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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatel | Structured version Visualization version GIF version |
Description: A nonzero vector in an atom determines the atom. (Contributed by NM, 25-Aug-2014.) |
Ref | Expression |
---|---|
lsatel.o | ⊢ 0 = (0g‘𝑊) |
lsatel.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lsatel.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lsatel.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lsatel.u | ⊢ (𝜑 → 𝑈 ∈ 𝐴) |
lsatel.x | ⊢ (𝜑 → 𝑋 ∈ 𝑈) |
lsatel.e | ⊢ (𝜑 → 𝑋 ≠ 0 ) |
Ref | Expression |
---|---|
lsatel | ⊢ (𝜑 → 𝑈 = (𝑁‘{𝑋})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2778 | . . . 4 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
2 | lsatel.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
3 | lsatel.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
4 | lveclmod 19505 | . . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) |
6 | lsatel.a | . . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
7 | lsatel.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝐴) | |
8 | 1, 6, 5, 7 | lsatlssel 35156 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑊)) |
9 | lsatel.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑈) | |
10 | 1, 2, 5, 8, 9 | lspsnel5a 19395 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ 𝑈) |
11 | eqid 2778 | . . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
12 | 11, 1 | lssel 19334 | . . . . . 6 ⊢ ((𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (Base‘𝑊)) |
13 | 8, 9, 12 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑊)) |
14 | lsatel.e | . . . . 5 ⊢ (𝜑 → 𝑋 ≠ 0 ) | |
15 | lsatel.o | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
16 | 11, 2, 15, 6 | lsatlspsn2 35151 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ (Base‘𝑊) ∧ 𝑋 ≠ 0 ) → (𝑁‘{𝑋}) ∈ 𝐴) |
17 | 5, 13, 14, 16 | syl3anc 1439 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝐴) |
18 | 6, 3, 17, 7 | lsatcmp 35162 | . . 3 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊆ 𝑈 ↔ (𝑁‘{𝑋}) = 𝑈)) |
19 | 10, 18 | mpbid 224 | . 2 ⊢ (𝜑 → (𝑁‘{𝑋}) = 𝑈) |
20 | 19 | eqcomd 2784 | 1 ⊢ (𝜑 → 𝑈 = (𝑁‘{𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ⊆ wss 3792 {csn 4398 ‘cfv 6137 Basecbs 16259 0gc0g 16490 LModclmod 19259 LSubSpclss 19328 LSpanclspn 19370 LVecclvec 19501 LSAtomsclsa 35133 |
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-int 4713 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-tpos 7636 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-3 11443 df-ndx 16262 df-slot 16263 df-base 16265 df-sets 16266 df-ress 16267 df-plusg 16355 df-mulr 16356 df-0g 16492 df-mgm 17632 df-sgrp 17674 df-mnd 17685 df-grp 17816 df-minusg 17817 df-sbg 17818 df-cmn 18585 df-abl 18586 df-mgp 18881 df-ur 18893 df-ring 18940 df-oppr 19014 df-dvdsr 19032 df-unit 19033 df-invr 19063 df-drng 19145 df-lmod 19261 df-lss 19329 df-lsp 19371 df-lvec 19502 df-lsatoms 35135 |
This theorem is referenced by: lsatelbN 35165 lsat2el 35166 dihpN 37495 dochsnkr 37631 lcfrlem25 37726 lcfrlem35 37736 mapdpglem20 37850 |
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