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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemkuel | Structured version Visualization version GIF version | ||
| Description: Part of proof of Lemma K of [Crawley] p. 118. Conditions for the sigma1 (p) function to be a translation. TODO: combine cdlemkj 40376? (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·) |
| cdlemk1.u | β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘π·)))))) |
| Ref | Expression |
|---|---|
| cdlemkuel | β’ ((((πΎ β HL β§ π β π») β§ (π βπΉ) = (π βπ) β§ πΊ β π) β§ (πΉ β π β§ π· β π β§ π β π) β§ (((π βπ·) β (π βπΉ) β§ (π βπ·) β (π βπΊ)) β§ (πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅) β§ π· β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β (πβπΊ) β π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp13 1202 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ (π βπΉ) = (π βπ) β§ πΊ β π) β§ (πΉ β π β§ π· β π β§ π β π) β§ (((π βπ·) β (π βπΉ) β§ (π βπ·) β (π βπΊ)) β§ (πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅) β§ π· β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β πΊ β π) | |
| 2 | cdlemk1.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
| 3 | cdlemk1.l | . . . 4 β’ β€ = (leβπΎ) | |
| 4 | cdlemk1.j | . . . 4 β’ β¨ = (joinβπΎ) | |
| 5 | cdlemk1.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
| 6 | cdlemk1.h | . . . 4 β’ π» = (LHypβπΎ) | |
| 7 | cdlemk1.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
| 8 | cdlemk1.r | . . . 4 β’ π = ((trLβπΎ)βπ) | |
| 9 | cdlemk1.m | . . . 4 β’ β§ = (meetβπΎ) | |
| 10 | cdlemk1.u | . . . 4 β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘π·)))))) | |
| 11 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | cdlemksv 40357 | . . 3 β’ (πΊ β π β (πβπΊ) = (β©π β π (πβπ) = ((π β¨ (π βπΊ)) β§ ((πβπ) β¨ (π β(πΊ β β‘π·)))))) |
| 12 | 1, 11 | syl 17 | . 2 β’ ((((πΎ β HL β§ π β π») β§ (π βπΉ) = (π βπ) β§ πΊ β π) β§ (πΉ β π β§ π· β π β§ π β π) β§ (((π βπ·) β (π βπΉ) β§ (π βπ·) β (π βπΊ)) β§ (πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅) β§ π· β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β (πβπΊ) = (β©π β π (πβπ) = ((π β¨ (π βπΊ)) β§ ((πβπ) β¨ (π β(πΊ β β‘π·)))))) |
| 13 | cdlemk1.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)))))) | |
| 14 | cdlemk1.o | . . 3 β’ π = (πβπ·) | |
| 15 | eqid 2728 | . . 3 β’ (β©π β π (πβπ) = ((π β¨ (π βπΊ)) β§ ((πβπ) β¨ (π β(πΊ β β‘π·))))) = (β©π β π (πβπ) = ((π β¨ (π βπΊ)) β§ ((πβπ) β¨ (π β(πΊ β β‘π·))))) | |
| 16 | 2, 3, 4, 9, 5, 6, 7, 8, 13, 14, 15 | cdlemkj 40376 | . 2 β’ ((((πΎ β HL β§ π β π») β§ (π βπΉ) = (π βπ) β§ πΊ β π) β§ (πΉ β π β§ π· β π β§ π β π) β§ (((π βπ·) β (π βπΉ) β§ (π βπ·) β (π βπΊ)) β§ (πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅) β§ π· β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β (β©π β π (πβπ) = ((π β¨ (π βπΊ)) β§ ((πβπ) β¨ (π β(πΊ β β‘π·))))) β π) |
| 17 | 12, 16 | eqeltrd 2829 | 1 β’ ((((πΎ β HL β§ π β π») β§ (π βπΉ) = (π βπ) β§ πΊ β π) β§ (πΉ β π β§ π· β π β§ π β π) β§ (((π βπ·) β (π βπΉ) β§ (π βπ·) β (π βπΊ)) β§ (πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅) β§ π· β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β (πβπΊ) β π) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2937 class class class wbr 5152 β¦ cmpt 5235 I cid 5579 β‘ccnv 5681 βΎ cres 5684 β ccom 5686 βcfv 6553 β©crio 7381 (class class class)co 7426 Basecbs 17189 lecple 17249 joincjn 18312 meetcmee 18313 Atomscatm 38775 HLchlt 38862 LHypclh 39497 LTrncltrn 39614 trLctrl 39671 |
| 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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-riotaBAD 38465 |
| 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 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 8001 df-2nd 8002 df-undef 8287 df-map 8855 df-proset 18296 df-poset 18314 df-plt 18331 df-lub 18347 df-glb 18348 df-join 18349 df-meet 18350 df-p0 18426 df-p1 18427 df-lat 18433 df-clat 18500 df-oposet 38688 df-ol 38690 df-oml 38691 df-covers 38778 df-ats 38779 df-atl 38810 df-cvlat 38834 df-hlat 38863 df-llines 39011 df-lplanes 39012 df-lvols 39013 df-lines 39014 df-psubsp 39016 df-pmap 39017 df-padd 39309 df-lhyp 39501 df-laut 39502 df-ldil 39617 df-ltrn 39618 df-trl 39672 |
| This theorem is referenced by: cdlemkuat 40379 cdlemkuv2 40380 cdlemk19 40382 cdlemk12u 40385 cdlemkuel-2N 40397 cdlemkuel-3 40411 |
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