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Mirrors > Home > MPE Home > Th. List > islinds4 | Structured version Visualization version GIF version |
Description: A set is independent in a vector space iff it is a subset of some basis. This is an axiom of choice equivalent. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
Ref | Expression |
---|---|
islinds4.j | β’ π½ = (LBasisβπ) |
Ref | Expression |
---|---|
islinds4 | β’ (π β LVec β (π β (LIndSβπ) β βπ β π½ π β π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 482 | . . . 4 β’ ((π β LVec β§ π β (LIndSβπ)) β π β LVec) | |
2 | eqid 2726 | . . . . . 6 β’ (Baseβπ) = (Baseβπ) | |
3 | 2 | linds1 21705 | . . . . 5 β’ (π β (LIndSβπ) β π β (Baseβπ)) |
4 | 3 | adantl 481 | . . . 4 β’ ((π β LVec β§ π β (LIndSβπ)) β π β (Baseβπ)) |
5 | lveclmod 20954 | . . . . . . 7 β’ (π β LVec β π β LMod) | |
6 | 5 | ad2antrr 723 | . . . . . 6 β’ (((π β LVec β§ π β (LIndSβπ)) β§ π₯ β π) β π β LMod) |
7 | eqid 2726 | . . . . . . . . 9 β’ (Scalarβπ) = (Scalarβπ) | |
8 | 7 | lvecdrng 20953 | . . . . . . . 8 β’ (π β LVec β (Scalarβπ) β DivRing) |
9 | drngnzr 20607 | . . . . . . . 8 β’ ((Scalarβπ) β DivRing β (Scalarβπ) β NzRing) | |
10 | 8, 9 | syl 17 | . . . . . . 7 β’ (π β LVec β (Scalarβπ) β NzRing) |
11 | 10 | ad2antrr 723 | . . . . . 6 β’ (((π β LVec β§ π β (LIndSβπ)) β§ π₯ β π) β (Scalarβπ) β NzRing) |
12 | simplr 766 | . . . . . 6 β’ (((π β LVec β§ π β (LIndSβπ)) β§ π₯ β π) β π β (LIndSβπ)) | |
13 | simpr 484 | . . . . . 6 β’ (((π β LVec β§ π β (LIndSβπ)) β§ π₯ β π) β π₯ β π) | |
14 | eqid 2726 | . . . . . . 7 β’ (LSpanβπ) = (LSpanβπ) | |
15 | 14, 7 | lindsind2 21714 | . . . . . 6 β’ (((π β LMod β§ (Scalarβπ) β NzRing) β§ π β (LIndSβπ) β§ π₯ β π) β Β¬ π₯ β ((LSpanβπ)β(π β {π₯}))) |
16 | 6, 11, 12, 13, 15 | syl211anc 1373 | . . . . 5 β’ (((π β LVec β§ π β (LIndSβπ)) β§ π₯ β π) β Β¬ π₯ β ((LSpanβπ)β(π β {π₯}))) |
17 | 16 | ralrimiva 3140 | . . . 4 β’ ((π β LVec β§ π β (LIndSβπ)) β βπ₯ β π Β¬ π₯ β ((LSpanβπ)β(π β {π₯}))) |
18 | islinds4.j | . . . . 5 β’ π½ = (LBasisβπ) | |
19 | 18, 2, 14 | lbsext 21014 | . . . 4 β’ ((π β LVec β§ π β (Baseβπ) β§ βπ₯ β π Β¬ π₯ β ((LSpanβπ)β(π β {π₯}))) β βπ β π½ π β π) |
20 | 1, 4, 17, 19 | syl3anc 1368 | . . 3 β’ ((π β LVec β§ π β (LIndSβπ)) β βπ β π½ π β π) |
21 | 20 | ex 412 | . 2 β’ (π β LVec β (π β (LIndSβπ) β βπ β π½ π β π)) |
22 | 5 | ad2antrr 723 | . . . 4 β’ (((π β LVec β§ π β π½) β§ π β π) β π β LMod) |
23 | 18 | lbslinds 21728 | . . . . . 6 β’ π½ β (LIndSβπ) |
24 | 23 | sseli 3973 | . . . . 5 β’ (π β π½ β π β (LIndSβπ)) |
25 | 24 | ad2antlr 724 | . . . 4 β’ (((π β LVec β§ π β π½) β§ π β π) β π β (LIndSβπ)) |
26 | simpr 484 | . . . 4 β’ (((π β LVec β§ π β π½) β§ π β π) β π β π) | |
27 | lindsss 21719 | . . . 4 β’ ((π β LMod β§ π β (LIndSβπ) β§ π β π) β π β (LIndSβπ)) | |
28 | 22, 25, 26, 27 | syl3anc 1368 | . . 3 β’ (((π β LVec β§ π β π½) β§ π β π) β π β (LIndSβπ)) |
29 | 28 | rexlimdva2 3151 | . 2 β’ (π β LVec β (βπ β π½ π β π β π β (LIndSβπ))) |
30 | 21, 29 | impbid 211 | 1 β’ (π β LVec β (π β (LIndSβπ) β βπ β π½ π β π)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 βwral 3055 βwrex 3064 β cdif 3940 β wss 3943 {csn 4623 βcfv 6537 Basecbs 17153 Scalarcsca 17209 NzRingcnzr 20414 DivRingcdr 20587 LModclmod 20706 LSpanclspn 20818 LBasisclbs 20922 LVecclvec 20950 LIndSclinds 21700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-ac2 10460 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-rpss 7710 df-om 7853 df-1st 7974 df-2nd 7975 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-oadd 8471 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-dju 9898 df-card 9936 df-ac 10113 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-sbg 18868 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-nzr 20415 df-drng 20589 df-lmod 20708 df-lss 20779 df-lsp 20819 df-lbs 20923 df-lvec 20951 df-lindf 21701 df-linds 21702 |
This theorem is referenced by: lssdimle 33210 dimkerim 33230 |
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