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Mirrors > Home > MPE Home > Th. List > Mathboxes > lincolss | Structured version Visualization version GIF version |
Description: According to the statement in [Lang] p. 129, the set (LSubSp‘𝑀) of all linear combinations of a set of vectors V is a submodule (generated by V) of the module M. The elements of V are called generators of (LSubSp‘𝑀). (Contributed by AV, 12-Apr-2019.) |
Ref | Expression |
---|---|
lincolss | ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo 𝑉) ∈ (LSubSp‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2623 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (Scalar‘𝑀) = (Scalar‘𝑀)) | |
2 | eqidd 2623 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))) | |
3 | eqidd 2623 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (Base‘𝑀) = (Base‘𝑀)) | |
4 | eqidd 2623 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (+g‘𝑀) = (+g‘𝑀)) | |
5 | eqidd 2623 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)) | |
6 | eqidd 2623 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (LSubSp‘𝑀) = (LSubSp‘𝑀)) | |
7 | eqid 2622 | . . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
8 | eqid 2622 | . . . . 5 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀) | |
9 | eqid 2622 | . . . . 5 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀)) | |
10 | 7, 8, 9 | lcoval 42201 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑣 ∈ (𝑀 LinCo 𝑉) ↔ (𝑣 ∈ (Base‘𝑀) ∧ ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑠( linC ‘𝑀)𝑉))))) |
11 | simpl 473 | . . . 4 ⊢ ((𝑣 ∈ (Base‘𝑀) ∧ ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑠( linC ‘𝑀)𝑉))) → 𝑣 ∈ (Base‘𝑀)) | |
12 | 10, 11 | syl6bi 243 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑣 ∈ (𝑀 LinCo 𝑉) → 𝑣 ∈ (Base‘𝑀))) |
13 | 12 | ssrdv 3609 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo 𝑉) ⊆ (Base‘𝑀)) |
14 | lcoel0 42217 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (0g‘𝑀) ∈ (𝑀 LinCo 𝑉)) | |
15 | ne0i 3921 | . . 3 ⊢ ((0g‘𝑀) ∈ (𝑀 LinCo 𝑉) → (𝑀 LinCo 𝑉) ≠ ∅) | |
16 | 14, 15 | syl 17 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo 𝑉) ≠ ∅) |
17 | eqid 2622 | . . 3 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀) | |
18 | eqid 2622 | . . 3 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
19 | 17, 9, 18 | lincsumscmcl 42222 | . 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑎 ∈ (𝑀 LinCo 𝑉) ∧ 𝑏 ∈ (𝑀 LinCo 𝑉))) → ((𝑥( ·𝑠 ‘𝑀)𝑎)(+g‘𝑀)𝑏) ∈ (𝑀 LinCo 𝑉)) |
20 | 1, 2, 3, 4, 5, 6, 13, 16, 19 | islssd 18936 | 1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo 𝑉) ∈ (LSubSp‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∃wrex 2913 ∅c0 3915 𝒫 cpw 4158 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 finSupp cfsupp 8275 Basecbs 15857 +gcplusg 15941 Scalarcsca 15944 ·𝑠 cvsca 15945 0gc0g 16100 LModclmod 18863 LSubSpclss 18932 linC clinc 42193 LinCo clinco 42194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-0g 16102 df-gsum 16103 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-lmod 18865 df-lss 18933 df-linc 42195 df-lco 42196 |
This theorem is referenced by: lspsslco 42226 |
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