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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemk42yN | Structured version Visualization version GIF version | ||
| Description: Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 20-Jul-2013.) (New usage is discouraged.) |
| 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)))) |
| cdlemk5.x | β’ π = (β©π§ β π βπ β π ((π β ( I βΎ π΅) β§ (π βπ) β (π βπΉ) β§ (π βπ) β (π βπ)) β (π§βπ) = π)) |
| Ref | Expression |
|---|---|
| cdlemk42yN | 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¦πβπ) = ((π β¨ (π βπΊ)) β§ (π β¨ (π β(πΊ β β‘π))))) |
| 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 | cdlemk5.x | . . 3 β’ π = (β©π§ β π βπ β π ((π β ( I βΎ π΅) β§ (π βπ) β (π βπΉ) β§ (π βπ) β (π βπ)) β (π§βπ) = π)) | |
| 12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | cdlemk42 39800 | . 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¦πβπ) = β¦πΊ / πβ¦π) |
| 13 | simp13l 1288 | . . 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 πΊ β π) | |
| 14 | 10 | cdlemk41 39779 | . . 3 β’ (πΊ β π β β¦πΊ / πβ¦π = ((π β¨ (π βπΊ)) β§ (π β¨ (π β(πΊ β β‘π))))) |
| 15 | 13, 14 | syl 17 | . 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 = ((π β¨ (π βπΊ)) β§ (π β¨ (π β(πΊ β β‘π))))) |
| 16 | 12, 15 | eqtrd 2772 | 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¦πβπ) = ((π β¨ (π βπΊ)) β§ (π β¨ (π β(πΊ β 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 3892 class class class wbr 5147 I cid 5572 β‘ccnv 5674 βΎ cres 5677 β ccom 5679 βcfv 6540 β©crio 7360 (class class class)co 7405 Basecbs 17140 lecple 17200 joincjn 18260 meetcmee 18261 Atomscatm 38121 HLchlt 38208 LHypclh 38843 LTrncltrn 38960 trLctrl 39017 |
| 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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-riotaBAD 37811 |
| 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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7971 df-2nd 7972 df-undef 8254 df-map 8818 df-proset 18244 df-poset 18262 df-plt 18279 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-p0 18374 df-p1 18375 df-lat 18381 df-clat 18448 df-oposet 38034 df-ol 38036 df-oml 38037 df-covers 38124 df-ats 38125 df-atl 38156 df-cvlat 38180 df-hlat 38209 df-llines 38357 df-lplanes 38358 df-lvols 38359 df-lines 38360 df-psubsp 38362 df-pmap 38363 df-padd 38655 df-lhyp 38847 df-laut 38848 df-ldil 38963 df-ltrn 38964 df-trl 39018 |
| This theorem is referenced by: cdlemkyyN 39821 |
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