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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 2819 | . . . 4 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
2 | lsatel.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
3 | lsatel.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
4 | lveclmod 19870 | . . . . 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 36125 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑊)) |
9 | lsatel.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑈) | |
10 | 1, 2, 5, 8, 9 | lspsnel5a 19760 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ 𝑈) |
11 | eqid 2819 | . . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
12 | 11, 1 | lssel 19701 | . . . . . 6 ⊢ ((𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (Base‘𝑊)) |
13 | 8, 9, 12 | syl2anc 586 | . . . . 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 36120 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ (Base‘𝑊) ∧ 𝑋 ≠ 0 ) → (𝑁‘{𝑋}) ∈ 𝐴) |
17 | 5, 13, 14, 16 | syl3anc 1366 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝐴) |
18 | 6, 3, 17, 7 | lsatcmp 36131 | . . 3 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊆ 𝑈 ↔ (𝑁‘{𝑋}) = 𝑈)) |
19 | 10, 18 | mpbid 234 | . 2 ⊢ (𝜑 → (𝑁‘{𝑋}) = 𝑈) |
20 | 19 | eqcomd 2825 | 1 ⊢ (𝜑 → 𝑈 = (𝑁‘{𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 ⊆ wss 3934 {csn 4559 ‘cfv 6348 Basecbs 16475 0gc0g 16705 LModclmod 19626 LSubSpclss 19695 LSpanclspn 19735 LVecclvec 19866 LSAtomsclsa 36102 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-tpos 7884 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-cmn 18900 df-abl 18901 df-mgp 19232 df-ur 19244 df-ring 19291 df-oppr 19365 df-dvdsr 19383 df-unit 19384 df-invr 19414 df-drng 19496 df-lmod 19628 df-lss 19696 df-lsp 19736 df-lvec 19867 df-lsatoms 36104 |
This theorem is referenced by: lsatelbN 36134 lsat2el 36135 dihpN 38464 dochsnkr 38600 lcfrlem25 38695 lcfrlem35 38705 mapdpglem20 38819 |
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