Nearby theorems
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Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > djhlsmat | Structured version Visualization version GIF version | ||
| Description: The sum of two subspace atoms equals their join. TODO: seems convoluted to go via dihprrn 40383; should we directly use dihjat 40380? (Contributed by NM, 13-Aug-2014.) |
| Ref | Expression |
|---|---|
| djhlsmat.h | β’ π» = (LHypβπΎ) |
| djhlsmat.u | β’ π = ((DVecHβπΎ)βπ) |
| djhlsmat.v | β’ π = (Baseβπ) |
| djhlsmat.p | β’ β = (LSSumβπ) |
| djhlsmat.n | β’ π = (LSpanβπ) |
| djhlsmat.i | β’ πΌ = ((DIsoHβπΎ)βπ) |
| djhlsmat.j | β’ β¨ = ((joinHβπΎ)βπ) |
| djhlsmat.k | β’ (π β (πΎ β HL β§ π β π»)) |
| djhlsmat.x | β’ (π β π β π) |
| djhlsmat.y | β’ (π β π β π) |
| Ref | Expression |
|---|---|
| djhlsmat | β’ (π β ((πβ{π}) β (πβ{π})) = ((πβ{π}) β¨ (πβ{π}))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | djhlsmat.h | . . . . . 6 β’ π» = (LHypβπΎ) | |
| 2 | djhlsmat.u | . . . . . 6 β’ π = ((DVecHβπΎ)βπ) | |
| 3 | djhlsmat.k | . . . . . 6 β’ (π β (πΎ β HL β§ π β π»)) | |
| 4 | 1, 2, 3 | dvhlmod 40067 | . . . . 5 β’ (π β π β LMod) |
| 5 | djhlsmat.x | . . . . . 6 β’ (π β π β π) | |
| 6 | 5 | snssd 4812 | . . . . 5 β’ (π β {π} β π) |
| 7 | djhlsmat.y | . . . . . 6 β’ (π β π β π) | |
| 8 | 7 | snssd 4812 | . . . . 5 β’ (π β {π} β π) |
| 9 | djhlsmat.v | . . . . . 6 β’ π = (Baseβπ) | |
| 10 | djhlsmat.n | . . . . . 6 β’ π = (LSpanβπ) | |
| 11 | djhlsmat.p | . . . . . 6 β’ β = (LSSumβπ) | |
| 12 | 9, 10, 11 | lsmsp2 20703 | . . . . 5 β’ ((π β LMod β§ {π} β π β§ {π} β π) β ((πβ{π}) β (πβ{π})) = (πβ({π} βͺ {π}))) |
| 13 | 4, 6, 8, 12 | syl3anc 1371 | . . . 4 β’ (π β ((πβ{π}) β (πβ{π})) = (πβ({π} βͺ {π}))) |
| 14 | df-pr 4631 | . . . . 5 β’ {π, π} = ({π} βͺ {π}) | |
| 15 | 14 | fveq2i 6894 | . . . 4 β’ (πβ{π, π}) = (πβ({π} βͺ {π})) |
| 16 | 13, 15 | eqtr4di 2790 | . . 3 β’ (π β ((πβ{π}) β (πβ{π})) = (πβ{π, π})) |
| 17 | djhlsmat.i | . . . 4 β’ πΌ = ((DIsoHβπΎ)βπ) | |
| 18 | 1, 2, 9, 10, 17, 3, 5, 7 | dihprrn 40383 | . . 3 β’ (π β (πβ{π, π}) β ran πΌ) |
| 19 | 16, 18 | eqeltrd 2833 | . 2 β’ (π β ((πβ{π}) β (πβ{π})) β ran πΌ) |
| 20 | eqid 2732 | . . 3 β’ (LSubSpβπ) = (LSubSpβπ) | |
| 21 | djhlsmat.j | . . 3 β’ β¨ = ((joinHβπΎ)βπ) | |
| 22 | 9, 20, 10 | lspsncl 20593 | . . . 4 β’ ((π β LMod β§ π β π) β (πβ{π}) β (LSubSpβπ)) |
| 23 | 4, 5, 22 | syl2anc 584 | . . 3 β’ (π β (πβ{π}) β (LSubSpβπ)) |
| 24 | 9, 20, 10 | lspsncl 20593 | . . . 4 β’ ((π β LMod β§ π β π) β (πβ{π}) β (LSubSpβπ)) |
| 25 | 4, 7, 24 | syl2anc 584 | . . 3 β’ (π β (πβ{π}) β (LSubSpβπ)) |
| 26 | 1, 2, 9, 20, 11, 17, 21, 3, 23, 25 | djhlsmcl 40371 | . 2 β’ (π β (((πβ{π}) β (πβ{π})) β ran πΌ β ((πβ{π}) β (πβ{π})) = ((πβ{π}) β¨ (πβ{π})))) |
| 27 | 19, 26 | mpbid 231 | 1 β’ (π β ((πβ{π}) β (πβ{π})) = ((πβ{π}) β¨ (πβ{π}))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βͺ cun 3946 β wss 3948 {csn 4628 {cpr 4630 ran crn 5677 βcfv 6543 (class class class)co 7411 Basecbs 17146 LSSumclsm 19504 LModclmod 20475 LSubSpclss 20547 LSpanclspn 20587 HLchlt 38306 LHypclh 38941 DVecHcdvh 40035 DIsoHcdih 40185 joinHcdjh 40351 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-riotaBAD 37909 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-tpos 8213 df-undef 8260 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-n0 12475 df-z 12561 df-uz 12825 df-fz 13487 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-sca 17215 df-vsca 17216 df-0g 17389 df-proset 18250 df-poset 18268 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-p1 18381 df-lat 18387 df-clat 18454 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-submnd 18674 df-grp 18824 df-minusg 18825 df-sbg 18826 df-subg 19005 df-cntz 19183 df-lsm 19506 df-cmn 19652 df-abl 19653 df-mgp 19990 df-ur 20007 df-ring 20060 df-oppr 20154 df-dvdsr 20175 df-unit 20176 df-invr 20206 df-dvr 20219 df-drng 20363 df-lmod 20477 df-lss 20548 df-lsp 20588 df-lvec 20719 df-lsatoms 37932 df-oposet 38132 df-ol 38134 df-oml 38135 df-covers 38222 df-ats 38223 df-atl 38254 df-cvlat 38278 df-hlat 38307 df-llines 38455 df-lplanes 38456 df-lvols 38457 df-lines 38458 df-psubsp 38460 df-pmap 38461 df-padd 38753 df-lhyp 38945 df-laut 38946 df-ldil 39061 df-ltrn 39062 df-trl 39116 df-tgrp 39700 df-tendo 39712 df-edring 39714 df-dveca 39960 df-disoa 39986 df-dvech 40036 df-dib 40096 df-dic 40130 df-dih 40186 df-doch 40305 df-djh 40352 |
| This theorem is referenced by: dihjat1lem 40385 |
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