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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 4792 | . . . . 5 β’ (π β ((Baseβ(Scalarβπ)) β {(0gβ(Scalarβπ))}) β π β (0gβ(Scalarβπ))) | |
| 2 | 1 | adantl 480 | . . . 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 1191 | . . . 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 2730 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
| 5 | eqid 2730 | . . . . 5 β’ ( Β·π βπ) = ( Β·π βπ) | |
| 6 | eqid 2730 | . . . . 5 β’ (Scalarβπ) = (Scalarβπ) | |
| 7 | eqid 2730 | . . . . 5 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
| 8 | eqid 2730 | . . . . 5 β’ (0gβ(Scalarβπ)) = (0gβ(Scalarβπ)) | |
| 9 | eqid 2730 | . . . . 5 β’ (0gβπ) = (0gβπ) | |
| 10 | simpl1 1189 | . . . . 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 4125 | . . . . . 6 β’ (π β ((Baseβ(Scalarβπ)) β {(0gβ(Scalarβπ))}) β π β (Baseβ(Scalarβπ))) | |
| 12 | 11 | adantl 480 | . . . . 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 1190 | . . . . 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 20867 | . . . 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 709 | . . 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 3144 | . 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 20861 | . . . . 5 β’ (π β LVec β π β LMod) | |
| 18 | 17 | anim1i 613 | . . . 4 β’ ((π β LVec β§ π β (Baseβπ)) β (π β LMod β§ π β (Baseβπ))) |
| 19 | 18 | 3adant3 1130 | . . 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 47239 | . . 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 394 β§ w3a 1085 β wcel 2104 β wne 2938 βwral 3059 β cdif 3944 {csn 4627 class class class wbr 5147 βcfv 6542 (class class class)co 7411 Basecbs 17148 Scalarcsca 17204 Β·π cvsca 17205 0gc0g 17389 LModclmod 20614 LVecclvec 20857 linIndS clininds 47208 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 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 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-oi 9507 df-card 9936 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-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-fzo 13632 df-seq 13971 df-hash 14295 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-0g 17391 df-gsum 17392 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-minusg 18859 df-mulg 18987 df-cntz 19222 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-oppr 20225 df-dvdsr 20248 df-unit 20249 df-invr 20279 df-drng 20502 df-lmod 20616 df-lvec 20858 df-linc 47174 df-lininds 47210 |
| This theorem is referenced by: (None) |
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