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Mirrors > Home > MPE Home > Th. List > Mathboxes > llnexch2N | Structured version Visualization version GIF version |
Description: Line exchange property (compare cvlatexch2 37802 for atoms). (Contributed by NM, 18-Nov-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
llnexch.l | β’ β€ = (leβπΎ) |
llnexch.j | β’ β¨ = (joinβπΎ) |
llnexch.m | β’ β§ = (meetβπΎ) |
llnexch.a | β’ π΄ = (AtomsβπΎ) |
llnexch.n | β’ π = (LLinesβπΎ) |
Ref | Expression |
---|---|
llnexch2N | β’ ((πΎ β HL β§ (π β π β§ π β π β§ π β π) β§ ((π β§ π) β π΄ β§ π β π)) β ((π β§ π) β€ π β (π β§ π) β€ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | llnexch.l | . . 3 β’ β€ = (leβπΎ) | |
2 | llnexch.j | . . 3 β’ β¨ = (joinβπΎ) | |
3 | llnexch.m | . . 3 β’ β§ = (meetβπΎ) | |
4 | llnexch.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
5 | llnexch.n | . . 3 β’ π = (LLinesβπΎ) | |
6 | 1, 2, 3, 4, 5 | llnexchb2 38335 | . 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 β π)) β ((π β§ π) β€ π β (π β§ π) = (π β§ π))) |
7 | hllat 37828 | . . . . 5 β’ (πΎ β HL β πΎ β Lat) | |
8 | 7 | 3ad2ant1 1134 | . . . 4 β’ ((πΎ β HL β§ (π β π β§ π β π β§ π β π) β§ ((π β§ π) β π΄ β§ π β π)) β πΎ β Lat) |
9 | simp21 1207 | . . . . 5 β’ ((πΎ β HL β§ (π β π β§ π β π β§ π β π) β§ ((π β§ π) β π΄ β§ π β π)) β π β π) | |
10 | eqid 2737 | . . . . . 6 β’ (BaseβπΎ) = (BaseβπΎ) | |
11 | 10, 5 | llnbase 37975 | . . . . 5 β’ (π β π β π β (BaseβπΎ)) |
12 | 9, 11 | syl 17 | . . . 4 β’ ((πΎ β HL β§ (π β π β§ π β π β§ π β π) β§ ((π β§ π) β π΄ β§ π β π)) β π β (BaseβπΎ)) |
13 | simp22 1208 | . . . . 5 β’ ((πΎ β HL β§ (π β π β§ π β π β§ π β π) β§ ((π β§ π) β π΄ β§ π β π)) β π β π) | |
14 | 10, 5 | llnbase 37975 | . . . . 5 β’ (π β π β π β (BaseβπΎ)) |
15 | 13, 14 | syl 17 | . . . 4 β’ ((πΎ β HL β§ (π β π β§ π β π β§ π β π) β§ ((π β§ π) β π΄ β§ π β π)) β π β (BaseβπΎ)) |
16 | 10, 1, 3 | latmle2 18355 | . . . 4 β’ ((πΎ β Lat β§ π β (BaseβπΎ) β§ π β (BaseβπΎ)) β (π β§ π) β€ π) |
17 | 8, 12, 15, 16 | 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΄ β§ π β π)) β (π β§ π) β€ π) |
18 | breq1 5109 | . . 3 β’ ((π β§ π) = (π β§ π) β ((π β§ π) β€ π β (π β§ π) β€ π)) | |
19 | 17, 18 | syl5ibcom 244 | . 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 β π)) β ((π β§ π) = (π β§ π) β (π β§ π) β€ π)) |
20 | 6, 19 | sylbid 239 | 1 β’ ((πΎ β HL β§ (π β π β§ π β π β§ π β π) β§ ((π β§ π) β π΄ 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 β wne 2944 class class class wbr 5106 βcfv 6497 (class class class)co 7358 Basecbs 17084 lecple 17141 joincjn 18201 meetcmee 18202 Latclat 18321 Atomscatm 37728 HLchlt 37815 LLinesclln 37957 |
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-iin 4958 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-mpo 7363 df-1st 7922 df-2nd 7923 df-proset 18185 df-poset 18203 df-plt 18220 df-lub 18236 df-glb 18237 df-join 18238 df-meet 18239 df-p0 18315 df-lat 18322 df-clat 18389 df-oposet 37641 df-ol 37643 df-oml 37644 df-covers 37731 df-ats 37732 df-atl 37763 df-cvlat 37787 df-hlat 37816 df-llines 37964 df-psubsp 37969 df-pmap 37970 df-padd 38262 |
This theorem is referenced by: (None) |
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