Nearby theorems
|
|
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
| 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 32121 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 20996 | . . . . . . 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 20847 | . . . . . 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 3981 | . . . 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 20866 | . . . . . 6 β’ ((π β LMod β§ π β π) β (πβ{π}) β π) |
| 15 | 5, 11, 14 | syl2anc 582 | . . . . 5 β’ (π β (πβ{π}) β π) |
| 16 | 8, 15 | sseldd 3981 | . . . 4 β’ (π β (πβ{π}) β (SubGrpβπ)) |
| 17 | 2, 10, 16 | lssnle 19634 | . . 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 479 | . . . . . 6 β’ ((π β§ π₯ β π) β π β LVec) |
| 23 | 9 | adantr 479 | . . . . . 6 β’ ((π β§ π₯ β π) β π β π) |
| 24 | simpr 483 | . . . . . 6 β’ ((π β§ π₯ β π) β π₯ β π) | |
| 25 | 11 | adantr 479 | . . . . . 6 β’ ((π β§ π₯ β π) β π β π) |
| 26 | 12, 6, 13, 2, 22, 23, 24, 25 | lsmcv 21034 | . . . . 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 3141 | . 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 20973 | . . . 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 38498 | . 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 711 | 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 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3057 β wss 3947 β wpss 3948 {csn 4630 class class class wbr 5150 βcfv 6551 (class class class)co 7424 Basecbs 17185 SubGrpcsubg 19080 LSSumclsm 19594 LModclmod 20748 LSubSpclss 20820 LSpanclspn 20860 LVecclvec 20992 βL clcv 38494 |
| 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 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 |
| 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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-tpos 8236 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-2 12311 df-3 12312 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-mulr 17252 df-0g 17428 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-submnd 18746 df-grp 18898 df-minusg 18899 df-sbg 18900 df-subg 19083 df-cntz 19273 df-lsm 19596 df-cmn 19742 df-abl 19743 df-mgp 20080 df-rng 20098 df-ur 20127 df-ring 20180 df-oppr 20278 df-dvdsr 20301 df-unit 20302 df-invr 20332 df-drng 20631 df-lmod 20750 df-lss 20821 df-lsp 20861 df-lvec 20993 df-lcv 38495 |
| This theorem is referenced by: lcv1 38517 |
| Copyright terms: Public domain | W3C validator |