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Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > djhlsmcl | Structured version Visualization version GIF version | ||
| Description: A closed subspace sum equals subspace join. (shjshseli 31319 analog.) (Contributed by NM, 13-Aug-2014.) |
| Ref | Expression |
|---|---|
| djhlsmcl.h | β’ π» = (LHypβπΎ) |
| djhlsmcl.u | β’ π = ((DVecHβπΎ)βπ) |
| djhlsmcl.v | β’ π = (Baseβπ) |
| djhlsmcl.s | β’ π = (LSubSpβπ) |
| djhlsmcl.p | β’ β = (LSSumβπ) |
| djhlsmcl.i | β’ πΌ = ((DIsoHβπΎ)βπ) |
| djhlsmcl.j | β’ β¨ = ((joinHβπΎ)βπ) |
| djhlsmcl.k | β’ (π β (πΎ β HL β§ π β π»)) |
| djhlsmcl.x | β’ (π β π β π) |
| djhlsmcl.y | β’ (π β π β π) |
| Ref | Expression |
|---|---|
| djhlsmcl | β’ (π β ((π β π) β ran πΌ β (π β π) = (π β¨ π))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | djhlsmcl.k | . . . . . 6 β’ (π β (πΎ β HL β§ π β π»)) | |
| 2 | 1 | adantr 479 | . . . . 5 β’ ((π β§ (π β π) β ran πΌ) β (πΎ β HL β§ π β π»)) |
| 3 | djhlsmcl.x | . . . . . . 7 β’ (π β π β π) | |
| 4 | djhlsmcl.v | . . . . . . . 8 β’ π = (Baseβπ) | |
| 5 | djhlsmcl.s | . . . . . . . 8 β’ π = (LSubSpβπ) | |
| 6 | 4, 5 | lssss 20822 | . . . . . . 7 β’ (π β π β π β π) |
| 7 | 3, 6 | syl 17 | . . . . . 6 β’ (π β π β π) |
| 8 | 7 | adantr 479 | . . . . 5 β’ ((π β§ (π β π) β ran πΌ) β π β π) |
| 9 | djhlsmcl.y | . . . . . . 7 β’ (π β π β π) | |
| 10 | 4, 5 | lssss 20822 | . . . . . . 7 β’ (π β π β π β π) |
| 11 | 9, 10 | syl 17 | . . . . . 6 β’ (π β π β π) |
| 12 | 11 | adantr 479 | . . . . 5 β’ ((π β§ (π β π) β ran πΌ) β π β π) |
| 13 | djhlsmcl.h | . . . . . 6 β’ π» = (LHypβπΎ) | |
| 14 | djhlsmcl.u | . . . . . 6 β’ π = ((DVecHβπΎ)βπ) | |
| 15 | eqid 2725 | . . . . . 6 β’ ((ocHβπΎ)βπ) = ((ocHβπΎ)βπ) | |
| 16 | djhlsmcl.j | . . . . . 6 β’ β¨ = ((joinHβπΎ)βπ) | |
| 17 | 13, 14, 4, 15, 16 | djhval2 40900 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ π β π β§ π β π) β (π β¨ π) = (((ocHβπΎ)βπ)β(((ocHβπΎ)βπ)β(π βͺ π)))) |
| 18 | 2, 8, 12, 17 | syl3anc 1368 | . . . 4 β’ ((π β§ (π β π) β ran πΌ) β (π β¨ π) = (((ocHβπΎ)βπ)β(((ocHβπΎ)βπ)β(π βͺ π)))) |
| 19 | 13, 14, 1 | dvhlmod 40611 | . . . . . . . . 9 β’ (π β π β LMod) |
| 20 | 19 | adantr 479 | . . . . . . . 8 β’ ((π β§ (π β π) β ran πΌ) β π β LMod) |
| 21 | 3 | adantr 479 | . . . . . . . 8 β’ ((π β§ (π β π) β ran πΌ) β π β π) |
| 22 | 9 | adantr 479 | . . . . . . . 8 β’ ((π β§ (π β π) β ran πΌ) β π β π) |
| 23 | eqid 2725 | . . . . . . . . 9 β’ (LSpanβπ) = (LSpanβπ) | |
| 24 | djhlsmcl.p | . . . . . . . . 9 β’ β = (LSSumβπ) | |
| 25 | 5, 23, 24 | lsmsp 20973 | . . . . . . . 8 β’ ((π β LMod β§ π β π β§ π β π) β (π β π) = ((LSpanβπ)β(π βͺ π))) |
| 26 | 20, 21, 22, 25 | syl3anc 1368 | . . . . . . 7 β’ ((π β§ (π β π) β ran πΌ) β (π β π) = ((LSpanβπ)β(π βͺ π))) |
| 27 | 26 | fveq2d 6894 | . . . . . 6 β’ ((π β§ (π β π) β ran πΌ) β (((ocHβπΎ)βπ)β(π β π)) = (((ocHβπΎ)βπ)β((LSpanβπ)β(π βͺ π)))) |
| 28 | 7, 11 | unssd 4178 | . . . . . . . 8 β’ (π β (π βͺ π) β π) |
| 29 | 28 | adantr 479 | . . . . . . 7 β’ ((π β§ (π β π) β ran πΌ) β (π βͺ π) β π) |
| 30 | 13, 14, 15, 4, 23, 2, 29 | dochocsp 40880 | . . . . . 6 β’ ((π β§ (π β π) β ran πΌ) β (((ocHβπΎ)βπ)β((LSpanβπ)β(π βͺ π))) = (((ocHβπΎ)βπ)β(π βͺ π))) |
| 31 | 27, 30 | eqtrd 2765 | . . . . 5 β’ ((π β§ (π β π) β ran πΌ) β (((ocHβπΎ)βπ)β(π β π)) = (((ocHβπΎ)βπ)β(π βͺ π))) |
| 32 | 31 | fveq2d 6894 | . . . 4 β’ ((π β§ (π β π) β ran πΌ) β (((ocHβπΎ)βπ)β(((ocHβπΎ)βπ)β(π β π))) = (((ocHβπΎ)βπ)β(((ocHβπΎ)βπ)β(π βͺ π)))) |
| 33 | djhlsmcl.i | . . . . . 6 β’ πΌ = ((DIsoHβπΎ)βπ) | |
| 34 | 13, 33, 15 | dochoc 40868 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ (π β π) β ran πΌ) β (((ocHβπΎ)βπ)β(((ocHβπΎ)βπ)β(π β π))) = (π β π)) |
| 35 | 1, 34 | sylan 578 | . . . 4 β’ ((π β§ (π β π) β ran πΌ) β (((ocHβπΎ)βπ)β(((ocHβπΎ)βπ)β(π β π))) = (π β π)) |
| 36 | 18, 32, 35 | 3eqtr2rd 2772 | . . 3 β’ ((π β§ (π β π) β ran πΌ) β (π β π) = (π β¨ π)) |
| 37 | 36 | ex 411 | . 2 β’ (π β ((π β π) β ran πΌ β (π β π) = (π β¨ π))) |
| 38 | 13, 33, 14, 4, 16 | djhcl 40901 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π β§ π β π)) β (π β¨ π) β ran πΌ) |
| 39 | 1, 7, 11, 38 | syl12anc 835 | . . 3 β’ (π β (π β¨ π) β ran πΌ) |
| 40 | eleq1a 2820 | . . 3 β’ ((π β¨ π) β ran πΌ β ((π β π) = (π β¨ π) β (π β π) β ran πΌ)) | |
| 41 | 39, 40 | syl 17 | . 2 β’ (π β ((π β π) = (π β¨ π) β (π β π) β ran πΌ)) |
| 42 | 37, 41 | impbid 211 | 1 β’ (π β ((π β π) β ran πΌ β (π β π) = (π β¨ π))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 βͺ cun 3937 β wss 3939 ran crn 5671 βcfv 6541 (class class class)co 7414 Basecbs 17177 LSSumclsm 19591 LModclmod 20745 LSubSpclss 20817 LSpanclspn 20857 HLchlt 38850 LHypclh 39485 DVecHcdvh 40579 DIsoHcdih 40729 ocHcoch 40848 joinHcdjh 40895 |
| 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 5278 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-riotaBAD 38453 |
| 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 3958 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-int 4943 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7989 df-2nd 7990 df-tpos 8228 df-undef 8275 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-n0 12501 df-z 12587 df-uz 12851 df-fz 13515 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-0g 17420 df-proset 18284 df-poset 18302 df-plt 18319 df-lub 18335 df-glb 18336 df-join 18337 df-meet 18338 df-p0 18414 df-p1 18415 df-lat 18421 df-clat 18488 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18738 df-grp 18895 df-minusg 18896 df-sbg 18897 df-subg 19080 df-cntz 19270 df-lsm 19593 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-oppr 20275 df-dvdsr 20298 df-unit 20299 df-invr 20329 df-dvr 20342 df-drng 20628 df-lmod 20747 df-lss 20818 df-lsp 20858 df-lvec 20990 df-lsatoms 38476 df-oposet 38676 df-ol 38678 df-oml 38679 df-covers 38766 df-ats 38767 df-atl 38798 df-cvlat 38822 df-hlat 38851 df-llines 38999 df-lplanes 39000 df-lvols 39001 df-lines 39002 df-psubsp 39004 df-pmap 39005 df-padd 39297 df-lhyp 39489 df-laut 39490 df-ldil 39605 df-ltrn 39606 df-trl 39660 df-tendo 40256 df-edring 40258 df-disoa 40530 df-dvech 40580 df-dib 40640 df-dic 40674 df-dih 40730 df-doch 40849 df-djh 40896 |
| This theorem is referenced by: djhlsmat 40928 dihsmatrn 40937 lclkrslem2 41039 |
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