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Mirrors > Home > MPE Home > Th. List > asplss | Structured version Visualization version GIF version |
Description: The algebraic span of a set of vectors is a vector subspace. (Contributed by Mario Carneiro, 7-Jan-2015.) |
Ref | Expression |
---|---|
aspval.a | β’ π΄ = (AlgSpanβπ) |
aspval.v | β’ π = (Baseβπ) |
aspval.l | β’ πΏ = (LSubSpβπ) |
Ref | Expression |
---|---|
asplss | β’ ((π β AssAlg β§ π β π) β (π΄βπ) β πΏ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aspval.a | . . 3 β’ π΄ = (AlgSpanβπ) | |
2 | aspval.v | . . 3 β’ π = (Baseβπ) | |
3 | aspval.l | . . 3 β’ πΏ = (LSubSpβπ) | |
4 | 1, 2, 3 | aspval 21763 | . 2 β’ ((π β AssAlg β§ π β π) β (π΄βπ) = β© {π‘ β ((SubRingβπ) β© πΏ) β£ π β π‘}) |
5 | assalmod 21751 | . . . 4 β’ (π β AssAlg β π β LMod) | |
6 | 5 | adantr 480 | . . 3 β’ ((π β AssAlg β§ π β π) β π β LMod) |
7 | ssrab2 4072 | . . . . 5 β’ {π‘ β ((SubRingβπ) β© πΏ) β£ π β π‘} β ((SubRingβπ) β© πΏ) | |
8 | inss2 4224 | . . . . 5 β’ ((SubRingβπ) β© πΏ) β πΏ | |
9 | 7, 8 | sstri 3986 | . . . 4 β’ {π‘ β ((SubRingβπ) β© πΏ) β£ π β π‘} β πΏ |
10 | 9 | a1i 11 | . . 3 β’ ((π β AssAlg β§ π β π) β {π‘ β ((SubRingβπ) β© πΏ) β£ π β π‘} β πΏ) |
11 | fvex 6897 | . . . . 5 β’ (π΄βπ) β V | |
12 | 4, 11 | eqeltrrdi 2836 | . . . 4 β’ ((π β AssAlg β§ π β π) β β© {π‘ β ((SubRingβπ) β© πΏ) β£ π β π‘} β V) |
13 | intex 5330 | . . . 4 β’ ({π‘ β ((SubRingβπ) β© πΏ) β£ π β π‘} β β β β© {π‘ β ((SubRingβπ) β© πΏ) β£ π β π‘} β V) | |
14 | 12, 13 | sylibr 233 | . . 3 β’ ((π β AssAlg β§ π β π) β {π‘ β ((SubRingβπ) β© πΏ) β£ π β π‘} β β ) |
15 | 3 | lssintcl 20809 | . . 3 β’ ((π β LMod β§ {π‘ β ((SubRingβπ) β© πΏ) β£ π β π‘} β πΏ β§ {π‘ β ((SubRingβπ) β© πΏ) β£ π β π‘} β β ) β β© {π‘ β ((SubRingβπ) β© πΏ) β£ π β π‘} β πΏ) |
16 | 6, 10, 14, 15 | syl3anc 1368 | . 2 β’ ((π β AssAlg β§ π β π) β β© {π‘ β ((SubRingβπ) β© πΏ) β£ π β π‘} β πΏ) |
17 | 4, 16 | eqeltrd 2827 | 1 β’ ((π β AssAlg β§ π β π) β (π΄βπ) β πΏ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wne 2934 {crab 3426 Vcvv 3468 β© cin 3942 β wss 3943 β c0 4317 β© cint 4943 βcfv 6536 Basecbs 17151 SubRingcsubrg 20467 LModclmod 20704 LSubSpclss 20776 AssAlgcasa 21741 AlgSpancasp 21742 |
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-rep 5278 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-int 4944 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-1st 7971 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 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mgp 20038 df-ur 20085 df-ring 20138 df-subrg 20469 df-lmod 20706 df-lss 20777 df-assa 21744 df-asp 21745 |
This theorem is referenced by: mplbas2 21935 |
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