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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatelbN | Structured version Visualization version GIF version |
Description: A nonzero vector in an atom determines the atom. (Contributed by NM, 3-Feb-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lsatelb.v | ⊢ 𝑉 = (Base‘𝑊) |
lsatelb.o | ⊢ 0 = (0g‘𝑊) |
lsatelb.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lsatelb.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lsatelb.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lsatelb.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
lsatelb.u | ⊢ (𝜑 → 𝑈 ∈ 𝐴) |
Ref | Expression |
---|---|
lsatelbN | ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ 𝑈 = (𝑁‘{𝑋}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatelb.o | . . 3 ⊢ 0 = (0g‘𝑊) | |
2 | lsatelb.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
3 | lsatelb.a | . . 3 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
4 | lsatelb.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
5 | 4 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑊 ∈ LVec) |
6 | lsatelb.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐴) | |
7 | 6 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑈 ∈ 𝐴) |
8 | simpr 477 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑈) | |
9 | lsatelb.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
10 | eldifsn 4593 | . . . . . 6 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) | |
11 | 9, 10 | sylib 210 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) |
12 | 11 | simprd 488 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 0 ) |
13 | 12 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 ≠ 0 ) |
14 | 1, 2, 3, 5, 7, 8, 13 | lsatel 35583 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑈 = (𝑁‘{𝑋})) |
15 | eqimss2 3915 | . . . 4 ⊢ (𝑈 = (𝑁‘{𝑋}) → (𝑁‘{𝑋}) ⊆ 𝑈) | |
16 | 15 | adantl 474 | . . 3 ⊢ ((𝜑 ∧ 𝑈 = (𝑁‘{𝑋})) → (𝑁‘{𝑋}) ⊆ 𝑈) |
17 | lsatelb.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
18 | eqid 2779 | . . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
19 | lveclmod 19600 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
20 | 4, 19 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
21 | 18, 3, 20, 6 | lsatlssel 35575 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑊)) |
22 | 9 | eldifad 3842 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
23 | 17, 18, 2, 20, 21, 22 | lspsnel5 19489 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑁‘{𝑋}) ⊆ 𝑈)) |
24 | 23 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝑈 = (𝑁‘{𝑋})) → (𝑋 ∈ 𝑈 ↔ (𝑁‘{𝑋}) ⊆ 𝑈)) |
25 | 16, 24 | mpbird 249 | . 2 ⊢ ((𝜑 ∧ 𝑈 = (𝑁‘{𝑋})) → 𝑋 ∈ 𝑈) |
26 | 14, 25 | impbida 788 | 1 ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ 𝑈 = (𝑁‘{𝑋}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ≠ wne 2968 ∖ cdif 3827 ⊆ wss 3830 {csn 4441 ‘cfv 6188 Basecbs 16339 0gc0g 16569 LModclmod 19356 LSubSpclss 19425 LSpanclspn 19465 LVecclvec 19596 LSAtomsclsa 35552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-tpos 7695 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-3 11504 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-mulr 16435 df-0g 16571 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-grp 17894 df-minusg 17895 df-sbg 17896 df-cmn 18668 df-abl 18669 df-mgp 18963 df-ur 18975 df-ring 19022 df-oppr 19096 df-dvdsr 19114 df-unit 19115 df-invr 19145 df-drng 19227 df-lmod 19358 df-lss 19426 df-lsp 19466 df-lvec 19597 df-lsatoms 35554 |
This theorem is referenced by: (None) |
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