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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lindssnlvec | Structured version Visualization version GIF version | ||
| Description: A singleton not containing the zero element of a vector space is always linearly independent. (Contributed by AV, 16-Apr-2019.) (Revised by AV, 28-Apr-2019.) |
| Ref | Expression |
|---|---|
| lindssnlvec | β’ ((π β LVec β§ π β (Baseβπ) β§ π β (0gβπ)) β {π} linIndS π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eldifsni 4770 | . . . . 5 β’ (π β ((Baseβ(Scalarβπ)) β {(0gβ(Scalarβπ))}) β π β (0gβ(Scalarβπ))) | |
| 2 | 1 | adantl 482 | . . . 4 β’ (((π β LVec β§ π β (Baseβπ) β§ π β (0gβπ)) β§ π β ((Baseβ(Scalarβπ)) β {(0gβ(Scalarβπ))})) β π β (0gβ(Scalarβπ))) |
| 3 | simpl3 1193 | . . . 4 β’ (((π β LVec β§ π β (Baseβπ) β§ π β (0gβπ)) β§ π β ((Baseβ(Scalarβπ)) β {(0gβ(Scalarβπ))})) β π β (0gβπ)) | |
| 4 | eqid 2731 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
| 5 | eqid 2731 | . . . . 5 β’ ( Β·π βπ) = ( Β·π βπ) | |
| 6 | eqid 2731 | . . . . 5 β’ (Scalarβπ) = (Scalarβπ) | |
| 7 | eqid 2731 | . . . . 5 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
| 8 | eqid 2731 | . . . . 5 β’ (0gβ(Scalarβπ)) = (0gβ(Scalarβπ)) | |
| 9 | eqid 2731 | . . . . 5 β’ (0gβπ) = (0gβπ) | |
| 10 | simpl1 1191 | . . . . 5 β’ (((π β LVec β§ π β (Baseβπ) β§ π β (0gβπ)) β§ π β ((Baseβ(Scalarβπ)) β {(0gβ(Scalarβπ))})) β π β LVec) | |
| 11 | eldifi 4106 | . . . . . 6 β’ (π β ((Baseβ(Scalarβπ)) β {(0gβ(Scalarβπ))}) β π β (Baseβ(Scalarβπ))) | |
| 12 | 11 | adantl 482 | . . . . 5 β’ (((π β LVec β§ π β (Baseβπ) β§ π β (0gβπ)) β§ π β ((Baseβ(Scalarβπ)) β {(0gβ(Scalarβπ))})) β π β (Baseβ(Scalarβπ))) |
| 13 | simpl2 1192 | . . . . 5 β’ (((π β LVec β§ π β (Baseβπ) β§ π β (0gβπ)) β§ π β ((Baseβ(Scalarβπ)) β {(0gβ(Scalarβπ))})) β π β (Baseβπ)) | |
| 14 | 4, 5, 6, 7, 8, 9, 10, 12, 13 | lvecvsn0 20644 | . . . 4 β’ (((π β LVec β§ π β (Baseβπ) β§ π β (0gβπ)) β§ π β ((Baseβ(Scalarβπ)) β {(0gβ(Scalarβπ))})) β ((π ( Β·π βπ)π) β (0gβπ) β (π β (0gβ(Scalarβπ)) β§ π β (0gβπ)))) |
| 15 | 2, 3, 14 | mpbir2and 711 | . . 3 β’ (((π β LVec β§ π β (Baseβπ) β§ π β (0gβπ)) β§ π β ((Baseβ(Scalarβπ)) β {(0gβ(Scalarβπ))})) β (π ( Β·π βπ)π) β (0gβπ)) |
| 16 | 15 | ralrimiva 3145 | . 2 β’ ((π β LVec β§ π β (Baseβπ) β§ π β (0gβπ)) β βπ β ((Baseβ(Scalarβπ)) β {(0gβ(Scalarβπ))})(π ( Β·π βπ)π) β (0gβπ)) |
| 17 | lveclmod 20639 | . . . . 5 β’ (π β LVec β π β LMod) | |
| 18 | 17 | anim1i 615 | . . . 4 β’ ((π β LVec β§ π β (Baseβπ)) β (π β LMod β§ π β (Baseβπ))) |
| 19 | 18 | 3adant3 1132 | . . 3 β’ ((π β LVec β§ π β (Baseβπ) β§ π β (0gβπ)) β (π β LMod β§ π β (Baseβπ))) |
| 20 | 4, 6, 7, 8, 9, 5 | snlindsntor 46705 | . . 3 β’ ((π β LMod β§ π β (Baseβπ)) β (βπ β ((Baseβ(Scalarβπ)) β {(0gβ(Scalarβπ))})(π ( Β·π βπ)π) β (0gβπ) β {π} linIndS π)) |
| 21 | 19, 20 | syl 17 | . 2 β’ ((π β LVec β§ π β (Baseβπ) β§ π β (0gβπ)) β (βπ β ((Baseβ(Scalarβπ)) β {(0gβ(Scalarβπ))})(π ( Β·π βπ)π) β (0gβπ) β {π} linIndS π)) |
| 22 | 16, 21 | mpbid 231 | 1 β’ ((π β LVec β§ π β (Baseβπ) β§ π β (0gβπ)) β {π} linIndS π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 396 β§ w3a 1087 β wcel 2106 β wne 2939 βwral 3060 β cdif 3925 {csn 4606 class class class wbr 5125 βcfv 6516 (class class class)co 7377 Basecbs 17109 Scalarcsca 17165 Β·π cvsca 17166 0gc0g 17350 LModclmod 20393 LVecclvec 20635 linIndS clininds 46674 |
| 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 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 |
| 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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3364 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-pss 3947 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-int 4928 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-tr 5243 df-id 5551 df-eprel 5557 df-po 5565 df-so 5566 df-fr 5608 df-se 5609 df-we 5610 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-pred 6273 df-ord 6340 df-on 6341 df-lim 6342 df-suc 6343 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-om 7823 df-1st 7941 df-2nd 7942 df-supp 8113 df-tpos 8177 df-frecs 8232 df-wrecs 8263 df-recs 8337 df-rdg 8376 df-1o 8432 df-er 8670 df-map 8789 df-en 8906 df-dom 8907 df-sdom 8908 df-fin 8909 df-fsupp 9328 df-oi 9470 df-card 9899 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11411 df-neg 11412 df-nn 12178 df-2 12240 df-3 12241 df-n0 12438 df-z 12524 df-uz 12788 df-fz 13450 df-fzo 13593 df-seq 13932 df-hash 14256 df-sets 17062 df-slot 17080 df-ndx 17092 df-base 17110 df-ress 17139 df-plusg 17175 df-mulr 17176 df-0g 17352 df-gsum 17353 df-mgm 18526 df-sgrp 18575 df-mnd 18586 df-grp 18780 df-minusg 18781 df-mulg 18902 df-cntz 19126 df-mgp 19926 df-ur 19943 df-ring 19995 df-oppr 20078 df-dvdsr 20099 df-unit 20100 df-invr 20130 df-drng 20242 df-lmod 20395 df-lvec 20636 df-linc 46640 df-lininds 46676 |
| This theorem is referenced by: (None) |
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