![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dihord3 | Structured version Visualization version GIF version |
Description: The isomorphism H for a lattice πΎ is order-preserving in the region under co-atom π. (Contributed by NM, 6-Mar-2014.) |
Ref | Expression |
---|---|
dihord3.b | β’ π΅ = (BaseβπΎ) |
dihord3.l | β’ β€ = (leβπΎ) |
dihord3.h | β’ π» = (LHypβπΎ) |
dihord3.i | β’ πΌ = ((DIsoHβπΎ)βπ) |
Ref | Expression |
---|---|
dihord3 | 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 ((πΌβπ) β (πΌβπ) β π β€ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dihord3.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
2 | dihord3.l | . . . . 5 β’ β€ = (leβπΎ) | |
3 | dihord3.h | . . . . 5 β’ π» = (LHypβπΎ) | |
4 | dihord3.i | . . . . 5 β’ πΌ = ((DIsoHβπΎ)βπ) | |
5 | eqid 2726 | . . . . 5 β’ ((DIsoBβπΎ)βπ) = ((DIsoBβπΎ)βπ) | |
6 | 1, 2, 3, 4, 5 | dihvalb 40620 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π)) β (πΌβπ) = (((DIsoBβπΎ)βπ)βπ)) |
7 | 6 | 3adant3 1129 | . . 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 (πΌβπ) = (((DIsoBβπΎ)βπ)βπ)) |
8 | 1, 2, 3, 4, 5 | dihvalb 40620 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π)) β (πΌβπ) = (((DIsoBβπΎ)βπ)βπ)) |
9 | 8 | 3adant2 1128 | . . 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 (πΌβπ) = (((DIsoBβπΎ)βπ)βπ)) |
10 | 7, 9 | sseq12d 4010 | . 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 ((πΌβπ) β (πΌβπ) β (((DIsoBβπΎ)βπ)βπ) β (((DIsoBβπΎ)βπ)βπ))) |
11 | 1, 2, 3, 5 | dibord 40542 | . 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 ((((DIsoBβπΎ)βπ)βπ) β (((DIsoBβπΎ)βπ)βπ) β π β€ π)) |
12 | 10, 11 | bitrd 279 | 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 ((πΌβπ) β (πΌβπ) β π β€ π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wss 3943 class class class wbr 5141 βcfv 6536 Basecbs 17150 lecple 17210 HLchlt 38732 LHypclh 39367 DIsoBcdib 40521 DIsoHcdih 40611 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-riotaBAD 38335 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 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 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-undef 8256 df-map 8821 df-proset 18257 df-poset 18275 df-plt 18292 df-lub 18308 df-glb 18309 df-join 18310 df-meet 18311 df-p0 18387 df-p1 18388 df-lat 18394 df-clat 18461 df-oposet 38558 df-ol 38560 df-oml 38561 df-covers 38648 df-ats 38649 df-atl 38680 df-cvlat 38704 df-hlat 38733 df-llines 38881 df-lplanes 38882 df-lvols 38883 df-lines 38884 df-psubsp 38886 df-pmap 38887 df-padd 39179 df-lhyp 39371 df-laut 39372 df-ldil 39487 df-ltrn 39488 df-trl 39542 df-disoa 40412 df-dib 40522 df-dih 40612 |
This theorem is referenced by: dihord 40647 |
Copyright terms: Public domain | W3C validator |