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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > djhj | Structured version Visualization version GIF version |
Description: DVecH vector space closed subspace join in terms of lattice join. (Contributed by NM, 17-Aug-2014.) |
Ref | Expression |
---|---|
djhj.k | β’ β¨ = (joinβπΎ) |
djhj.h | β’ π» = (LHypβπΎ) |
djhj.i | β’ πΌ = ((DIsoHβπΎ)βπ) |
djhj.j | β’ π½ = ((joinHβπΎ)βπ) |
djhj.w | β’ (π β (πΎ β HL β§ π β π»)) |
djhj.x | β’ (π β π β ran πΌ) |
djhj.y | β’ (π β π β ran πΌ) |
Ref | Expression |
---|---|
djhj | β’ (π β (β‘πΌβ(ππ½π)) = ((β‘πΌβπ) β¨ (β‘πΌβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djhj.k | . . . 4 β’ β¨ = (joinβπΎ) | |
2 | djhj.h | . . . 4 β’ π» = (LHypβπΎ) | |
3 | djhj.i | . . . 4 β’ πΌ = ((DIsoHβπΎ)βπ) | |
4 | djhj.j | . . . 4 β’ π½ = ((joinHβπΎ)βπ) | |
5 | djhj.w | . . . 4 β’ (π β (πΎ β HL β§ π β π»)) | |
6 | djhj.x | . . . 4 β’ (π β π β ran πΌ) | |
7 | djhj.y | . . . 4 β’ (π β π β ran πΌ) | |
8 | 1, 2, 3, 4, 5, 6, 7 | djhjlj 40870 | . . 3 β’ (π β (ππ½π) = (πΌβ((β‘πΌβπ) β¨ (β‘πΌβπ)))) |
9 | 8 | fveq2d 6895 | . 2 β’ (π β (β‘πΌβ(ππ½π)) = (β‘πΌβ(πΌβ((β‘πΌβπ) β¨ (β‘πΌβπ))))) |
10 | 5 | simpld 494 | . . . . 5 β’ (π β πΎ β HL) |
11 | 10 | hllatd 38830 | . . . 4 β’ (π β πΎ β Lat) |
12 | eqid 2728 | . . . . . 6 β’ (BaseβπΎ) = (BaseβπΎ) | |
13 | 12, 2, 3 | dihcnvcl 40738 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ π β ran πΌ) β (β‘πΌβπ) β (BaseβπΎ)) |
14 | 5, 6, 13 | syl2anc 583 | . . . 4 β’ (π β (β‘πΌβπ) β (BaseβπΎ)) |
15 | 12, 2, 3 | dihcnvcl 40738 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ π β ran πΌ) β (β‘πΌβπ) β (BaseβπΎ)) |
16 | 5, 7, 15 | syl2anc 583 | . . . 4 β’ (π β (β‘πΌβπ) β (BaseβπΎ)) |
17 | 12, 1 | latjcl 18424 | . . . 4 β’ ((πΎ β Lat β§ (β‘πΌβπ) β (BaseβπΎ) β§ (β‘πΌβπ) β (BaseβπΎ)) β ((β‘πΌβπ) β¨ (β‘πΌβπ)) β (BaseβπΎ)) |
18 | 11, 14, 16, 17 | syl3anc 1369 | . . 3 β’ (π β ((β‘πΌβπ) β¨ (β‘πΌβπ)) β (BaseβπΎ)) |
19 | 12, 2, 3 | dihcnvid1 40739 | . . 3 β’ (((πΎ β HL β§ π β π») β§ ((β‘πΌβπ) β¨ (β‘πΌβπ)) β (BaseβπΎ)) β (β‘πΌβ(πΌβ((β‘πΌβπ) β¨ (β‘πΌβπ)))) = ((β‘πΌβπ) β¨ (β‘πΌβπ))) |
20 | 5, 18, 19 | syl2anc 583 | . 2 β’ (π β (β‘πΌβ(πΌβ((β‘πΌβπ) β¨ (β‘πΌβπ)))) = ((β‘πΌβπ) β¨ (β‘πΌβπ))) |
21 | 9, 20 | eqtrd 2768 | 1 β’ (π β (β‘πΌβ(ππ½π)) = ((β‘πΌβπ) β¨ (β‘πΌβπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 β‘ccnv 5671 ran crn 5673 βcfv 6542 (class class class)co 7414 Basecbs 17173 joincjn 18296 Latclat 18416 HLchlt 38816 LHypclh 39451 DIsoHcdih 40695 joinHcdjh 40861 |
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 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-riotaBAD 38419 |
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 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 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 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-tpos 8225 df-undef 8272 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-n0 12497 df-z 12583 df-uz 12847 df-fz 13511 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-sca 17242 df-vsca 17243 df-0g 17416 df-proset 18280 df-poset 18298 df-plt 18315 df-lub 18331 df-glb 18332 df-join 18333 df-meet 18334 df-p0 18410 df-p1 18411 df-lat 18417 df-clat 18484 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-submnd 18734 df-grp 18886 df-minusg 18887 df-sbg 18888 df-subg 19071 df-cntz 19261 df-lsm 19584 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-oppr 20266 df-dvdsr 20289 df-unit 20290 df-invr 20320 df-dvr 20333 df-drng 20619 df-lmod 20738 df-lss 20809 df-lsp 20849 df-lvec 20981 df-lsatoms 38442 df-oposet 38642 df-ol 38644 df-oml 38645 df-covers 38732 df-ats 38733 df-atl 38764 df-cvlat 38788 df-hlat 38817 df-llines 38965 df-lplanes 38966 df-lvols 38967 df-lines 38968 df-psubsp 38970 df-pmap 38971 df-padd 39263 df-lhyp 39455 df-laut 39456 df-ldil 39571 df-ltrn 39572 df-trl 39626 df-tendo 40222 df-edring 40224 df-disoa 40496 df-dvech 40546 df-dib 40606 df-dic 40640 df-dih 40696 df-doch 40815 df-djh 40862 |
This theorem is referenced by: djhcvat42 40882 |
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