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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 20057. (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 2739 | . . 3 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
4 | 1, 2, 3 | islbs2 20057 | . 2 ⊢ (𝑊 ∈ LVec → (𝑆 ∈ 𝐽 ↔ (𝑆 ⊆ 𝑋 ∧ ((LSpan‘𝑊)‘𝑆) = 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ ((LSpan‘𝑊)‘(𝑆 ∖ {𝑥}))))) |
5 | lveclmod 20009 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
6 | lbsacsbs.1 | . . . . . . 7 ⊢ 𝐴 = (LSubSp‘𝑊) | |
7 | lbsacsbs.2 | . . . . . . 7 ⊢ 𝑁 = (mrCls‘𝐴) | |
8 | 6, 3, 7 | mrclsp 19892 | . . . . . 6 ⊢ (𝑊 ∈ LMod → (LSpan‘𝑊) = 𝑁) |
9 | 5, 8 | syl 17 | . . . . 5 ⊢ (𝑊 ∈ LVec → (LSpan‘𝑊) = 𝑁) |
10 | 9 | fveq1d 6688 | . . . 4 ⊢ (𝑊 ∈ LVec → ((LSpan‘𝑊)‘𝑆) = (𝑁‘𝑆)) |
11 | 10 | eqeq1d 2741 | . . 3 ⊢ (𝑊 ∈ LVec → (((LSpan‘𝑊)‘𝑆) = 𝑋 ↔ (𝑁‘𝑆) = 𝑋)) |
12 | 9 | fveq1d 6688 | . . . . . 6 ⊢ (𝑊 ∈ LVec → ((LSpan‘𝑊)‘(𝑆 ∖ {𝑥})) = (𝑁‘(𝑆 ∖ {𝑥}))) |
13 | 12 | eleq2d 2819 | . . . . 5 ⊢ (𝑊 ∈ LVec → (𝑥 ∈ ((LSpan‘𝑊)‘(𝑆 ∖ {𝑥})) ↔ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
14 | 13 | notbid 321 | . . . 4 ⊢ (𝑊 ∈ LVec → (¬ 𝑥 ∈ ((LSpan‘𝑊)‘(𝑆 ∖ {𝑥})) ↔ ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
15 | 14 | ralbidv 3110 | . . 3 ⊢ (𝑊 ∈ LVec → (∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ ((LSpan‘𝑊)‘(𝑆 ∖ {𝑥})) ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
16 | 11, 15 | 3anbi23d 1440 | . 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 1098 | . . 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 19869 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝐴 ∈ (Moore‘𝑋)) |
19 | lbsacsbs.4 | . . . . . 6 ⊢ 𝐼 = (mrInd‘𝐴) | |
20 | 7, 19 | ismri 17017 | . . . . 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 293 | . 2 ⊢ (𝑊 ∈ LVec → ((𝑆 ⊆ 𝑋 ∧ (𝑁‘𝑆) = 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))) ↔ (𝑆 ∈ 𝐼 ∧ (𝑁‘𝑆) = 𝑋))) |
24 | 4, 16, 23 | 3bitrd 308 | 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 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∀wral 3054 ∖ cdif 3850 ⊆ wss 3853 {csn 4526 ‘cfv 6349 Basecbs 16598 Moorecmre 16968 mrClscmrc 16969 mrIndcmri 16970 LModclmod 19765 LSubSpclss 19834 LSpanclspn 19874 LBasisclbs 19977 LVecclvec 20005 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-1st 7726 df-2nd 7727 df-tpos 7933 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-nn 11729 df-2 11791 df-3 11792 df-ndx 16601 df-slot 16602 df-base 16604 df-sets 16605 df-ress 16606 df-plusg 16693 df-mulr 16694 df-0g 16830 df-mre 16972 df-mrc 16973 df-mri 16974 df-mgm 17980 df-sgrp 18029 df-mnd 18040 df-grp 18234 df-minusg 18235 df-sbg 18236 df-mgp 19371 df-ur 19383 df-ring 19430 df-oppr 19507 df-dvdsr 19525 df-unit 19526 df-invr 19556 df-drng 19635 df-lmod 19767 df-lss 19835 df-lsp 19875 df-lbs 19978 df-lvec 20006 |
This theorem is referenced by: lvecdim 20060 lvecdimfi 31267 |
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