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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 37487 | . . 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 3950 | 1 β’ (π β π β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β wss 3915 βcfv 6501 LModclmod 20338 LSubSpclss 20408 LSAtomsclsa 37465 |
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-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-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-plusg 17153 df-0g 17330 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-grp 18758 df-minusg 18759 df-sbg 18760 df-mgp 19904 df-ur 19921 df-ring 19973 df-lmod 20340 df-lss 20409 df-lsp 20449 df-lsatoms 37467 |
This theorem is referenced by: lsatssv 37489 lsatssn0 37493 lsatcmp 37494 lsatel 37496 lsatelbN 37497 lrelat 37505 lcvat 37521 lsatcv0 37522 lsatcveq0 37523 lcvp 37531 lcv1 37532 lcv2 37533 lsatexch 37534 lsatnem0 37536 lsatexch1 37537 lsatcv0eq 37538 lsatcv1 37539 lsatcvatlem 37540 lsatcvat 37541 lsatcvat2 37542 lsatcvat3 37543 l1cvat 37546 dochsat 39875 dihsmatrn 39928 dvh3dimatN 39931 dvh2dimatN 39932 dochsatshp 39943 dochexmidlem1 39952 dochexmidlem4 39955 dochexmidlem5 39956 dochexmidlem6 39957 dochexmidlem7 39958 lcfrlem29 40063 lcfrlem35 40069 mapd1dim2lem1N 40136 mapdcnvatN 40158 mapdat 40159 |
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