΎ β HL β§ π β π»)) | |
2 | cdlemk1.o | . . 3 β’ π = (πβπ·) | |
3 | cdlemk1.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
4 | cdlemk1.l | . . . 4 β’ β€ = (leβπΎ) | |
5 | cdlemk1.j | . . . 4 β’ β¨ = (joinβπΎ) | |
6 | cdlemk1.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
7 | cdlemk1.h | . . . 4 β’ π» = (LHypβπΎ) | |
8 | cdlemk1.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
9 | cdlemk1.r | . . . 4 β’ π = ((trLβπΎ)βπ) | |
10 | cdlemk1.m | . . . 4 β’ β§ = (meetβπΎ) | |
11 | cdlemk1.s | . . . 4 β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) | |
12 | 3, 4, 5, 6, 7, 8, 9, 10, 11 | cdlemksel 39311 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ (π βπ·) β (π βπΉ))) β (πβπ·) β π) |
13 | 2, 12 | eqeltrid 2842 | . 2 β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ (π βπ·) β (π βπΉ))) β π β π) |
14 | simp22 1208 | . 2 β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ (π βπ·) β (π βπΉ))) β (π β π΄ β§ Β¬ π β€ π)) | |
15 | 4, 6, 7, 8 | ltrnel 38605 | . 2 β’ (((πΎ β HL β§ π β π») β§ π β π β§ (π β π΄ β§ Β¬ π β€ π)) β ((πβπ) β π΄ β§ Β¬ (πβπ) β€ π)) |
16 | 1, 13, 14, 15 | syl3anc 1372 | 1 β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ (π βπ·) β (π βπΉ))) β ((πβπ) β π΄ β§ Β¬ (πβπ) β€ π)) |
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 class class class wbr 5106 β¦ cmpt 5189 I cid 5531 β‘ccnv 5633 βΎ cres 5636 β ccom 5638 βcfv 6497 β©crio 7313 (class class class)co 7358 Basecbs 17084 lecple 17141 joincjn 18201 meetcmee 18202 Atomscatm 37728 HLchlt 37815 LHypclh 38450 LTrncltrn 38567 trLctrl 38624 |
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 ax-riotaBAD 37418 |
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 3354 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-iin 4958 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-mpo 7363 df-1st 7922 df-2nd 7923 df-undef 8205 df-map 8768 df-proset 18185 df-poset 18203 df-plt 18220 df-lub 18236 df-glb 18237 df-join 18238 df-meet 18239 df-p0 18315 df-p1 18316 df-lat 18322 df-clat 18389 df-oposet 37641 df-ol 37643 df-oml 37644 df-covers 37731 df-ats 37732 df-atl 37763 df-cvlat 37787 df-hlat 37816 df-llines 37964 df-lplanes 37965 df-lvols 37966 df-lines 37967 df-psubsp 37969 df-pmap 37970 df-padd 38262 df-lhyp 38454 df-laut 38455 df-ldil 38570 df-ltrn 38571 df-trl 38625 |
This theorem is referenced by: cdlemk14 39320 cdlemk16a 39322 cdlemk1u 39325 cdlemk5u 39327 cdlemk6u 39328 cdlemk7u 39336 cdlemk12u 39338 cdlemkoatnle-2N 39341 |
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