![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
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 | ⊢ 𝑆 = (LSubSp‘𝑊) |
lsatlss.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lssatssel.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lssatssel.u | ⊢ (𝜑 → 𝑈 ∈ 𝐴) |
Ref | Expression |
---|---|
lsatlssel | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lssatssel.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
2 | lsatlss.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | lsatlss.a | . . . 4 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
4 | 2, 3 | lsatlss 35070 | . . 3 ⊢ (𝑊 ∈ LMod → 𝐴 ⊆ 𝑆) |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝑆) |
6 | lssatssel.u | . 2 ⊢ (𝜑 → 𝑈 ∈ 𝐴) | |
7 | 5, 6 | sseldd 3827 | 1 ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ⊆ wss 3797 ‘cfv 6122 LModclmod 19218 LSubSpclss 19287 LSAtomsclsa 35048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-om 7326 df-1st 7427 df-2nd 7428 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-er 8008 df-en 8222 df-dom 8223 df-sdom 8224 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-nn 11350 df-2 11413 df-ndx 16224 df-slot 16225 df-base 16227 df-sets 16228 df-plusg 16317 df-0g 16454 df-mgm 17594 df-sgrp 17636 df-mnd 17647 df-grp 17778 df-minusg 17779 df-sbg 17780 df-mgp 18843 df-ur 18855 df-ring 18902 df-lmod 19220 df-lss 19288 df-lsp 19330 df-lsatoms 35050 |
This theorem is referenced by: lsatssv 35072 lsatssn0 35076 lsatcmp 35077 lsatel 35079 lsatelbN 35080 lrelat 35088 lcvat 35104 lsatcv0 35105 lsatcveq0 35106 lcvp 35114 lcv1 35115 lcv2 35116 lsatexch 35117 lsatnem0 35119 lsatexch1 35120 lsatcv0eq 35121 lsatcv1 35122 lsatcvatlem 35123 lsatcvat 35124 lsatcvat2 35125 lsatcvat3 35126 l1cvat 35129 dochsat 37457 dihsmatrn 37510 dvh3dimatN 37513 dvh2dimatN 37514 dochsatshp 37525 dochexmidlem1 37534 dochexmidlem4 37537 dochexmidlem5 37538 dochexmidlem6 37539 dochexmidlem7 37540 lcfrlem29 37645 lcfrlem35 37651 mapd1dim2lem1N 37718 mapdcnvatN 37740 mapdat 37741 |
Copyright terms: Public domain | W3C validator |