Nearby theorems
|
|
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme0aa | Structured version Visualization version GIF version | ||
| Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.) |
| Ref | Expression |
|---|---|
| cdleme0.l | β’ β€ = (leβπΎ) |
| cdleme0.j | β’ β¨ = (joinβπΎ) |
| cdleme0.m | β’ β§ = (meetβπΎ) |
| cdleme0.a | β’ π΄ = (AtomsβπΎ) |
| cdleme0.h | β’ π» = (LHypβπΎ) |
| cdleme0.u | β’ π = ((π β¨ π) β§ π) |
| cdleme0.b | β’ π΅ = (BaseβπΎ) |
| Ref | Expression |
|---|---|
| cdleme0aa | 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 | cdleme0.u | . 2 β’ π = ((π β¨ π) β§ π) | |
| 2 | simp1l 1194 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ π β π΄ β§ π β π΄) β πΎ β HL) | |
| 3 | 2 | hllatd 38892 | . . 3 β’ (((πΎ β HL β§ π β π») β§ π β π΄ β§ π β π΄) β πΎ β Lat) |
| 4 | cdleme0.b | . . . . . 6 β’ π΅ = (BaseβπΎ) | |
| 5 | cdleme0.a | . . . . . 6 β’ π΄ = (AtomsβπΎ) | |
| 6 | 4, 5 | atbase 38817 | . . . . 5 β’ (π β π΄ β π β π΅) |
| 7 | 6 | 3ad2ant2 1131 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ π β π΄ β§ π β π΄) β π β π΅) |
| 8 | 4, 5 | atbase 38817 | . . . . 5 β’ (π β π΄ β π β π΅) |
| 9 | 8 | 3ad2ant3 1132 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ π β π΄ β§ π β π΄) β π β π΅) |
| 10 | cdleme0.j | . . . . 5 β’ β¨ = (joinβπΎ) | |
| 11 | 4, 10 | latjcl 18430 | . . . 4 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β¨ π) β π΅) |
| 12 | 3, 7, 9, 11 | 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΅) |
| 13 | simp1r 1195 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ π β π΄ β§ π β π΄) β π β π») | |
| 14 | cdleme0.h | . . . . 5 β’ π» = (LHypβπΎ) | |
| 15 | 4, 14 | lhpbase 39527 | . . . 4 β’ (π β π» β π β π΅) |
| 16 | 13, 15 | syl 17 | . . 3 β’ (((πΎ β HL β§ π β π») β§ π β π΄ β§ π β π΄) β π β π΅) |
| 17 | cdleme0.m | . . . 4 β’ β§ = (meetβπΎ) | |
| 18 | 4, 17 | latmcl 18431 | . . 3 β’ ((πΎ β Lat β§ (π β¨ π) β π΅ β§ π β π΅) β ((π β¨ π) β§ π) β π΅) |
| 19 | 3, 12, 16, 18 | 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΅) |
| 20 | 1, 19 | eqeltrid 2829 | 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 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6543 (class class class)co 7416 Basecbs 17179 lecple 17239 joincjn 18302 meetcmee 18303 Latclat 18422 Atomscatm 38791 HLchlt 38878 LHypclh 39513 |
| 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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
| 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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-lub 18337 df-glb 18338 df-join 18339 df-meet 18340 df-lat 18423 df-ats 38795 df-atl 38826 df-cvlat 38850 df-hlat 38879 df-lhyp 39517 |
| This theorem is referenced by: cdleme1b 39755 cdleme5 39769 cdleme9 39782 cdleme11g 39794 cdleme11 39799 cdleme35fnpq 39978 cdleme42e 40008 cdlemeg46frv 40054 cdlemg2fv2 40129 cdlemg2m 40133 |
| Copyright terms: Public domain | W3C validator |