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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemkyu | Structured version Visualization version GIF version |
Description: Convert between function and explicit forms. πΆ represents π in cdlemkuu 39404. TODO: Clean all this up. (Contributed by NM, 21-Jul-2013.) |
Ref | Expression |
---|---|
cdlemk5.b | β’ π΅ = (BaseβπΎ) |
cdlemk5.l | β’ β€ = (leβπΎ) |
cdlemk5.j | β’ β¨ = (joinβπΎ) |
cdlemk5.m | β’ β§ = (meetβπΎ) |
cdlemk5.a | β’ π΄ = (AtomsβπΎ) |
cdlemk5.h | β’ π» = (LHypβπΎ) |
cdlemk5.t | β’ π = ((LTrnβπΎ)βπ) |
cdlemk5.r | β’ π = ((trLβπΎ)βπ) |
cdlemk5.z | β’ π = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))) |
cdlemk5.y | β’ π = ((π β¨ (π βπ)) β§ (π β¨ (π β(π β β‘π)))) |
cdlemk5b.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)))))) |
cdlemk5b.u1 | β’ π = (π β π, π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ (((πβπ)βπ) β¨ (π β(π β β‘π)))))) |
cdlemk5.o2 | β’ π = (πβπ) |
cdlemk5.u2 | β’ πΆ = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘π)))))) |
Ref | Expression |
---|---|
cdlemkyu | β’ ((((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΉ β ( I βΎ π΅)) β§ (πΊ β π β§ πΊ β ( I βΎ π΅))) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (π β π β§ (π β ( I βΎ π΅) β§ (π βπ) β (π βπΉ) β§ (π βπ) β (π βπΊ)))) β β¦πΊ / πβ¦π = ((πΆβπΊ)βπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemk5.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | cdlemk5.l | . . 3 β’ β€ = (leβπΎ) | |
3 | cdlemk5.j | . . 3 β’ β¨ = (joinβπΎ) | |
4 | cdlemk5.m | . . 3 β’ β§ = (meetβπΎ) | |
5 | cdlemk5.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
6 | cdlemk5.h | . . 3 β’ π» = (LHypβπΎ) | |
7 | cdlemk5.t | . . 3 β’ π = ((LTrnβπΎ)βπ) | |
8 | cdlemk5.r | . . 3 β’ π = ((trLβπΎ)βπ) | |
9 | cdlemk5.z | . . 3 β’ π = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))) | |
10 | cdlemk5.y | . . 3 β’ π = ((π β¨ (π βπ)) β§ (π β¨ (π β(π β β‘π)))) | |
11 | cdlemk5b.s | . . 3 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 | cdlemk5b.u1 | . . 3 β’ π = (π β π, π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ (((πβπ)βπ) β¨ (π β(π β β‘π)))))) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | cdlemky 39435 | . 2 β’ ((((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΉ β ( I βΎ π΅)) β§ (πΊ β π β§ πΊ β ( I βΎ π΅))) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (π β π β§ (π β ( I βΎ π΅) β§ (π βπ) β (π βπΉ) β§ (π βπ) β (π βπΊ)))) β β¦πΊ / πβ¦π = ((πππΊ)βπ)) |
14 | simp3l 1202 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΉ β ( I βΎ π΅)) β§ (πΊ β π β§ πΊ β ( I βΎ π΅))) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (π β π β§ (π β ( I βΎ π΅) β§ (π βπ) β (π βπΉ) β§ (π βπ) β (π βπΊ)))) β π β π) | |
15 | simp13l 1289 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΉ β ( I βΎ π΅)) β§ (πΊ β π β§ πΊ β ( I βΎ π΅))) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (π β π β§ (π β ( I βΎ π΅) β§ (π βπ) β (π βπΉ) β§ (π βπ) β (π βπΊ)))) β πΊ β π) | |
16 | cdlemk5.o2 | . . . . 5 β’ π = (πβπ) | |
17 | cdlemk5.u2 | . . . . 5 β’ πΆ = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘π)))))) | |
18 | 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 16, 17 | cdlemkuu 39404 | . . . 4 β’ ((π β π β§ πΊ β π) β (πππΊ) = (πΆβπΊ)) |
19 | 14, 15, 18 | syl2anc 585 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΉ β ( I βΎ π΅)) β§ (πΊ β π β§ πΊ β ( I βΎ π΅))) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (π β π β§ (π β ( I βΎ π΅) β§ (π βπ) β (π βπΉ) β§ (π βπ) β (π βπΊ)))) β (πππΊ) = (πΆβπΊ)) |
20 | 19 | fveq1d 6845 | . 2 β’ ((((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΉ β ( I βΎ π΅)) β§ (πΊ β π β§ πΊ β ( I βΎ π΅))) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (π β π β§ (π β ( I βΎ π΅) β§ (π βπ) β (π βπΉ) β§ (π βπ) β (π βπΊ)))) β ((πππΊ)βπ) = ((πΆβπΊ)βπ)) |
21 | 13, 20 | eqtrd 2773 | 1 β’ ((((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΉ β ( I βΎ π΅)) β§ (πΊ β π β§ πΊ β ( I βΎ π΅))) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (π β π β§ (π β ( I βΎ π΅) β§ (π βπ) β (π βπΉ) β§ (π βπ) β (π βπΊ)))) β β¦πΊ / πβ¦π = ((πΆβπΊ)βπ)) |
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 2940 β¦csb 3856 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 β cmpo 7360 Basecbs 17088 lecple 17145 joincjn 18205 meetcmee 18206 Atomscatm 37771 HLchlt 37858 LHypclh 38493 LTrncltrn 38610 trLctrl 38667 |
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 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-riotaBAD 37461 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 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 8770 df-proset 18189 df-poset 18207 df-plt 18224 df-lub 18240 df-glb 18241 df-join 18242 df-meet 18243 df-p0 18319 df-p1 18320 df-lat 18326 df-clat 18393 df-oposet 37684 df-ol 37686 df-oml 37687 df-covers 37774 df-ats 37775 df-atl 37806 df-cvlat 37830 df-hlat 37859 df-llines 38007 df-lplanes 38008 df-lvols 38009 df-lines 38010 df-psubsp 38012 df-pmap 38013 df-padd 38305 df-lhyp 38497 df-laut 38498 df-ldil 38613 df-ltrn 38614 df-trl 38668 |
This theorem is referenced by: cdlemkyuu 39437 |
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