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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemefs45ee | Structured version Visualization version GIF version |
Description: Explicit expansion of cdlemefs45 39392. TODO: use to shorten cdlemefs45 39392 uses? Should ((π β¨ π) β§ (π β¨ ((π β¨ π) β§ π))) be assigned to a hypothesis letter? TODO: FIX COMMENT. (Contributed by NM, 10-Apr-2013.) |
Ref | Expression |
---|---|
cdlemef45.b | β’ π΅ = (BaseβπΎ) |
cdlemef45.l | β’ β€ = (leβπΎ) |
cdlemef45.j | β’ β¨ = (joinβπΎ) |
cdlemef45.m | β’ β§ = (meetβπΎ) |
cdlemef45.a | β’ π΄ = (AtomsβπΎ) |
cdlemef45.h | β’ π» = (LHypβπΎ) |
cdlemef45.u | β’ π = ((π β¨ π) β§ π) |
cdlemef45.d | β’ π· = ((π‘ β¨ π) β§ (π β¨ ((π β¨ π‘) β§ π))) |
cdlemef45.f | β’ πΉ = (π₯ β π΅ β¦ if((π β π β§ Β¬ π₯ β€ π), (β©π§ β π΅ βπ β π΄ ((Β¬ π β€ π β§ (π β¨ (π₯ β§ π)) = π₯) β π§ = (if(π β€ (π β¨ π), (β©π¦ β π΅ βπ‘ β π΄ ((Β¬ π‘ β€ π β§ Β¬ π‘ β€ (π β¨ π)) β π¦ = πΈ)), β¦π / π‘β¦π·) β¨ (π₯ β§ π)))), π₯)) |
cdlemefs45.e | β’ πΈ = ((π β¨ π) β§ (π· β¨ ((π β¨ π‘) β§ π))) |
Ref | Expression |
---|---|
cdlemefs45ee | 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§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β€ (π β¨ π) β§ Β¬ π β€ (π β¨ π))) β (πΉβπ ) = ((π β¨ π) β§ (((π β¨ π) β§ (π β¨ ((π β¨ π) β§ π))) β¨ ((π β¨ π) β§ π)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemef45.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | cdlemef45.l | . . 3 β’ β€ = (leβπΎ) | |
3 | cdlemef45.j | . . 3 β’ β¨ = (joinβπΎ) | |
4 | cdlemef45.m | . . 3 β’ β§ = (meetβπΎ) | |
5 | cdlemef45.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
6 | cdlemef45.h | . . 3 β’ π» = (LHypβπΎ) | |
7 | cdlemef45.u | . . 3 β’ π = ((π β¨ π) β§ π) | |
8 | cdlemef45.d | . . 3 β’ π· = ((π‘ β¨ π) β§ (π β¨ ((π β¨ π‘) β§ π))) | |
9 | cdlemef45.f | . . 3 β’ πΉ = (π₯ β π΅ β¦ if((π β π β§ Β¬ π₯ β€ π), (β©π§ β π΅ βπ β π΄ ((Β¬ π β€ π β§ (π β¨ (π₯ β§ π)) = π₯) β π§ = (if(π β€ (π β¨ π), (β©π¦ β π΅ βπ‘ β π΄ ((Β¬ π‘ β€ π β§ Β¬ π‘ β€ (π β¨ π)) β π¦ = πΈ)), β¦π / π‘β¦π·) β¨ (π₯ β§ π)))), π₯)) | |
10 | cdlemefs45.e | . . 3 β’ πΈ = ((π β¨ π) β§ (π· β¨ ((π β¨ π‘) β§ π))) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | cdlemefs45 39392 | . 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§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β€ (π β¨ π) β§ Β¬ π β€ (π β¨ π))) β (πΉβπ ) = β¦π / π β¦β¦π / π‘β¦πΈ) |
12 | simp22l 1292 | . . 3 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§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β€ (π β¨ π) β§ Β¬ π β€ (π β¨ π))) β π β π΄) | |
13 | simp23l 1294 | . . 3 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§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β€ (π β¨ π) β§ Β¬ π β€ (π β¨ π))) β π β π΄) | |
14 | eqid 2732 | . . . 4 β’ ((π β¨ π) β§ (π β¨ ((π β¨ π) β§ π))) = ((π β¨ π) β§ (π β¨ ((π β¨ π) β§ π))) | |
15 | eqid 2732 | . . . 4 β’ ((π β¨ π) β§ (((π β¨ π) β§ (π β¨ ((π β¨ π) β§ π))) β¨ ((π β¨ π) β§ π))) = ((π β¨ π) β§ (((π β¨ π) β§ (π β¨ ((π β¨ π) β§ π))) β¨ ((π β¨ π) β§ π))) | |
16 | 8, 10, 14, 15 | cdleme31sde 39348 | . . 3 β’ ((π β π΄ β§ π β π΄) β β¦π / π β¦β¦π / π‘β¦πΈ = ((π β¨ π) β§ (((π β¨ π) β§ (π β¨ ((π β¨ π) β§ π))) β¨ ((π β¨ π) β§ π)))) |
17 | 12, 13, 16 | syl2anc 584 | . 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§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β€ (π β¨ π) β§ Β¬ π β€ (π β¨ π))) β β¦π / π β¦β¦π / π‘β¦πΈ = ((π β¨ π) β§ (((π β¨ π) β§ (π β¨ ((π β¨ π) β§ π))) β¨ ((π β¨ π) β§ π)))) |
18 | 11, 17 | eqtrd 2772 | 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§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β€ (π β¨ π) β§ Β¬ π β€ (π β¨ π))) β (πΉβπ ) = ((π β¨ π) β§ (((π β¨ π) β§ (π β¨ ((π β¨ π) β§ π))) β¨ ((π β¨ π) β§ π)))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β wne 2940 βwral 3061 β¦csb 3893 ifcif 4528 class class class wbr 5148 β¦ cmpt 5231 βcfv 6543 β©crio 7366 (class class class)co 7411 Basecbs 17146 lecple 17206 joincjn 18266 meetcmee 18267 Atomscatm 38225 HLchlt 38312 LHypclh 38947 |
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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-riotaBAD 37915 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-undef 8260 df-proset 18250 df-poset 18268 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-p1 18381 df-lat 18387 df-clat 18454 df-oposet 38138 df-ol 38140 df-oml 38141 df-covers 38228 df-ats 38229 df-atl 38260 df-cvlat 38284 df-hlat 38313 df-llines 38461 df-lplanes 38462 df-lvols 38463 df-lines 38464 df-psubsp 38466 df-pmap 38467 df-padd 38759 df-lhyp 38951 |
This theorem is referenced by: cdlemefs45eN 39394 cdleme50trn2a 39513 |
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