![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
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 1133 | . 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 1198 | . . 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 1134 | . . 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 40244 | . . 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 1368 | . 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 40307 | . . 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 1123 | . 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 2727 | . . 3 β’ (ocβπΎ) = (ocβπΎ) | |
17 | eqid 2727 | . . 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 40450 | . 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 1372 | 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 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2936 βwral 3057 ifcif 4530 β¦ cmpt 5233 I cid 5577 β‘ccnv 5679 βΎ cres 5682 β ccom 5684 βcfv 6551 β©crio 7379 (class class class)co 7424 Basecbs 17185 occoc 17246 joincjn 18308 meetcmee 18309 Atomscatm 38739 HLchlt 38826 LHypclh 39461 LTrncltrn 39578 trLctrl 39635 TEndoctendo 40229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-riotaBAD 38429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-iin 5001 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 7997 df-2nd 7998 df-undef 8283 df-map 8851 df-proset 18292 df-poset 18310 df-plt 18327 df-lub 18343 df-glb 18344 df-join 18345 df-meet 18346 df-p0 18422 df-p1 18423 df-lat 18429 df-clat 18496 df-oposet 38652 df-ol 38654 df-oml 38655 df-covers 38742 df-ats 38743 df-atl 38774 df-cvlat 38798 df-hlat 38827 df-llines 38975 df-lplanes 38976 df-lvols 38977 df-lines 38978 df-psubsp 38980 df-pmap 38981 df-padd 39273 df-lhyp 39465 df-laut 39466 df-ldil 39581 df-ltrn 39582 df-trl 39636 df-tendo 40232 |
This theorem is referenced by: cdleml7 40459 cdleml8 40460 erngdvlem4 40468 erngdvlem4-rN 40476 |
Copyright terms: Public domain | W3C validator |