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| Mirrors > Home > MPE Home > Th. List > lbsind2 | Structured version Visualization version GIF version | ||
| Description: A basis is linearly independent; that is, every element is not in the span of the remainder of the basis. (Contributed by Mario Carneiro, 25-Jun-2014.) (Revised by Mario Carneiro, 12-Jan-2015.) |
| Ref | Expression |
|---|---|
| lbsind2.j | β’ π½ = (LBasisβπ) |
| lbsind2.n | β’ π = (LSpanβπ) |
| lbsind2.f | β’ πΉ = (Scalarβπ) |
| lbsind2.o | β’ 1 = (1rβπΉ) |
| lbsind2.z | β’ 0 = (0gβπΉ) |
| Ref | Expression |
|---|---|
| lbsind2 | β’ (((π β LMod β§ 1 β 0 ) β§ π΅ β π½ β§ πΈ β π΅) β Β¬ πΈ β (πβ(π΅ β {πΈ}))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp1l 1194 | . . 3 β’ (((π β LMod β§ 1 β 0 ) β§ π΅ β π½ β§ πΈ β π΅) β π β LMod) | |
| 2 | simp2 1134 | . . . 4 β’ (((π β LMod β§ 1 β 0 ) β§ π΅ β π½ β§ πΈ β π΅) β π΅ β π½) | |
| 3 | simp3 1135 | . . . 4 β’ (((π β LMod β§ 1 β 0 ) β§ π΅ β π½ β§ πΈ β π΅) β πΈ β π΅) | |
| 4 | eqid 2728 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
| 5 | lbsind2.j | . . . . 5 β’ π½ = (LBasisβπ) | |
| 6 | 4, 5 | lbsel 20970 | . . . 4 β’ ((π΅ β π½ β§ πΈ β π΅) β πΈ β (Baseβπ)) |
| 7 | 2, 3, 6 | syl2anc 582 | . . 3 β’ (((π β LMod β§ 1 β 0 ) β§ π΅ β π½ β§ πΈ β π΅) β πΈ β (Baseβπ)) |
| 8 | lbsind2.f | . . . 4 β’ πΉ = (Scalarβπ) | |
| 9 | eqid 2728 | . . . 4 β’ ( Β·π βπ) = ( Β·π βπ) | |
| 10 | lbsind2.o | . . . 4 β’ 1 = (1rβπΉ) | |
| 11 | 4, 8, 9, 10 | lmodvs1 20780 | . . 3 β’ ((π β LMod β§ πΈ β (Baseβπ)) β ( 1 ( Β·π βπ)πΈ) = πΈ) |
| 12 | 1, 7, 11 | syl2anc 582 | . 2 β’ (((π β LMod β§ 1 β 0 ) β§ π΅ β π½ β§ πΈ β π΅) β ( 1 ( Β·π βπ)πΈ) = πΈ) |
| 13 | 8 | lmodring 20758 | . . . 4 β’ (π β LMod β πΉ β Ring) |
| 14 | eqid 2728 | . . . . 5 β’ (BaseβπΉ) = (BaseβπΉ) | |
| 15 | 14, 10 | ringidcl 20209 | . . . 4 β’ (πΉ β Ring β 1 β (BaseβπΉ)) |
| 16 | 1, 13, 15 | 3syl 18 | . . 3 β’ (((π β LMod β§ 1 β 0 ) β§ π΅ β π½ β§ πΈ β π΅) β 1 β (BaseβπΉ)) |
| 17 | simp1r 1195 | . . 3 β’ (((π β LMod β§ 1 β 0 ) β§ π΅ β π½ β§ πΈ β π΅) β 1 β 0 ) | |
| 18 | lbsind2.n | . . . 4 β’ π = (LSpanβπ) | |
| 19 | lbsind2.z | . . . 4 β’ 0 = (0gβπΉ) | |
| 20 | 4, 5, 18, 8, 9, 14, 19 | lbsind 20972 | . . 3 β’ (((π΅ β π½ β§ πΈ β π΅) β§ ( 1 β (BaseβπΉ) β§ 1 β 0 )) β Β¬ ( 1 ( Β·π βπ)πΈ) β (πβ(π΅ β {πΈ}))) |
| 21 | 2, 3, 16, 17, 20 | syl22anc 837 | . 2 β’ (((π β LMod β§ 1 β 0 ) β§ π΅ β π½ β§ πΈ β π΅) β Β¬ ( 1 ( Β·π βπ)πΈ) β (πβ(π΅ β {πΈ}))) |
| 22 | 12, 21 | eqneltrrd 2850 | 1 β’ (((π β LMod β§ 1 β 0 ) β§ π΅ β π½ β§ πΈ β π΅) β Β¬ πΈ β (πβ(π΅ β {πΈ}))) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2937 β cdif 3946 {csn 4632 βcfv 6553 (class class class)co 7426 Basecbs 17187 Scalarcsca 17243 Β·π cvsca 17244 0gc0g 17428 1rcur 20128 Ringcrg 20180 LModclmod 20750 LSpanclspn 20862 LBasisclbs 20966 |
| 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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mgp 20082 df-ur 20129 df-ring 20182 df-lmod 20752 df-lbs 20967 |
| This theorem is referenced by: lbspss 20974 islbs2 21049 |
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