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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme50laut | Structured version Visualization version GIF version |
Description: Part of proof of Lemma D in [Crawley] p. 113. πΉ is a lattice automorphism. TODO: fix comment. (Contributed by NM, 9-Apr-2013.) |
Ref | Expression |
---|---|
cdlemef50.b | β’ π΅ = (BaseβπΎ) |
cdlemef50.l | β’ β€ = (leβπΎ) |
cdlemef50.j | β’ β¨ = (joinβπΎ) |
cdlemef50.m | β’ β§ = (meetβπΎ) |
cdlemef50.a | β’ π΄ = (AtomsβπΎ) |
cdlemef50.h | β’ π» = (LHypβπΎ) |
cdlemef50.u | β’ π = ((π β¨ π) β§ π) |
cdlemef50.d | β’ π· = ((π‘ β¨ π) β§ (π β¨ ((π β¨ π‘) β§ π))) |
cdlemefs50.e | β’ πΈ = ((π β¨ π) β§ (π· β¨ ((π β¨ π‘) β§ π))) |
cdlemef50.f | β’ πΉ = (π₯ β π΅ β¦ if((π β π β§ Β¬ π₯ β€ π), (β©π§ β π΅ βπ β π΄ ((Β¬ π β€ π β§ (π β¨ (π₯ β§ π)) = π₯) β π§ = (if(π β€ (π β¨ π), (β©π¦ β π΅ βπ‘ β π΄ ((Β¬ π‘ β€ π β§ Β¬ π‘ β€ (π β¨ π)) β π¦ = πΈ)), β¦π / π‘β¦π·) β¨ (π₯ β§ π)))), π₯)) |
cdleme50laut.i | β’ πΌ = (LAutβπΎ) |
Ref | Expression |
---|---|
cdleme50laut | β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β πΉ β πΌ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemef50.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | cdlemef50.l | . . 3 β’ β€ = (leβπΎ) | |
3 | cdlemef50.j | . . 3 β’ β¨ = (joinβπΎ) | |
4 | cdlemef50.m | . . 3 β’ β§ = (meetβπΎ) | |
5 | cdlemef50.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
6 | cdlemef50.h | . . 3 β’ π» = (LHypβπΎ) | |
7 | cdlemef50.u | . . 3 β’ π = ((π β¨ π) β§ π) | |
8 | cdlemef50.d | . . 3 β’ π· = ((π‘ β¨ π) β§ (π β¨ ((π β¨ π‘) β§ π))) | |
9 | cdlemefs50.e | . . 3 β’ πΈ = ((π β¨ π) β§ (π· β¨ ((π β¨ π‘) β§ π))) | |
10 | cdlemef50.f | . . 3 β’ πΉ = (π₯ β π΅ β¦ if((π β π β§ Β¬ π₯ β€ π), (β©π§ β π΅ βπ β π΄ ((Β¬ π β€ π β§ (π β¨ (π₯ β§ π)) = π₯) β π§ = (if(π β€ (π β¨ π), (β©π¦ β π΅ βπ‘ β π΄ ((Β¬ π‘ β€ π β§ Β¬ π‘ β€ (π β¨ π)) β π¦ = πΈ)), β¦π / π‘β¦π·) β¨ (π₯ β§ π)))), π₯)) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | cdleme50f1o 39220 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β πΉ:π΅β1-1-ontoβπ΅) |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | cdleme50lebi 39214 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΅ β§ π β π΅)) β (π β€ π β (πΉβπ) β€ (πΉβπ))) |
13 | 12 | ralrimivva 3199 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β βπ β π΅ βπ β π΅ (π β€ π β (πΉβπ) β€ (πΉβπ))) |
14 | simp1l 1197 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β πΎ β HL) | |
15 | cdleme50laut.i | . . . 4 β’ πΌ = (LAutβπΎ) | |
16 | 1, 2, 15 | islaut 38757 | . . 3 β’ (πΎ β HL β (πΉ β πΌ β (πΉ:π΅β1-1-ontoβπ΅ β§ βπ β π΅ βπ β π΅ (π β€ π β (πΉβπ) β€ (πΉβπ))))) |
17 | 14, 16 | syl 17 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β (πΉ β πΌ β (πΉ:π΅β1-1-ontoβπ΅ β§ βπ β π΅ βπ β π΅ (π β€ π β (πΉβπ) β€ (πΉβπ))))) |
18 | 11, 13, 17 | mpbir2and 711 | 1 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β πΉ β πΌ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β wne 2939 βwral 3060 β¦csb 3889 ifcif 4522 class class class wbr 5141 β¦ cmpt 5224 β1-1-ontoβwf1o 6531 βcfv 6532 β©crio 7348 (class class class)co 7393 Basecbs 17126 lecple 17186 joincjn 18246 meetcmee 18247 Atomscatm 37936 HLchlt 38023 LHypclh 38658 LAutclaut 38659 |
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 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-riotaBAD 37626 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-1st 7957 df-2nd 7958 df-undef 8240 df-map 8805 df-proset 18230 df-poset 18248 df-plt 18265 df-lub 18281 df-glb 18282 df-join 18283 df-meet 18284 df-p0 18360 df-p1 18361 df-lat 18367 df-clat 18434 df-oposet 37849 df-ol 37851 df-oml 37852 df-covers 37939 df-ats 37940 df-atl 37971 df-cvlat 37995 df-hlat 38024 df-llines 38172 df-lplanes 38173 df-lvols 38174 df-lines 38175 df-psubsp 38177 df-pmap 38178 df-padd 38470 df-lhyp 38662 df-laut 38663 |
This theorem is referenced by: cdleme50ldil 39222 |
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