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 483 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑊 ∈ LVec) |
6 | lsatelb.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐴) | |
7 | 6 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑈 ∈ 𝐴) |
8 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑈) | |
9 | lsatelb.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
10 | eldifsn 4713 | . . . . . 6 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) | |
11 | 9, 10 | sylib 220 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) |
12 | 11 | simprd 498 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 0 ) |
13 | 12 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 ≠ 0 ) |
14 | 1, 2, 3, 5, 7, 8, 13 | lsatel 36135 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑈 = (𝑁‘{𝑋})) |
15 | eqimss2 4024 | . . . 4 ⊢ (𝑈 = (𝑁‘{𝑋}) → (𝑁‘{𝑋}) ⊆ 𝑈) | |
16 | 15 | adantl 484 | . . 3 ⊢ ((𝜑 ∧ 𝑈 = (𝑁‘{𝑋})) → (𝑁‘{𝑋}) ⊆ 𝑈) |
17 | lsatelb.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
18 | eqid 2821 | . . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
19 | lveclmod 19872 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
20 | 4, 19 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
21 | 18, 3, 20, 6 | lsatlssel 36127 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑊)) |
22 | 9 | eldifad 3948 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
23 | 17, 18, 2, 20, 21, 22 | lspsnel5 19761 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑁‘{𝑋}) ⊆ 𝑈)) |
24 | 23 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑈 = (𝑁‘{𝑋})) → (𝑋 ∈ 𝑈 ↔ (𝑁‘{𝑋}) ⊆ 𝑈)) |
25 | 16, 24 | mpbird 259 | . 2 ⊢ ((𝜑 ∧ 𝑈 = (𝑁‘{𝑋})) → 𝑋 ∈ 𝑈) |
26 | 14, 25 | impbida 799 | 1 ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ 𝑈 = (𝑁‘{𝑋}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∖ cdif 3933 ⊆ wss 3936 {csn 4561 ‘cfv 6350 Basecbs 16477 0gc0g 16707 LModclmod 19628 LSubSpclss 19697 LSpanclspn 19737 LVecclvec 19868 LSAtomsclsa 36104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-sbg 18102 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-drng 19498 df-lmod 19630 df-lss 19698 df-lsp 19738 df-lvec 19869 df-lsatoms 36106 |
This theorem is referenced by: (None) |
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