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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcvp | Structured version Visualization version GIF version |
Description: Covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 32100 analog.) (Contributed by NM, 10-Jan-2015.) |
Ref | Expression |
---|---|
lcvp.s | β’ π = (LSubSpβπ) |
lcvp.p | β’ β = (LSSumβπ) |
lcvp.o | β’ 0 = (0gβπ) |
lcvp.a | β’ π΄ = (LSAtomsβπ) |
lcvp.c | β’ πΆ = ( βL βπ) |
lcvp.w | β’ (π β π β LVec) |
lcvp.u | β’ (π β π β π) |
lcvp.q | β’ (π β π β π΄) |
Ref | Expression |
---|---|
lcvp | β’ (π β ((π β© π) = { 0 } β ππΆ(π β π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcvp.o | . . 3 β’ 0 = (0gβπ) | |
2 | lcvp.s | . . 3 β’ π = (LSubSpβπ) | |
3 | lcvp.a | . . 3 β’ π΄ = (LSAtomsβπ) | |
4 | lcvp.c | . . 3 β’ πΆ = ( βL βπ) | |
5 | lcvp.w | . . 3 β’ (π β π β LVec) | |
6 | lveclmod 20946 | . . . . 5 β’ (π β LVec β π β LMod) | |
7 | 5, 6 | syl 17 | . . . 4 β’ (π β π β LMod) |
8 | lcvp.u | . . . 4 β’ (π β π β π) | |
9 | lcvp.q | . . . . 5 β’ (π β π β π΄) | |
10 | 2, 3, 7, 9 | lsatlssel 38361 | . . . 4 β’ (π β π β π) |
11 | 2 | lssincl 20804 | . . . 4 β’ ((π β LMod β§ π β π β§ π β π) β (π β© π) β π) |
12 | 7, 8, 10, 11 | syl3anc 1368 | . . 3 β’ (π β (π β© π) β π) |
13 | 1, 2, 3, 4, 5, 12, 9 | lsatcveq0 38396 | . 2 β’ (π β ((π β© π)πΆπ β (π β© π) = { 0 })) |
14 | lcvp.p | . . 3 β’ β = (LSSumβπ) | |
15 | 2, 14, 4, 7, 8, 10 | lcvexch 38403 | . 2 β’ (π β ((π β© π)πΆπ β ππΆ(π β π))) |
16 | 13, 15 | bitr3d 281 | 1 β’ (π β ((π β© π) = { 0 } β ππΆ(π β π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1533 β wcel 2098 β© cin 3940 {csn 4621 class class class wbr 5139 βcfv 6534 (class class class)co 7402 0gc0g 17386 LSSumclsm 19546 LModclmod 20698 LSubSpclss 20770 LVecclvec 20942 LSAtomsclsa 38338 βL clcv 38382 |
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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-3 12274 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-mulr 17212 df-0g 17388 df-mre 17531 df-mrc 17532 df-acs 17534 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-subg 19042 df-cntz 19225 df-oppg 19254 df-lsm 19548 df-cmn 19694 df-abl 19695 df-mgp 20032 df-rng 20050 df-ur 20079 df-ring 20132 df-oppr 20228 df-dvdsr 20251 df-unit 20252 df-invr 20282 df-drng 20581 df-lmod 20700 df-lss 20771 df-lsp 20811 df-lvec 20943 df-lsatoms 38340 df-lcv 38383 |
This theorem is referenced by: lsatexch 38407 lsatnle 38408 lsatcv0eq 38411 lsatcvatlem 38413 |
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