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Mathbox for Norm Megill |
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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 1198 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ π β π΄ β§ π β π΄) β πΎ β HL) | |
3 | 2 | hllatd 37829 | . . 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 37754 | . . . . 5 β’ (π β π΄ β π β π΅) |
7 | 6 | 3ad2ant2 1135 | . . . 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 37754 | . . . . 5 β’ (π β π΄ β π β π΅) |
9 | 8 | 3ad2ant3 1136 | . . . 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 18329 | . . . 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 1372 | . . 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 1199 | . . . 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 38464 | . . . 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 18330 | . . 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 1372 | . 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 2842 | 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βcfv 6497 (class class class)co 7358 Basecbs 17084 lecple 17141 joincjn 18201 meetcmee 18202 Latclat 18321 Atomscatm 37728 HLchlt 37815 LHypclh 38450 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-lub 18236 df-glb 18237 df-join 18238 df-meet 18239 df-lat 18322 df-ats 37732 df-atl 37763 df-cvlat 37787 df-hlat 37816 df-lhyp 38454 |
This theorem is referenced by: cdleme1b 38692 cdleme5 38706 cdleme9 38719 cdleme11g 38731 cdleme11 38736 cdleme35fnpq 38915 cdleme42e 38945 cdlemeg46frv 38991 cdlemg2fv2 39066 cdlemg2m 39070 |
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