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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 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑊 ∈ LVec) |
6 | lsatelb.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐴) | |
7 | 6 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑈 ∈ 𝐴) |
8 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑈) | |
9 | lsatelb.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
10 | eldifsn 4791 | . . . . . 6 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) | |
11 | 9, 10 | sylib 217 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) |
12 | 11 | simprd 495 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 0 ) |
13 | 12 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 ≠ 0 ) |
14 | 1, 2, 3, 5, 7, 8, 13 | lsatel 38477 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑈 = (𝑁‘{𝑋})) |
15 | eqimss2 4039 | . . . 4 ⊢ (𝑈 = (𝑁‘{𝑋}) → (𝑁‘{𝑋}) ⊆ 𝑈) | |
16 | 15 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑈 = (𝑁‘{𝑋})) → (𝑁‘{𝑋}) ⊆ 𝑈) |
17 | lsatelb.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
18 | eqid 2728 | . . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
19 | lveclmod 20990 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
20 | 4, 19 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
21 | 18, 3, 20, 6 | lsatlssel 38469 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑊)) |
22 | 9 | eldifad 3959 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
23 | 17, 18, 2, 20, 21, 22 | lspsnel5 20878 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑁‘{𝑋}) ⊆ 𝑈)) |
24 | 23 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑈 = (𝑁‘{𝑋})) → (𝑋 ∈ 𝑈 ↔ (𝑁‘{𝑋}) ⊆ 𝑈)) |
25 | 16, 24 | mpbird 257 | . 2 ⊢ ((𝜑 ∧ 𝑈 = (𝑁‘{𝑋})) → 𝑋 ∈ 𝑈) |
26 | 14, 25 | impbida 800 | 1 ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ 𝑈 = (𝑁‘{𝑋}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∖ cdif 3944 ⊆ wss 3947 {csn 4629 ‘cfv 6548 Basecbs 17179 0gc0g 17420 LModclmod 20742 LSubSpclss 20814 LSpanclspn 20854 LVecclvec 20986 LSAtomsclsa 38446 |
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 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-tpos 8231 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18892 df-minusg 18893 df-sbg 18894 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-oppr 20272 df-dvdsr 20295 df-unit 20296 df-invr 20326 df-drng 20625 df-lmod 20744 df-lss 20815 df-lsp 20855 df-lvec 20987 df-lsatoms 38448 |
This theorem is referenced by: (None) |
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