![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
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 2726 | . . . 4 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
2 | lsatel.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
3 | lsatel.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
4 | lveclmod 21086 | . . . . 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 38697 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑊)) |
9 | lsatel.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑈) | |
10 | 1, 2, 5, 8, 9 | ellspsn5 20975 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ 𝑈) |
11 | eqid 2726 | . . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
12 | 11, 1 | lssel 20916 | . . . . . 6 ⊢ ((𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (Base‘𝑊)) |
13 | 8, 9, 12 | syl2anc 582 | . . . . 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 38692 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ (Base‘𝑊) ∧ 𝑋 ≠ 0 ) → (𝑁‘{𝑋}) ∈ 𝐴) |
17 | 5, 13, 14, 16 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝐴) |
18 | 6, 3, 17, 7 | lsatcmp 38703 | . . 3 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊆ 𝑈 ↔ (𝑁‘{𝑋}) = 𝑈)) |
19 | 10, 18 | mpbid 231 | . 2 ⊢ (𝜑 → (𝑁‘{𝑋}) = 𝑈) |
20 | 19 | eqcomd 2732 | 1 ⊢ (𝜑 → 𝑈 = (𝑁‘{𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ⊆ wss 3947 {csn 4633 ‘cfv 6556 Basecbs 17215 0gc0g 17456 LModclmod 20838 LSubSpclss 20910 LSpanclspn 20950 LVecclvec 21082 LSAtomsclsa 38674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-int 4957 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8005 df-2nd 8006 df-tpos 8243 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-er 8736 df-en 8977 df-dom 8978 df-sdom 8979 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-nn 12267 df-2 12329 df-3 12330 df-sets 17168 df-slot 17186 df-ndx 17198 df-base 17216 df-ress 17245 df-plusg 17281 df-mulr 17282 df-0g 17458 df-mgm 18635 df-sgrp 18714 df-mnd 18730 df-grp 18933 df-minusg 18934 df-sbg 18935 df-cmn 19782 df-abl 19783 df-mgp 20120 df-rng 20138 df-ur 20167 df-ring 20220 df-oppr 20318 df-dvdsr 20341 df-unit 20342 df-invr 20372 df-drng 20711 df-lmod 20840 df-lss 20911 df-lsp 20951 df-lvec 21083 df-lsatoms 38676 |
This theorem is referenced by: lsatelbN 38706 lsat2el 38707 dihpN 41037 dochsnkr 41173 lcfrlem25 41268 lcfrlem35 41278 mapdpglem20 41392 |
Copyright terms: Public domain | W3C validator |