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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemedb | Structured version Visualization version GIF version | ||
| Description: Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. π· represents s2. (Contributed by NM, 20-Nov-2012.) |
| Ref | Expression |
|---|---|
| cdlemeda.l | β’ β€ = (leβπΎ) |
| cdlemeda.j | β’ β¨ = (joinβπΎ) |
| cdlemeda.m | β’ β§ = (meetβπΎ) |
| cdlemeda.a | β’ π΄ = (AtomsβπΎ) |
| cdlemeda.h | β’ π» = (LHypβπΎ) |
| cdlemeda.d | β’ π· = ((π β¨ π) β§ π) |
| cdlemedb.b | β’ π΅ = (BaseβπΎ) |
| Ref | Expression |
|---|---|
| cdlemedb | β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β π΄)) β π· β π΅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cdlemeda.d | . 2 β’ π· = ((π β¨ π) β§ π) | |
| 2 | hllat 38875 | . . . 4 β’ (πΎ β HL β πΎ β Lat) | |
| 3 | 2 | ad2antrr 724 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β π΄)) β πΎ β Lat) |
| 4 | simpll 765 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β π΄)) β πΎ β HL) | |
| 5 | simprl 769 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β π΄)) β π β π΄) | |
| 6 | simprr 771 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β π΄)) β π β π΄) | |
| 7 | cdlemedb.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
| 8 | cdlemeda.j | . . . . 5 β’ β¨ = (joinβπΎ) | |
| 9 | cdlemeda.a | . . . . 5 β’ π΄ = (AtomsβπΎ) | |
| 10 | 7, 8, 9 | hlatjcl 38879 | . . . 4 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β (π β¨ π) β π΅) |
| 11 | 4, 5, 6, 10 | syl3anc 1368 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β π΄)) β (π β¨ π) β π΅) |
| 12 | cdlemeda.h | . . . . 5 β’ π» = (LHypβπΎ) | |
| 13 | 7, 12 | lhpbase 39511 | . . . 4 β’ (π β π» β π β π΅) |
| 14 | 13 | ad2antlr 725 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β π΄)) β π β π΅) |
| 15 | cdlemeda.m | . . . 4 β’ β§ = (meetβπΎ) | |
| 16 | 7, 15 | latmcl 18441 | . . 3 β’ ((πΎ β Lat β§ (π β¨ π) β π΅ β§ π β π΅) β ((π β¨ π) β§ π) β π΅) |
| 17 | 3, 11, 14, 16 | syl3anc 1368 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β π΄)) β ((π β¨ π) β§ π) β π΅) |
| 18 | 1, 17 | eqeltrid 2833 | 1 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β π΄)) β π· β π΅) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 βcfv 6553 (class class class)co 7426 Basecbs 17189 lecple 17249 joincjn 18312 meetcmee 18313 Latclat 18432 Atomscatm 38775 HLchlt 38862 LHypclh 39497 |
| 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 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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-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-lub 18347 df-glb 18348 df-join 18349 df-meet 18350 df-lat 18433 df-ats 38779 df-atl 38810 df-cvlat 38834 df-hlat 38863 df-lhyp 39501 |
| This theorem is referenced by: cdleme20k 39832 cdleme20l2 39834 cdleme20l 39835 cdleme20m 39836 |
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