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Mathbox for Norm Megill |
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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 2728 | . . . . 5 β’ ((DIsoBβπΎ)βπ) = ((DIsoBβπΎ)βπ) | |
6 | 1, 2, 3, 4, 5 | dihvalb 40742 | . . . 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 40742 | . . . 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 4015 | . 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 40664 | . 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 278 | 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 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wss 3949 class class class wbr 5152 βcfv 6553 Basecbs 17187 lecple 17247 HLchlt 38854 LHypclh 39489 DIsoBcdib 40643 DIsoHcdih 40733 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-riotaBAD 38457 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-undef 8285 df-map 8853 df-proset 18294 df-poset 18312 df-plt 18329 df-lub 18345 df-glb 18346 df-join 18347 df-meet 18348 df-p0 18424 df-p1 18425 df-lat 18431 df-clat 18498 df-oposet 38680 df-ol 38682 df-oml 38683 df-covers 38770 df-ats 38771 df-atl 38802 df-cvlat 38826 df-hlat 38855 df-llines 39003 df-lplanes 39004 df-lvols 39005 df-lines 39006 df-psubsp 39008 df-pmap 39009 df-padd 39301 df-lhyp 39493 df-laut 39494 df-ldil 39609 df-ltrn 39610 df-trl 39664 df-disoa 40534 df-dib 40644 df-dih 40734 |
This theorem is referenced by: dihord 40769 |
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