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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleml6 | Structured version Visualization version GIF version |
Description: Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.) |
Ref | Expression |
---|---|
cdleml6.b | β’ π΅ = (BaseβπΎ) |
cdleml6.j | β’ β¨ = (joinβπΎ) |
cdleml6.m | β’ β§ = (meetβπΎ) |
cdleml6.h | β’ π» = (LHypβπΎ) |
cdleml6.t | β’ π = ((LTrnβπΎ)βπ) |
cdleml6.r | β’ π = ((trLβπΎ)βπ) |
cdleml6.p | β’ π = ((ocβπΎ)βπ) |
cdleml6.z | β’ π = ((π β¨ (π βπ)) β§ ((ββπ) β¨ (π β(π β β‘(π ββ))))) |
cdleml6.y | β’ π = ((π β¨ (π βπ)) β§ (π β¨ (π β(π β β‘π)))) |
cdleml6.x | β’ π = (β©π§ β π βπ β π ((π β ( I βΎ π΅) β§ (π βπ) β (π β(π ββ)) β§ (π βπ) β (π βπ)) β (π§βπ) = π)) |
cdleml6.u | β’ π = (π β π β¦ if((π ββ) = β, π, π)) |
cdleml6.e | β’ πΈ = ((TEndoβπΎ)βπ) |
cdleml6.o | β’ 0 = (π β π β¦ ( I βΎ π΅)) |
Ref | Expression |
---|---|
cdleml6 | β’ (((πΎ β HL β§ π β π») β§ β β π β§ (π β πΈ β§ π β 0 )) β (π β πΈ β§ (πβ(π ββ)) = β)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1137 | . 2 β’ (((πΎ β HL β§ π β π») β§ β β π β§ (π β πΈ β§ π β 0 )) β (πΎ β HL β§ π β π»)) | |
2 | simp3l 1202 | . . 3 β’ (((πΎ β HL β§ π β π») β§ β β π β§ (π β πΈ β§ π β 0 )) β π β πΈ) | |
3 | simp2 1138 | . . 3 β’ (((πΎ β HL β§ π β π») β§ β β π β§ (π β πΈ β§ π β 0 )) β β β π) | |
4 | cdleml6.h | . . . 4 β’ π» = (LHypβπΎ) | |
5 | cdleml6.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
6 | cdleml6.e | . . . 4 β’ πΈ = ((TEndoβπΎ)βπ) | |
7 | 4, 5, 6 | tendocl 39280 | . . 3 β’ (((πΎ β HL β§ π β π») β§ π β πΈ β§ β β π) β (π ββ) β π) |
8 | 1, 2, 3, 7 | syl3anc 1372 | . 2 β’ (((πΎ β HL β§ π β π») β§ β β π β§ (π β πΈ β§ π β 0 )) β (π ββ) β π) |
9 | cdleml6.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
10 | cdleml6.r | . . . 4 β’ π = ((trLβπΎ)βπ) | |
11 | cdleml6.o | . . . 4 β’ 0 = (π β π β¦ ( I βΎ π΅)) | |
12 | 9, 4, 5, 10, 6, 11 | tendotr 39343 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β πΈ β§ π β 0 ) β§ β β π) β (π β(π ββ)) = (π ββ)) |
13 | 12 | 3com23 1127 | . 2 β’ (((πΎ β HL β§ π β π») β§ β β π β§ (π β πΈ β§ π β 0 )) β (π β(π ββ)) = (π ββ)) |
14 | cdleml6.j | . . 3 β’ β¨ = (joinβπΎ) | |
15 | cdleml6.m | . . 3 β’ β§ = (meetβπΎ) | |
16 | eqid 2733 | . . 3 β’ (ocβπΎ) = (ocβπΎ) | |
17 | eqid 2733 | . . 3 β’ (AtomsβπΎ) = (AtomsβπΎ) | |
18 | cdleml6.p | . . 3 β’ π = ((ocβπΎ)βπ) | |
19 | cdleml6.z | . . 3 β’ π = ((π β¨ (π βπ)) β§ ((ββπ) β¨ (π β(π β β‘(π ββ))))) | |
20 | cdleml6.y | . . 3 β’ π = ((π β¨ (π βπ)) β§ (π β¨ (π β(π β β‘π)))) | |
21 | cdleml6.x | . . 3 β’ π = (β©π§ β π βπ β π ((π β ( I βΎ π΅) β§ (π βπ) β (π β(π ββ)) β§ (π βπ) β (π βπ)) β (π§βπ) = π)) | |
22 | cdleml6.u | . . 3 β’ π = (π β π β¦ if((π ββ) = β, π, π)) | |
23 | 9, 14, 15, 16, 17, 4, 5, 10, 18, 19, 20, 21, 22, 6 | cdlemk56w 39486 | . 2 β’ (((πΎ β HL β§ π β π») β§ ((π ββ) β π β§ β β π) β§ (π β(π ββ)) = (π ββ)) β (π β πΈ β§ (πβ(π ββ)) = β)) |
24 | 1, 8, 3, 13, 23 | syl121anc 1376 | 1 β’ (((πΎ β HL β§ π β π») β§ β β π β§ (π β πΈ β§ π β 0 )) β (π β πΈ β§ (πβ(π ββ)) = β)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2940 βwral 3061 ifcif 4490 β¦ cmpt 5192 I cid 5534 β‘ccnv 5636 βΎ cres 5639 β ccom 5641 βcfv 6500 β©crio 7316 (class class class)co 7361 Basecbs 17091 occoc 17149 joincjn 18208 meetcmee 18209 Atomscatm 37775 HLchlt 37862 LHypclh 38497 LTrncltrn 38614 trLctrl 38671 TEndoctendo 39265 |
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 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-riotaBAD 37465 |
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 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7925 df-2nd 7926 df-undef 8208 df-map 8773 df-proset 18192 df-poset 18210 df-plt 18227 df-lub 18243 df-glb 18244 df-join 18245 df-meet 18246 df-p0 18322 df-p1 18323 df-lat 18329 df-clat 18396 df-oposet 37688 df-ol 37690 df-oml 37691 df-covers 37778 df-ats 37779 df-atl 37810 df-cvlat 37834 df-hlat 37863 df-llines 38011 df-lplanes 38012 df-lvols 38013 df-lines 38014 df-psubsp 38016 df-pmap 38017 df-padd 38309 df-lhyp 38501 df-laut 38502 df-ldil 38617 df-ltrn 38618 df-trl 38672 df-tendo 39268 |
This theorem is referenced by: cdleml7 39495 cdleml8 39496 erngdvlem4 39504 erngdvlem4-rN 39512 |
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