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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lsmcv2 | Structured version Visualization version GIF version | ||
| Description: Subspace sum has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (spansncv2 32051 analog.) (Contributed by NM, 10-Jan-2015.) |
| Ref | Expression |
|---|---|
| lsmcv2.v | β’ π = (Baseβπ) |
| lsmcv2.s | β’ π = (LSubSpβπ) |
| lsmcv2.n | β’ π = (LSpanβπ) |
| lsmcv2.p | β’ β = (LSSumβπ) |
| lsmcv2.c | β’ πΆ = ( βL βπ) |
| lsmcv2.w | β’ (π β π β LVec) |
| lsmcv2.u | β’ (π β π β π) |
| lsmcv2.x | β’ (π β π β π) |
| lsmcv2.l | β’ (π β Β¬ (πβ{π}) β π) |
| Ref | Expression |
|---|---|
| lsmcv2 | β’ (π β ππΆ(π β (πβ{π}))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lsmcv2.l | . . 3 β’ (π β Β¬ (πβ{π}) β π) | |
| 2 | lsmcv2.p | . . . 4 β’ β = (LSSumβπ) | |
| 3 | lsmcv2.w | . . . . . . 7 β’ (π β π β LVec) | |
| 4 | lveclmod 20952 | . . . . . . 7 β’ (π β LVec β π β LMod) | |
| 5 | 3, 4 | syl 17 | . . . . . 6 β’ (π β π β LMod) |
| 6 | lsmcv2.s | . . . . . . 7 β’ π = (LSubSpβπ) | |
| 7 | 6 | lsssssubg 20803 | . . . . . 6 β’ (π β LMod β π β (SubGrpβπ)) |
| 8 | 5, 7 | syl 17 | . . . . 5 β’ (π β π β (SubGrpβπ)) |
| 9 | lsmcv2.u | . . . . 5 β’ (π β π β π) | |
| 10 | 8, 9 | sseldd 3978 | . . . 4 β’ (π β π β (SubGrpβπ)) |
| 11 | lsmcv2.x | . . . . . 6 β’ (π β π β π) | |
| 12 | lsmcv2.v | . . . . . . 7 β’ π = (Baseβπ) | |
| 13 | lsmcv2.n | . . . . . . 7 β’ π = (LSpanβπ) | |
| 14 | 12, 6, 13 | lspsncl 20822 | . . . . . 6 β’ ((π β LMod β§ π β π) β (πβ{π}) β π) |
| 15 | 5, 11, 14 | syl2anc 583 | . . . . 5 β’ (π β (πβ{π}) β π) |
| 16 | 8, 15 | sseldd 3978 | . . . 4 β’ (π β (πβ{π}) β (SubGrpβπ)) |
| 17 | 2, 10, 16 | lssnle 19592 | . . 3 β’ (π β (Β¬ (πβ{π}) β π β π β (π β (πβ{π})))) |
| 18 | 1, 17 | mpbid 231 | . 2 β’ (π β π β (π β (πβ{π}))) |
| 19 | 3simpa 1145 | . . . . 5 β’ ((π β§ π₯ β π β§ (π β π₯ β§ π₯ β (π β (πβ{π})))) β (π β§ π₯ β π)) | |
| 20 | simp3l 1198 | . . . . 5 β’ ((π β§ π₯ β π β§ (π β π₯ β§ π₯ β (π β (πβ{π})))) β π β π₯) | |
| 21 | simp3r 1199 | . . . . 5 β’ ((π β§ π₯ β π β§ (π β π₯ β§ π₯ β (π β (πβ{π})))) β π₯ β (π β (πβ{π}))) | |
| 22 | 3 | adantr 480 | . . . . . 6 β’ ((π β§ π₯ β π) β π β LVec) |
| 23 | 9 | adantr 480 | . . . . . 6 β’ ((π β§ π₯ β π) β π β π) |
| 24 | simpr 484 | . . . . . 6 β’ ((π β§ π₯ β π) β π₯ β π) | |
| 25 | 11 | adantr 480 | . . . . . 6 β’ ((π β§ π₯ β π) β π β π) |
| 26 | 12, 6, 13, 2, 22, 23, 24, 25 | lsmcv 20990 | . . . . 5 β’ (((π β§ π₯ β π) β§ π β π₯ β§ π₯ β (π β (πβ{π}))) β π₯ = (π β (πβ{π}))) |
| 27 | 19, 20, 21, 26 | syl3anc 1368 | . . . 4 β’ ((π β§ π₯ β π β§ (π β π₯ β§ π₯ β (π β (πβ{π})))) β π₯ = (π β (πβ{π}))) |
| 28 | 27 | 3exp 1116 | . . 3 β’ (π β (π₯ β π β ((π β π₯ β§ π₯ β (π β (πβ{π}))) β π₯ = (π β (πβ{π}))))) |
| 29 | 28 | ralrimiv 3139 | . 2 β’ (π β βπ₯ β π ((π β π₯ β§ π₯ β (π β (πβ{π}))) β π₯ = (π β (πβ{π})))) |
| 30 | lsmcv2.c | . . 3 β’ πΆ = ( βL βπ) | |
| 31 | 6, 2 | lsmcl 20929 | . . . 4 β’ ((π β LMod β§ π β π β§ (πβ{π}) β π) β (π β (πβ{π})) β π) |
| 32 | 5, 9, 15, 31 | syl3anc 1368 | . . 3 β’ (π β (π β (πβ{π})) β π) |
| 33 | 6, 30, 3, 9, 32 | lcvbr2 38403 | . 2 β’ (π β (ππΆ(π β (πβ{π})) β (π β (π β (πβ{π})) β§ βπ₯ β π ((π β π₯ β§ π₯ β (π β (πβ{π}))) β π₯ = (π β (πβ{π})))))) |
| 34 | 18, 29, 33 | mpbir2and 710 | 1 β’ (π β ππΆ(π β (πβ{π}))) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3055 β wss 3943 β wpss 3944 {csn 4623 class class class wbr 5141 βcfv 6536 (class class class)co 7404 Basecbs 17151 SubGrpcsubg 19045 LSSumclsm 19552 LModclmod 20704 LSubSpclss 20776 LSpanclspn 20816 LVecclvec 20948 βL clcv 38399 |
| 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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8209 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19048 df-cntz 19231 df-lsm 19554 df-cmn 19700 df-abl 19701 df-mgp 20038 df-rng 20056 df-ur 20085 df-ring 20138 df-oppr 20234 df-dvdsr 20257 df-unit 20258 df-invr 20288 df-drng 20587 df-lmod 20706 df-lss 20777 df-lsp 20817 df-lvec 20949 df-lcv 38400 |
| This theorem is referenced by: lcv1 38422 |
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