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Mirrors > Home > MPE Home > Th. List > lbsacsbs | Structured version Visualization version GIF version |
Description: Being a basis in a vector space is equivalent to being a basis in the associated algebraic closure system. Equivalent to islbs2 19979. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
lbsacsbs.1 | ⊢ 𝐴 = (LSubSp‘𝑊) |
lbsacsbs.2 | ⊢ 𝑁 = (mrCls‘𝐴) |
lbsacsbs.3 | ⊢ 𝑋 = (Base‘𝑊) |
lbsacsbs.4 | ⊢ 𝐼 = (mrInd‘𝐴) |
lbsacsbs.5 | ⊢ 𝐽 = (LBasis‘𝑊) |
Ref | Expression |
---|---|
lbsacsbs | ⊢ (𝑊 ∈ LVec → (𝑆 ∈ 𝐽 ↔ (𝑆 ∈ 𝐼 ∧ (𝑁‘𝑆) = 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lbsacsbs.3 | . . 3 ⊢ 𝑋 = (Base‘𝑊) | |
2 | lbsacsbs.5 | . . 3 ⊢ 𝐽 = (LBasis‘𝑊) | |
3 | eqid 2759 | . . 3 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
4 | 1, 2, 3 | islbs2 19979 | . 2 ⊢ (𝑊 ∈ LVec → (𝑆 ∈ 𝐽 ↔ (𝑆 ⊆ 𝑋 ∧ ((LSpan‘𝑊)‘𝑆) = 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ ((LSpan‘𝑊)‘(𝑆 ∖ {𝑥}))))) |
5 | lveclmod 19931 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
6 | lbsacsbs.1 | . . . . . . 7 ⊢ 𝐴 = (LSubSp‘𝑊) | |
7 | lbsacsbs.2 | . . . . . . 7 ⊢ 𝑁 = (mrCls‘𝐴) | |
8 | 6, 3, 7 | mrclsp 19814 | . . . . . 6 ⊢ (𝑊 ∈ LMod → (LSpan‘𝑊) = 𝑁) |
9 | 5, 8 | syl 17 | . . . . 5 ⊢ (𝑊 ∈ LVec → (LSpan‘𝑊) = 𝑁) |
10 | 9 | fveq1d 6653 | . . . 4 ⊢ (𝑊 ∈ LVec → ((LSpan‘𝑊)‘𝑆) = (𝑁‘𝑆)) |
11 | 10 | eqeq1d 2761 | . . 3 ⊢ (𝑊 ∈ LVec → (((LSpan‘𝑊)‘𝑆) = 𝑋 ↔ (𝑁‘𝑆) = 𝑋)) |
12 | 9 | fveq1d 6653 | . . . . . 6 ⊢ (𝑊 ∈ LVec → ((LSpan‘𝑊)‘(𝑆 ∖ {𝑥})) = (𝑁‘(𝑆 ∖ {𝑥}))) |
13 | 12 | eleq2d 2836 | . . . . 5 ⊢ (𝑊 ∈ LVec → (𝑥 ∈ ((LSpan‘𝑊)‘(𝑆 ∖ {𝑥})) ↔ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
14 | 13 | notbid 322 | . . . 4 ⊢ (𝑊 ∈ LVec → (¬ 𝑥 ∈ ((LSpan‘𝑊)‘(𝑆 ∖ {𝑥})) ↔ ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
15 | 14 | ralbidv 3124 | . . 3 ⊢ (𝑊 ∈ LVec → (∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ ((LSpan‘𝑊)‘(𝑆 ∖ {𝑥})) ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
16 | 11, 15 | 3anbi23d 1437 | . 2 ⊢ (𝑊 ∈ LVec → ((𝑆 ⊆ 𝑋 ∧ ((LSpan‘𝑊)‘𝑆) = 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ ((LSpan‘𝑊)‘(𝑆 ∖ {𝑥}))) ↔ (𝑆 ⊆ 𝑋 ∧ (𝑁‘𝑆) = 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))) |
17 | 3anan32 1095 | . . 3 ⊢ ((𝑆 ⊆ 𝑋 ∧ (𝑁‘𝑆) = 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))) ↔ ((𝑆 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))) ∧ (𝑁‘𝑆) = 𝑋)) | |
18 | 1, 6 | lssmre 19791 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝐴 ∈ (Moore‘𝑋)) |
19 | lbsacsbs.4 | . . . . . 6 ⊢ 𝐼 = (mrInd‘𝐴) | |
20 | 7, 19 | ismri 16945 | . . . . 5 ⊢ (𝐴 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝐼 ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))) |
21 | 5, 18, 20 | 3syl 18 | . . . 4 ⊢ (𝑊 ∈ LVec → (𝑆 ∈ 𝐼 ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))) |
22 | 21 | anbi1d 633 | . . 3 ⊢ (𝑊 ∈ LVec → ((𝑆 ∈ 𝐼 ∧ (𝑁‘𝑆) = 𝑋) ↔ ((𝑆 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))) ∧ (𝑁‘𝑆) = 𝑋))) |
23 | 17, 22 | bitr4id 294 | . 2 ⊢ (𝑊 ∈ LVec → ((𝑆 ⊆ 𝑋 ∧ (𝑁‘𝑆) = 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))) ↔ (𝑆 ∈ 𝐼 ∧ (𝑁‘𝑆) = 𝑋))) |
24 | 4, 16, 23 | 3bitrd 309 | 1 ⊢ (𝑊 ∈ LVec → (𝑆 ∈ 𝐽 ↔ (𝑆 ∈ 𝐼 ∧ (𝑁‘𝑆) = 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1085 = wceq 1539 ∈ wcel 2112 ∀wral 3068 ∖ cdif 3851 ⊆ wss 3854 {csn 4515 ‘cfv 6328 Basecbs 16526 Moorecmre 16896 mrClscmrc 16897 mrIndcmri 16898 LModclmod 19687 LSubSpclss 19756 LSpanclspn 19796 LBasisclbs 19899 LVecclvec 19927 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-int 4832 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-1st 7686 df-2nd 7687 df-tpos 7895 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-nn 11660 df-2 11722 df-3 11723 df-ndx 16529 df-slot 16530 df-base 16532 df-sets 16533 df-ress 16534 df-plusg 16621 df-mulr 16622 df-0g 16758 df-mre 16900 df-mrc 16901 df-mri 16902 df-mgm 17903 df-sgrp 17952 df-mnd 17963 df-grp 18157 df-minusg 18158 df-sbg 18159 df-mgp 19293 df-ur 19305 df-ring 19352 df-oppr 19429 df-dvdsr 19447 df-unit 19448 df-invr 19478 df-drng 19557 df-lmod 19689 df-lss 19757 df-lsp 19797 df-lbs 19900 df-lvec 19928 |
This theorem is referenced by: lvecdim 19982 lvecdimfi 31189 |
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