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Mirrors > Home > MPE Home > Th. List > lbsext | Structured version Visualization version GIF version |
Description: For any linearly independent subset 𝐶 of 𝑉, there is a basis containing the vectors in 𝐶. (Contributed by Mario Carneiro, 25-Jun-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
Ref | Expression |
---|---|
lbsex.j | ⊢ 𝐽 = (LBasis‘𝑊) |
lbsex.v | ⊢ 𝑉 = (Base‘𝑊) |
lbsex.n | ⊢ 𝑁 = (LSpan‘𝑊) |
Ref | Expression |
---|---|
lbsext | ⊢ ((𝑊 ∈ LVec ∧ 𝐶 ⊆ 𝑉 ∧ ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))) → ∃𝑠 ∈ 𝐽 𝐶 ⊆ 𝑠) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lbsex.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
2 | 1 | fvexi 6853 | . . . . 5 ⊢ 𝑉 ∈ V |
3 | 2 | pwex 5333 | . . . 4 ⊢ 𝒫 𝑉 ∈ V |
4 | numth3 10364 | . . . 4 ⊢ (𝒫 𝑉 ∈ V → 𝒫 𝑉 ∈ dom card) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ 𝒫 𝑉 ∈ dom card |
6 | 5 | jctr 525 | . 2 ⊢ (𝑊 ∈ LVec → (𝑊 ∈ LVec ∧ 𝒫 𝑉 ∈ dom card)) |
7 | lbsex.j | . . 3 ⊢ 𝐽 = (LBasis‘𝑊) | |
8 | lbsex.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
9 | 7, 1, 8 | lbsextg 20570 | . 2 ⊢ (((𝑊 ∈ LVec ∧ 𝒫 𝑉 ∈ dom card) ∧ 𝐶 ⊆ 𝑉 ∧ ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))) → ∃𝑠 ∈ 𝐽 𝐶 ⊆ 𝑠) |
10 | 6, 9 | syl3an1 1163 | 1 ⊢ ((𝑊 ∈ LVec ∧ 𝐶 ⊆ 𝑉 ∧ ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))) → ∃𝑠 ∈ 𝐽 𝐶 ⊆ 𝑠) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3062 ∃wrex 3071 Vcvv 3443 ∖ cdif 3905 ⊆ wss 3908 𝒫 cpw 4558 {csn 4584 dom cdm 5631 ‘cfv 6493 cardccrd 9829 Basecbs 17037 LSpanclspn 20379 LBasisclbs 20482 LVecclvec 20510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-ac2 10357 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-rpss 7652 df-om 7795 df-1st 7913 df-2nd 7914 df-tpos 8149 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-oadd 8408 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-dju 9795 df-card 9833 df-ac 10010 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-3 12175 df-sets 16990 df-slot 17008 df-ndx 17020 df-base 17038 df-ress 17067 df-plusg 17100 df-mulr 17101 df-0g 17277 df-mgm 18451 df-sgrp 18500 df-mnd 18511 df-grp 18705 df-minusg 18706 df-sbg 18707 df-cmn 19517 df-abl 19518 df-mgp 19850 df-ur 19867 df-ring 19914 df-oppr 19996 df-dvdsr 20017 df-unit 20018 df-invr 20048 df-drng 20134 df-lmod 20271 df-lss 20340 df-lsp 20380 df-lbs 20483 df-lvec 20511 |
This theorem is referenced by: islinds4 21188 |
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