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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 484 | . . . 4 β’ ((π β LVec β§ π β (LIndSβπ)) β π β LVec) | |
2 | eqid 2733 | . . . . . 6 β’ (Baseβπ) = (Baseβπ) | |
3 | 2 | linds1 21232 | . . . . 5 β’ (π β (LIndSβπ) β π β (Baseβπ)) |
4 | 3 | adantl 483 | . . . 4 β’ ((π β LVec β§ π β (LIndSβπ)) β π β (Baseβπ)) |
5 | lveclmod 20582 | . . . . . . 7 β’ (π β LVec β π β LMod) | |
6 | 5 | ad2antrr 725 | . . . . . 6 β’ (((π β LVec β§ π β (LIndSβπ)) β§ π₯ β π) β π β LMod) |
7 | eqid 2733 | . . . . . . . . 9 β’ (Scalarβπ) = (Scalarβπ) | |
8 | 7 | lvecdrng 20581 | . . . . . . . 8 β’ (π β LVec β (Scalarβπ) β DivRing) |
9 | drngnzr 20748 | . . . . . . . 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 725 | . . . . . 6 β’ (((π β LVec β§ π β (LIndSβπ)) β§ π₯ β π) β (Scalarβπ) β NzRing) |
12 | simplr 768 | . . . . . 6 β’ (((π β LVec β§ π β (LIndSβπ)) β§ π₯ β π) β π β (LIndSβπ)) | |
13 | simpr 486 | . . . . . 6 β’ (((π β LVec β§ π β (LIndSβπ)) β§ π₯ β π) β π₯ β π) | |
14 | eqid 2733 | . . . . . . 7 β’ (LSpanβπ) = (LSpanβπ) | |
15 | 14, 7 | lindsind2 21241 | . . . . . 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 1377 | . . . . 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 20640 | . . . 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 1372 | . . 3 β’ ((π β LVec β§ π β (LIndSβπ)) β βπ β π½ π β π) |
21 | 20 | ex 414 | . 2 β’ (π β LVec β (π β (LIndSβπ) β βπ β π½ π β π)) |
22 | 5 | ad2antrr 725 | . . . 4 β’ (((π β LVec β§ π β π½) β§ π β π) β π β LMod) |
23 | 18 | lbslinds 21255 | . . . . . 6 β’ π½ β (LIndSβπ) |
24 | 23 | sseli 3941 | . . . . 5 β’ (π β π½ β π β (LIndSβπ)) |
25 | 24 | ad2antlr 726 | . . . 4 β’ (((π β LVec β§ π β π½) β§ π β π) β π β (LIndSβπ)) |
26 | simpr 486 | . . . 4 β’ (((π β LVec β§ π β π½) β§ π β π) β π β π) | |
27 | lindsss 21246 | . . . 4 β’ ((π β LMod β§ π β (LIndSβπ) β§ π β π) β π β (LIndSβπ)) | |
28 | 22, 25, 26, 27 | syl3anc 1372 | . . 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 397 = wceq 1542 β wcel 2107 βwral 3061 βwrex 3070 β cdif 3908 β wss 3911 {csn 4587 βcfv 6497 Basecbs 17088 Scalarcsca 17141 DivRingcdr 20197 LModclmod 20336 LSpanclspn 20447 LBasisclbs 20550 LVecclvec 20578 NzRingcnzr 20743 LIndSclinds 21227 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-ac2 10404 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-rpss 7661 df-om 7804 df-1st 7922 df-2nd 7923 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-oadd 8417 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-dju 9842 df-card 9880 df-ac 10057 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-minusg 18757 df-sbg 18758 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-ring 19971 df-oppr 20054 df-dvdsr 20075 df-unit 20076 df-invr 20106 df-drng 20199 df-lmod 20338 df-lss 20408 df-lsp 20448 df-lbs 20551 df-lvec 20579 df-nzr 20744 df-lindf 21228 df-linds 21229 |
This theorem is referenced by: lssdimle 32360 dimkerim 32379 |
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