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| 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 1197 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ π β π΄ β§ π β π΄) β πΎ β HL) | |
| 3 | 2 | hllatd 38229 | . . 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 38154 | . . . . 5 β’ (π β π΄ β π β π΅) |
| 7 | 6 | 3ad2ant2 1134 | . . . 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 38154 | . . . . 5 β’ (π β π΄ β π β π΅) |
| 9 | 8 | 3ad2ant3 1135 | . . . 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 18391 | . . . 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 1371 | . . 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 1198 | . . . 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 38864 | . . . 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 18392 | . . 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 1371 | . 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 2837 | 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 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βcfv 6543 (class class class)co 7408 Basecbs 17143 lecple 17203 joincjn 18263 meetcmee 18264 Latclat 18383 Atomscatm 38128 HLchlt 38215 LHypclh 38850 |
| 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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7364 df-ov 7411 df-oprab 7412 df-lub 18298 df-glb 18299 df-join 18300 df-meet 18301 df-lat 18384 df-ats 38132 df-atl 38163 df-cvlat 38187 df-hlat 38216 df-lhyp 38854 |
| This theorem is referenced by: cdleme1b 39092 cdleme5 39106 cdleme9 39119 cdleme11g 39131 cdleme11 39136 cdleme35fnpq 39315 cdleme42e 39345 cdlemeg46frv 39391 cdlemg2fv2 39466 cdlemg2m 39470 |
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