Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| 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 2726 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
| 5 | lbsind2.j | . . . . 5 β’ π½ = (LBasisβπ) | |
| 6 | 4, 5 | lbsel 20923 | . . . 4 β’ ((π΅ β π½ β§ πΈ β π΅) β πΈ β (Baseβπ)) |
| 7 | 2, 3, 6 | syl2anc 583 | . . 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 2726 | . . . 4 β’ ( Β·π βπ) = ( Β·π βπ) | |
| 10 | lbsind2.o | . . . 4 β’ 1 = (1rβπΉ) | |
| 11 | 4, 8, 9, 10 | lmodvs1 20733 | . . 3 β’ ((π β LMod β§ πΈ β (Baseβπ)) β ( 1 ( Β·π βπ)πΈ) = πΈ) |
| 12 | 1, 7, 11 | syl2anc 583 | . 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 20711 | . . . 4 β’ (π β LMod β πΉ β Ring) |
| 14 | eqid 2726 | . . . . 5 β’ (BaseβπΉ) = (BaseβπΉ) | |
| 15 | 14, 10 | ringidcl 20162 | . . . 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 20925 | . . 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 836 | . 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 2848 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 β cdif 3940 {csn 4623 βcfv 6536 (class class class)co 7404 Basecbs 17150 Scalarcsca 17206 Β·π cvsca 17207 0gc0g 17391 1rcur 20083 Ringcrg 20135 LModclmod 20703 LSpanclspn 20815 LBasisclbs 20919 |
| 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-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
| 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-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-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 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-plusg 17216 df-0g 17393 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-mgp 20037 df-ur 20084 df-ring 20137 df-lmod 20705 df-lbs 20920 |
| This theorem is referenced by: lbspss 20927 islbs2 21002 |
| Copyright terms: Public domain | W3C validator |