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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 | ⊢ 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 34601 | . . 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 3637 | 1 ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 ⊆ wss 3607 ‘cfv 5926 LModclmod 18911 LSubSpclss 18980 LSAtomsclsa 34579 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-plusg 16001 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-minusg 17473 df-sbg 17474 df-mgp 18536 df-ur 18548 df-ring 18595 df-lmod 18913 df-lss 18981 df-lsp 19020 df-lsatoms 34581 |
This theorem is referenced by: lsatssv 34603 lsatssn0 34607 lsatcmp 34608 lsatel 34610 lsatelbN 34611 lrelat 34619 lcvat 34635 lsatcv0 34636 lsatcveq0 34637 lcvp 34645 lcv1 34646 lcv2 34647 lsatexch 34648 lsatnem0 34650 lsatexch1 34651 lsatcv0eq 34652 lsatcv1 34653 lsatcvatlem 34654 lsatcvat 34655 lsatcvat2 34656 lsatcvat3 34657 l1cvat 34660 dochsat 36989 dihsmatrn 37042 dvh3dimatN 37045 dvh2dimatN 37046 dochsatshp 37057 dochexmidlem1 37066 dochexmidlem4 37069 dochexmidlem5 37070 dochexmidlem6 37071 dochexmidlem7 37072 lcfrlem29 37177 lcfrlem35 37183 mapd1dim2lem1N 37250 mapdcnvatN 37272 mapdat 37273 |
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