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Mathbox for Norm Megill |
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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 31277 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 20583 | . . . . . . 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 20435 | . . . . . 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 3950 | . . . 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 20454 | . . . . . 6 β’ ((π β LMod β§ π β π) β (πβ{π}) β π) |
15 | 5, 11, 14 | syl2anc 585 | . . . . 5 β’ (π β (πβ{π}) β π) |
16 | 8, 15 | sseldd 3950 | . . . 4 β’ (π β (πβ{π}) β (SubGrpβπ)) |
17 | 2, 10, 16 | lssnle 19463 | . . 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 1149 | . . . . 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 1202 | . . . . 5 β’ ((π β§ π₯ β π β§ (π β π₯ β§ π₯ β (π β (πβ{π})))) β π β π₯) | |
21 | simp3r 1203 | . . . . 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 482 | . . . . . 6 β’ ((π β§ π₯ β π) β π β LVec) |
23 | 9 | adantr 482 | . . . . . 6 β’ ((π β§ π₯ β π) β π β π) |
24 | simpr 486 | . . . . . 6 β’ ((π β§ π₯ β π) β π₯ β π) | |
25 | 11 | adantr 482 | . . . . . 6 β’ ((π β§ π₯ β π) β π β π) |
26 | 12, 6, 13, 2, 22, 23, 24, 25 | lsmcv 20618 | . . . . 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 1372 | . . . 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 1120 | . . 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 3143 | . 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 20560 | . . . 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 1372 | . . 3 β’ (π β (π β (πβ{π})) β π) |
33 | 6, 30, 3, 9, 32 | lcvbr2 37513 | . 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 712 | 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3065 β wss 3915 β wpss 3916 {csn 4591 class class class wbr 5110 βcfv 6501 (class class class)co 7362 Basecbs 17090 SubGrpcsubg 18929 LSSumclsm 19423 LModclmod 20338 LSubSpclss 20408 LSpanclspn 20448 LVecclvec 20579 βL clcv 37509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-tpos 8162 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-0g 17330 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-submnd 18609 df-grp 18758 df-minusg 18759 df-sbg 18760 df-subg 18932 df-cntz 19104 df-lsm 19425 df-cmn 19571 df-abl 19572 df-mgp 19904 df-ur 19921 df-ring 19973 df-oppr 20056 df-dvdsr 20077 df-unit 20078 df-invr 20108 df-drng 20201 df-lmod 20340 df-lss 20409 df-lsp 20449 df-lvec 20580 df-lcv 37510 |
This theorem is referenced by: lcv1 37532 |
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