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 4721 | . . . . . 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 36143 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑈 = (𝑁‘{𝑋})) |
15 | eqimss2 4026 | . . . 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 2823 | . . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
19 | lveclmod 19880 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
20 | 4, 19 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
21 | 18, 3, 20, 6 | lsatlssel 36135 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑊)) |
22 | 9 | eldifad 3950 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
23 | 17, 18, 2, 20, 21, 22 | lspsnel5 19769 | . . . 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 1537 ∈ wcel 2114 ≠ wne 3018 ∖ cdif 3935 ⊆ wss 3938 {csn 4569 ‘cfv 6357 Basecbs 16485 0gc0g 16715 LModclmod 19636 LSubSpclss 19705 LSpanclspn 19745 LVecclvec 19876 LSAtomsclsa 36112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-sbg 18110 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-drng 19506 df-lmod 19638 df-lss 19706 df-lsp 19746 df-lvec 19877 df-lsatoms 36114 |
This theorem is referenced by: (None) |
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