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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihglblem4 | Structured version Visualization version GIF version |
Description: Isomorphism H of a lattice glb. (Contributed by NM, 21-Mar-2014.) |
Ref | Expression |
---|---|
dihglblem.b | β’ π΅ = (BaseβπΎ) |
dihglblem.l | β’ β€ = (leβπΎ) |
dihglblem.m | β’ β§ = (meetβπΎ) |
dihglblem.g | β’ πΊ = (glbβπΎ) |
dihglblem.h | β’ π» = (LHypβπΎ) |
dihglblem.t | β’ π = {π’ β π΅ β£ βπ£ β π π’ = (π£ β§ π)} |
dihglblem.i | β’ π½ = ((DIsoBβπΎ)βπ) |
dihglblem.ih | β’ πΌ = ((DIsoHβπΎ)βπ) |
Ref | Expression |
---|---|
dihglblem4 | β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β (πΌβ(πΊβπ)) β β© π₯ β π (πΌβπ₯)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlclat 38886 | . . . . . 6 β’ (πΎ β HL β πΎ β CLat) | |
2 | 1 | ad3antrrr 728 | . . . . 5 β’ ((((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β§ π₯ β π) β πΎ β CLat) |
3 | simplrl 775 | . . . . 5 β’ ((((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β§ π₯ β π) β π β π΅) | |
4 | simpr 483 | . . . . 5 β’ ((((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β§ π₯ β π) β π₯ β π) | |
5 | dihglblem.b | . . . . . 6 β’ π΅ = (BaseβπΎ) | |
6 | dihglblem.l | . . . . . 6 β’ β€ = (leβπΎ) | |
7 | dihglblem.g | . . . . . 6 β’ πΊ = (glbβπΎ) | |
8 | 5, 6, 7 | clatglble 18508 | . . . . 5 β’ ((πΎ β CLat β§ π β π΅ β§ π₯ β π) β (πΊβπ) β€ π₯) |
9 | 2, 3, 4, 8 | syl3anc 1368 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β§ π₯ β π) β (πΊβπ) β€ π₯) |
10 | simpll 765 | . . . . 5 β’ ((((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β§ π₯ β π) β (πΎ β HL β§ π β π»)) | |
11 | 5, 7 | clatglbcl 18496 | . . . . . 6 β’ ((πΎ β CLat β§ π β π΅) β (πΊβπ) β π΅) |
12 | 2, 3, 11 | syl2anc 582 | . . . . 5 β’ ((((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β§ π₯ β π) β (πΊβπ) β π΅) |
13 | 3, 4 | sseldd 3973 | . . . . 5 β’ ((((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β§ π₯ β π) β π₯ β π΅) |
14 | dihglblem.h | . . . . . 6 β’ π» = (LHypβπΎ) | |
15 | dihglblem.ih | . . . . . 6 β’ πΌ = ((DIsoHβπΎ)βπ) | |
16 | 5, 6, 14, 15 | dihord 40793 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ (πΊβπ) β π΅ β§ π₯ β π΅) β ((πΌβ(πΊβπ)) β (πΌβπ₯) β (πΊβπ) β€ π₯)) |
17 | 10, 12, 13, 16 | syl3anc 1368 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β§ π₯ β π) β ((πΌβ(πΊβπ)) β (πΌβπ₯) β (πΊβπ) β€ π₯)) |
18 | 9, 17 | mpbird 256 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β§ π₯ β π) β (πΌβ(πΊβπ)) β (πΌβπ₯)) |
19 | 18 | ralrimiva 3136 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β βπ₯ β π (πΌβ(πΊβπ)) β (πΌβπ₯)) |
20 | ssiin 5053 | . 2 β’ ((πΌβ(πΊβπ)) β β© π₯ β π (πΌβπ₯) β βπ₯ β π (πΌβ(πΊβπ)) β (πΌβπ₯)) | |
21 | 19, 20 | sylibr 233 | 1 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β )) β (πΌβ(πΊβπ)) β β© π₯ β π (πΌβπ₯)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2930 βwral 3051 βwrex 3060 {crab 3419 β wss 3939 β c0 4318 β© ciin 4992 class class class wbr 5143 βcfv 6543 (class class class)co 7416 Basecbs 17179 lecple 17239 glbcglb 18301 meetcmee 18303 CLatccla 18489 HLchlt 38878 LHypclh 39513 DIsoBcdib 40667 DIsoHcdih 40757 |
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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-riotaBAD 38481 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-tpos 8230 df-undef 8277 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-0g 17422 df-proset 18286 df-poset 18304 df-plt 18321 df-lub 18337 df-glb 18338 df-join 18339 df-meet 18340 df-p0 18416 df-p1 18417 df-lat 18423 df-clat 18490 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-grp 18897 df-minusg 18898 df-sbg 18899 df-subg 19082 df-cntz 19272 df-lsm 19595 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-oppr 20277 df-dvdsr 20300 df-unit 20301 df-invr 20331 df-dvr 20344 df-drng 20630 df-lmod 20749 df-lss 20820 df-lsp 20860 df-lvec 20992 df-oposet 38704 df-ol 38706 df-oml 38707 df-covers 38794 df-ats 38795 df-atl 38826 df-cvlat 38850 df-hlat 38879 df-llines 39027 df-lplanes 39028 df-lvols 39029 df-lines 39030 df-psubsp 39032 df-pmap 39033 df-padd 39325 df-lhyp 39517 df-laut 39518 df-ldil 39633 df-ltrn 39634 df-trl 39688 df-tendo 40284 df-edring 40286 df-disoa 40558 df-dvech 40608 df-dib 40668 df-dic 40702 df-dih 40758 |
This theorem is referenced by: dihglblem6 40869 |
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