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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dih0 | Structured version Visualization version GIF version |
Description: The value of isomorphism H at the lattice zero is the singleton of the zero vector i.e. the zero subspace. (Contributed by NM, 9-Mar-2014.) |
Ref | Expression |
---|---|
dih0.z | β’ 0 = (0.βπΎ) |
dih0.h | β’ π» = (LHypβπΎ) |
dih0.i | β’ πΌ = ((DIsoHβπΎ)βπ) |
dih0.u | β’ π = ((DVecHβπΎ)βπ) |
dih0.o | β’ π = (0gβπ) |
Ref | Expression |
---|---|
dih0 | β’ ((πΎ β HL β§ π β π») β (πΌβ 0 ) = {π}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 β’ ((πΎ β HL β§ π β π») β (πΎ β HL β§ π β π»)) | |
2 | hlop 38834 | . . . . 5 β’ (πΎ β HL β πΎ β OP) | |
3 | 2 | adantr 480 | . . . 4 β’ ((πΎ β HL β§ π β π») β πΎ β OP) |
4 | eqid 2728 | . . . . 5 β’ (BaseβπΎ) = (BaseβπΎ) | |
5 | dih0.z | . . . . 5 β’ 0 = (0.βπΎ) | |
6 | 4, 5 | op0cl 38656 | . . . 4 β’ (πΎ β OP β 0 β (BaseβπΎ)) |
7 | 3, 6 | syl 17 | . . 3 β’ ((πΎ β HL β§ π β π») β 0 β (BaseβπΎ)) |
8 | dih0.h | . . . . 5 β’ π» = (LHypβπΎ) | |
9 | 4, 8 | lhpbase 39471 | . . . 4 β’ (π β π» β π β (BaseβπΎ)) |
10 | eqid 2728 | . . . . 5 β’ (leβπΎ) = (leβπΎ) | |
11 | 4, 10, 5 | op0le 38658 | . . . 4 β’ ((πΎ β OP β§ π β (BaseβπΎ)) β 0 (leβπΎ)π) |
12 | 2, 9, 11 | syl2an 595 | . . 3 β’ ((πΎ β HL β§ π β π») β 0 (leβπΎ)π) |
13 | dih0.i | . . . 4 β’ πΌ = ((DIsoHβπΎ)βπ) | |
14 | eqid 2728 | . . . 4 β’ ((DIsoBβπΎ)βπ) = ((DIsoBβπΎ)βπ) | |
15 | 4, 10, 8, 13, 14 | dihvalb 40710 | . . 3 β’ (((πΎ β HL β§ π β π») β§ ( 0 β (BaseβπΎ) β§ 0 (leβπΎ)π)) β (πΌβ 0 ) = (((DIsoBβπΎ)βπ)β 0 )) |
16 | 1, 7, 12, 15 | syl12anc 836 | . 2 β’ ((πΎ β HL β§ π β π») β (πΌβ 0 ) = (((DIsoBβπΎ)βπ)β 0 )) |
17 | dih0.u | . . 3 β’ π = ((DVecHβπΎ)βπ) | |
18 | dih0.o | . . 3 β’ π = (0gβπ) | |
19 | 5, 8, 14, 17, 18 | dib0 40637 | . 2 β’ ((πΎ β HL β§ π β π») β (((DIsoBβπΎ)βπ)β 0 ) = {π}) |
20 | 16, 19 | eqtrd 2768 | 1 β’ ((πΎ β HL β§ π β π») β (πΌβ 0 ) = {π}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 {csn 4629 class class class wbr 5148 βcfv 6548 Basecbs 17179 lecple 17239 0gc0g 17420 0.cp0 18414 OPcops 38644 HLchlt 38822 LHypclh 39457 DVecHcdvh 40551 DIsoBcdib 40611 DIsoHcdih 40701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 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 38425 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-tpos 8231 df-undef 8278 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 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-grp 18892 df-minusg 18893 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-oppr 20272 df-dvdsr 20295 df-unit 20296 df-invr 20326 df-dvr 20339 df-drng 20625 df-lmod 20744 df-lvec 20987 df-oposet 38648 df-ol 38650 df-oml 38651 df-covers 38738 df-ats 38739 df-atl 38770 df-cvlat 38794 df-hlat 38823 df-llines 38971 df-lplanes 38972 df-lvols 38973 df-lines 38974 df-psubsp 38976 df-pmap 38977 df-padd 39269 df-lhyp 39461 df-laut 39462 df-ldil 39577 df-ltrn 39578 df-trl 39632 df-tendo 40228 df-edring 40230 df-disoa 40502 df-dvech 40552 df-dib 40612 df-dih 40702 |
This theorem is referenced by: dih0bN 40754 dih0vbN 40755 dih0cnv 40756 dih0rn 40757 dihmeetlem4preN 40779 dihmeetlem18N 40797 dihlspsnssN 40805 dihlspsnat 40806 dihatexv 40811 doch1 40832 dochnoncon 40864 |
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