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Mirrors > Home > MPE Home > Th. List > lss0cl | Structured version Visualization version GIF version |
Description: The zero vector belongs to every subspace. (Contributed by NM, 12-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lss0cl.z | ⊢ 0 = (0g‘𝑊) |
lss0cl.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
Ref | Expression |
---|---|
lss0cl | ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 0 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lss0cl.s | . . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊) | |
2 | 1 | lssn0 19337 | . . . 4 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ≠ ∅) |
3 | n0 4159 | . . . 4 ⊢ (𝑈 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑈) | |
4 | 2, 3 | sylib 210 | . . 3 ⊢ (𝑈 ∈ 𝑆 → ∃𝑥 𝑥 ∈ 𝑈) |
5 | 4 | adantl 475 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → ∃𝑥 𝑥 ∈ 𝑈) |
6 | simp1 1127 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑥 ∈ 𝑈) → 𝑊 ∈ LMod) | |
7 | eqid 2778 | . . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
8 | 7, 1 | lssel 19334 | . . . . . . 7 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ (Base‘𝑊)) |
9 | 8 | 3adant1 1121 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ (Base‘𝑊)) |
10 | lss0cl.z | . . . . . . 7 ⊢ 0 = (0g‘𝑊) | |
11 | eqid 2778 | . . . . . . 7 ⊢ (-g‘𝑊) = (-g‘𝑊) | |
12 | 7, 10, 11 | lmodsubid 19319 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑥(-g‘𝑊)𝑥) = 0 ) |
13 | 6, 9, 12 | syl2anc 579 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑥 ∈ 𝑈) → (𝑥(-g‘𝑊)𝑥) = 0 ) |
14 | 11, 1 | lssvsubcl 19340 | . . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑥 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈)) → (𝑥(-g‘𝑊)𝑥) ∈ 𝑈) |
15 | 14 | anabsan2 664 | . . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ 𝑈) → (𝑥(-g‘𝑊)𝑥) ∈ 𝑈) |
16 | 15 | 3impa 1097 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑥 ∈ 𝑈) → (𝑥(-g‘𝑊)𝑥) ∈ 𝑈) |
17 | 13, 16 | eqeltrrd 2860 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑥 ∈ 𝑈) → 0 ∈ 𝑈) |
18 | 17 | 3expia 1111 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑥 ∈ 𝑈 → 0 ∈ 𝑈)) |
19 | 18 | exlimdv 1976 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (∃𝑥 𝑥 ∈ 𝑈 → 0 ∈ 𝑈)) |
20 | 5, 19 | mpd 15 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 0 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∃wex 1823 ∈ wcel 2107 ≠ wne 2969 ∅c0 4141 ‘cfv 6137 (class class class)co 6924 Basecbs 16259 0gc0g 16490 -gcsg 17815 LModclmod 19259 LSubSpclss 19328 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11379 df-2 11442 df-ndx 16262 df-slot 16263 df-base 16265 df-sets 16266 df-plusg 16355 df-0g 16492 df-mgm 17632 df-sgrp 17674 df-mnd 17685 df-grp 17816 df-minusg 17817 df-sbg 17818 df-mgp 18881 df-ur 18893 df-ring 18940 df-lmod 19261 df-lss 19329 |
This theorem is referenced by: lss0ss 19345 lssvneln0 19348 lssneln0OLD 19350 lssssr 19351 lssssrOLD 19352 lssvscl 19354 lssintcl 19363 lssvs0or 19509 lspsolvlem 19542 lidl0cl 19613 frlmgsum 20519 frlmsslsp 20543 0ellsp 30435 lssats 35171 dia2dimlem7 37229 dochfl1 37635 lcfr 37744 mapdval2N 37789 mapdrvallem2 37804 mapdpglem6 37837 mapdpglem12 37842 |
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