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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihmeetlem12N | Structured version Visualization version GIF version |
Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dihmeetlem9.b | β’ π΅ = (BaseβπΎ) |
dihmeetlem9.l | β’ β€ = (leβπΎ) |
dihmeetlem9.h | β’ π» = (LHypβπΎ) |
dihmeetlem9.j | β’ β¨ = (joinβπΎ) |
dihmeetlem9.m | β’ β§ = (meetβπΎ) |
dihmeetlem9.a | β’ π΄ = (AtomsβπΎ) |
dihmeetlem9.u | β’ π = ((DVecHβπΎ)βπ) |
dihmeetlem9.s | β’ β = (LSSumβπ) |
dihmeetlem9.i | β’ πΌ = ((DIsoHβπΎ)βπ) |
Ref | Expression |
---|---|
dihmeetlem12N | β’ ((((πΎ β HL β§ π β π») β§ π β π΅ β§ π β π΅) β§ ((π β π΄ β§ Β¬ π β€ π) β§ π β€ π β§ (π β§ π) β€ π)) β ((πΌβ(π β§ π)) β ((πΌβπ) β© (πΌβπ))) = ((πΌβπ) β© (πΌβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1192 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ π β π΅ β§ π β π΅) β§ ((π β π΄ β§ Β¬ π β€ π) β§ π β€ π β§ (π β§ π) β€ π)) β (πΎ β HL β§ π β π»)) | |
2 | simpl2 1193 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ π β π΅ 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 π΅) | |
3 | simpl3 1194 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ π β π΅ 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 π΅) | |
4 | simpr1 1195 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ π β π΅ 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 π΄ β§ Β¬ π β€ π)) | |
5 | simpr2 1196 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ π β π΅ 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€ π) | |
6 | simpr3 1197 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ π β π΅ 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§ π) β€ π) | |
7 | dihmeetlem9.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
8 | dihmeetlem9.l | . . . . 5 β’ β€ = (leβπΎ) | |
9 | dihmeetlem9.h | . . . . 5 β’ π» = (LHypβπΎ) | |
10 | dihmeetlem9.j | . . . . 5 β’ β¨ = (joinβπΎ) | |
11 | dihmeetlem9.m | . . . . 5 β’ β§ = (meetβπΎ) | |
12 | dihmeetlem9.a | . . . . 5 β’ π΄ = (AtomsβπΎ) | |
13 | dihmeetlem9.u | . . . . 5 β’ π = ((DVecHβπΎ)βπ) | |
14 | dihmeetlem9.s | . . . . 5 β’ β = (LSSumβπ) | |
15 | dihmeetlem9.i | . . . . 5 β’ πΌ = ((DIsoHβπΎ)βπ) | |
16 | 7, 8, 9, 10, 11, 12, 13, 14, 15 | dihmeetlem8N 39823 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ π β π΅ β§ π β π΅) β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β€ π β§ (π β§ π) β€ π)) β (πΌβ((π β§ π) β¨ π)) = ((πΌβπ) β (πΌβ(π β§ π)))) |
17 | 1, 2, 3, 4, 5, 6, 16 | syl312anc 1392 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ π β π΅ β§ π β π΅) β§ ((π β π΄ β§ Β¬ π β€ π) β§ π β€ π β§ (π β§ π) β€ π)) β (πΌβ((π β§ π) β¨ π)) = ((πΌβπ) β (πΌβ(π β§ π)))) |
18 | 17 | ineq1d 4172 | . 2 β’ ((((πΎ β HL β§ π β π») β§ π β π΅ β§ π β π΅) β§ ((π β π΄ β§ Β¬ π β€ π) β§ π β€ π β§ (π β§ π) β€ π)) β ((πΌβ((π β§ π) β¨ π)) β© (πΌβπ)) = (((πΌβπ) β (πΌβ(π β§ π))) β© (πΌβπ))) |
19 | 7, 8, 9, 10, 11, 12, 13, 14, 15 | dihmeetlem11N 39826 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ π β π΅ β§ π β π΅) β§ ((π β π΄ β§ Β¬ π β€ π) β§ π β€ π)) β ((πΌβ((π β§ π) β¨ π)) β© (πΌβπ)) = ((πΌβπ) β© (πΌβπ))) |
20 | 19 | 3adantr3 1172 | . 2 β’ ((((πΎ β HL β§ π β π») β§ π β π΅ β§ π β π΅) β§ ((π β π΄ β§ Β¬ π β€ π) β§ π β€ π β§ (π β§ π) β€ π)) β ((πΌβ((π β§ π) β¨ π)) β© (πΌβπ)) = ((πΌβπ) β© (πΌβπ))) |
21 | simpr1l 1231 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ π β π΅ 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 π΄) | |
22 | 7, 8, 9, 10, 11, 12, 13, 14, 15 | dihmeetlem9N 39824 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β π΅) β§ π β π΄) β (((πΌβπ) β (πΌβ(π β§ π))) β© (πΌβπ)) = ((πΌβ(π β§ π)) β ((πΌβπ) β© (πΌβπ)))) |
23 | 1, 2, 3, 21, 22 | syl121anc 1376 | . 2 β’ ((((πΎ β HL β§ π β π») β§ π β π΅ β§ π β π΅) β§ ((π β π΄ β§ Β¬ π β€ π) β§ π β€ π β§ (π β§ π) β€ π)) β (((πΌβπ) β (πΌβ(π β§ π))) β© (πΌβπ)) = ((πΌβ(π β§ π)) β ((πΌβπ) β© (πΌβπ)))) |
24 | 18, 20, 23 | 3eqtr3rd 2782 | 1 β’ ((((πΎ β HL β§ π β π») β§ π β π΅ β§ π β π΅) β§ ((π β π΄ β§ Β¬ π β€ π) β§ π β€ π β§ (π β§ π) β€ π)) β ((πΌβ(π β§ π)) β ((πΌβπ) β© (πΌβπ))) = ((πΌβπ) β© (πΌβπ))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β© cin 3910 class class class wbr 5106 βcfv 6497 (class class class)co 7358 Basecbs 17088 lecple 17145 joincjn 18205 meetcmee 18206 LSSumclsm 19421 Atomscatm 37771 HLchlt 37858 LHypclh 38493 DVecHcdvh 39587 DIsoHcdih 39737 |
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 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-riotaBAD 37461 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 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-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 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-om 7804 df-1st 7922 df-2nd 7923 df-tpos 8158 df-undef 8205 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-sca 17154 df-vsca 17155 df-0g 17328 df-mre 17471 df-mrc 17472 df-acs 17474 df-proset 18189 df-poset 18207 df-plt 18224 df-lub 18240 df-glb 18241 df-join 18242 df-meet 18243 df-p0 18319 df-p1 18320 df-lat 18326 df-clat 18393 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-submnd 18607 df-grp 18756 df-minusg 18757 df-sbg 18758 df-subg 18930 df-cntz 19102 df-lsm 19423 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-ring 19971 df-oppr 20054 df-dvdsr 20075 df-unit 20076 df-invr 20106 df-dvr 20117 df-drng 20199 df-lmod 20338 df-lss 20408 df-lsp 20448 df-lvec 20579 df-oposet 37684 df-ol 37686 df-oml 37687 df-covers 37774 df-ats 37775 df-atl 37806 df-cvlat 37830 df-hlat 37859 df-llines 38007 df-lplanes 38008 df-lvols 38009 df-lines 38010 df-psubsp 38012 df-pmap 38013 df-padd 38305 df-lhyp 38497 df-laut 38498 df-ldil 38613 df-ltrn 38614 df-trl 38668 df-tendo 39264 df-edring 39266 df-disoa 39538 df-dvech 39588 df-dib 39648 df-dic 39682 df-dih 39738 |
This theorem is referenced by: dihmeetlem14N 39829 dihmeetlem19N 39834 |
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