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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatlssel | Structured version Visualization version GIF version |
Description: An atom is a subspace. (Contributed by NM, 25-Aug-2014.) |
Ref | Expression |
---|---|
lsatlss.s | β’ π = (LSubSpβπ) |
lsatlss.a | β’ π΄ = (LSAtomsβπ) |
lssatssel.w | β’ (π β π β LMod) |
lssatssel.u | β’ (π β π β π΄) |
Ref | Expression |
---|---|
lsatlssel | β’ (π β π β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lssatssel.w | . . 3 β’ (π β π β LMod) | |
2 | lsatlss.s | . . . 4 β’ π = (LSubSpβπ) | |
3 | lsatlss.a | . . . 4 β’ π΄ = (LSAtomsβπ) | |
4 | 2, 3 | lsatlss 38524 | . . 3 β’ (π β LMod β π΄ β π) |
5 | 1, 4 | syl 17 | . 2 β’ (π β π΄ β π) |
6 | lssatssel.u | . 2 β’ (π β π β π΄) | |
7 | 5, 6 | sseldd 3973 | 1 β’ (π β π β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wss 3939 βcfv 6543 LModclmod 20747 LSubSpclss 20819 LSAtomsclsa 38502 |
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 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
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 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-minusg 18898 df-sbg 18899 df-mgp 20079 df-ur 20126 df-ring 20179 df-lmod 20749 df-lss 20820 df-lsp 20860 df-lsatoms 38504 |
This theorem is referenced by: lsatssv 38526 lsatssn0 38530 lsatcmp 38531 lsatel 38533 lsatelbN 38534 lrelat 38542 lcvat 38558 lsatcv0 38559 lsatcveq0 38560 lcvp 38568 lcv1 38569 lcv2 38570 lsatexch 38571 lsatnem0 38573 lsatexch1 38574 lsatcv0eq 38575 lsatcv1 38576 lsatcvatlem 38577 lsatcvat 38578 lsatcvat2 38579 lsatcvat3 38580 l1cvat 38583 dochsat 40912 dihsmatrn 40965 dvh3dimatN 40968 dvh2dimatN 40969 dochsatshp 40980 dochexmidlem1 40989 dochexmidlem4 40992 dochexmidlem5 40993 dochexmidlem6 40994 dochexmidlem7 40995 lcfrlem29 41100 lcfrlem35 41106 mapd1dim2lem1N 41173 mapdcnvatN 41195 mapdat 41196 |
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