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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 | ⊢ ((𝑀 ∈ LVec ∧ 𝑆 ∈ (Base‘𝑀) ∧ 𝑆 ≠ (0g‘𝑀)) → {𝑆} linIndS 𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifsni 4683 | . . . . 5 ⊢ (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ∖ {(0g‘(Scalar‘𝑀))}) → 𝑠 ≠ (0g‘(Scalar‘𝑀))) | |
2 | 1 | adantl 485 | . . . 4 ⊢ (((𝑀 ∈ LVec ∧ 𝑆 ∈ (Base‘𝑀) ∧ 𝑆 ≠ (0g‘𝑀)) ∧ 𝑠 ∈ ((Base‘(Scalar‘𝑀)) ∖ {(0g‘(Scalar‘𝑀))})) → 𝑠 ≠ (0g‘(Scalar‘𝑀))) |
3 | simpl3 1190 | . . . 4 ⊢ (((𝑀 ∈ LVec ∧ 𝑆 ∈ (Base‘𝑀) ∧ 𝑆 ≠ (0g‘𝑀)) ∧ 𝑠 ∈ ((Base‘(Scalar‘𝑀)) ∖ {(0g‘(Scalar‘𝑀))})) → 𝑆 ≠ (0g‘𝑀)) | |
4 | eqid 2798 | . . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
5 | eqid 2798 | . . . . 5 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀) | |
6 | eqid 2798 | . . . . 5 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀) | |
7 | eqid 2798 | . . . . 5 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀)) | |
8 | eqid 2798 | . . . . 5 ⊢ (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀)) | |
9 | eqid 2798 | . . . . 5 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
10 | simpl1 1188 | . . . . 5 ⊢ (((𝑀 ∈ LVec ∧ 𝑆 ∈ (Base‘𝑀) ∧ 𝑆 ≠ (0g‘𝑀)) ∧ 𝑠 ∈ ((Base‘(Scalar‘𝑀)) ∖ {(0g‘(Scalar‘𝑀))})) → 𝑀 ∈ LVec) | |
11 | eldifi 4054 | . . . . . 6 ⊢ (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ∖ {(0g‘(Scalar‘𝑀))}) → 𝑠 ∈ (Base‘(Scalar‘𝑀))) | |
12 | 11 | adantl 485 | . . . . 5 ⊢ (((𝑀 ∈ LVec ∧ 𝑆 ∈ (Base‘𝑀) ∧ 𝑆 ≠ (0g‘𝑀)) ∧ 𝑠 ∈ ((Base‘(Scalar‘𝑀)) ∖ {(0g‘(Scalar‘𝑀))})) → 𝑠 ∈ (Base‘(Scalar‘𝑀))) |
13 | simpl2 1189 | . . . . 5 ⊢ (((𝑀 ∈ LVec ∧ 𝑆 ∈ (Base‘𝑀) ∧ 𝑆 ≠ (0g‘𝑀)) ∧ 𝑠 ∈ ((Base‘(Scalar‘𝑀)) ∖ {(0g‘(Scalar‘𝑀))})) → 𝑆 ∈ (Base‘𝑀)) | |
14 | 4, 5, 6, 7, 8, 9, 10, 12, 13 | lvecvsn0 19874 | . . . 4 ⊢ (((𝑀 ∈ LVec ∧ 𝑆 ∈ (Base‘𝑀) ∧ 𝑆 ≠ (0g‘𝑀)) ∧ 𝑠 ∈ ((Base‘(Scalar‘𝑀)) ∖ {(0g‘(Scalar‘𝑀))})) → ((𝑠( ·𝑠 ‘𝑀)𝑆) ≠ (0g‘𝑀) ↔ (𝑠 ≠ (0g‘(Scalar‘𝑀)) ∧ 𝑆 ≠ (0g‘𝑀)))) |
15 | 2, 3, 14 | mpbir2and 712 | . . 3 ⊢ (((𝑀 ∈ LVec ∧ 𝑆 ∈ (Base‘𝑀) ∧ 𝑆 ≠ (0g‘𝑀)) ∧ 𝑠 ∈ ((Base‘(Scalar‘𝑀)) ∖ {(0g‘(Scalar‘𝑀))})) → (𝑠( ·𝑠 ‘𝑀)𝑆) ≠ (0g‘𝑀)) |
16 | 15 | ralrimiva 3149 | . 2 ⊢ ((𝑀 ∈ LVec ∧ 𝑆 ∈ (Base‘𝑀) ∧ 𝑆 ≠ (0g‘𝑀)) → ∀𝑠 ∈ ((Base‘(Scalar‘𝑀)) ∖ {(0g‘(Scalar‘𝑀))})(𝑠( ·𝑠 ‘𝑀)𝑆) ≠ (0g‘𝑀)) |
17 | lveclmod 19871 | . . . . 5 ⊢ (𝑀 ∈ LVec → 𝑀 ∈ LMod) | |
18 | 17 | anim1i 617 | . . . 4 ⊢ ((𝑀 ∈ LVec ∧ 𝑆 ∈ (Base‘𝑀)) → (𝑀 ∈ LMod ∧ 𝑆 ∈ (Base‘𝑀))) |
19 | 18 | 3adant3 1129 | . . 3 ⊢ ((𝑀 ∈ LVec ∧ 𝑆 ∈ (Base‘𝑀) ∧ 𝑆 ≠ (0g‘𝑀)) → (𝑀 ∈ LMod ∧ 𝑆 ∈ (Base‘𝑀))) |
20 | 4, 6, 7, 8, 9, 5 | snlindsntor 44880 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ (Base‘𝑀)) → (∀𝑠 ∈ ((Base‘(Scalar‘𝑀)) ∖ {(0g‘(Scalar‘𝑀))})(𝑠( ·𝑠 ‘𝑀)𝑆) ≠ (0g‘𝑀) ↔ {𝑆} linIndS 𝑀)) |
21 | 19, 20 | syl 17 | . 2 ⊢ ((𝑀 ∈ LVec ∧ 𝑆 ∈ (Base‘𝑀) ∧ 𝑆 ≠ (0g‘𝑀)) → (∀𝑠 ∈ ((Base‘(Scalar‘𝑀)) ∖ {(0g‘(Scalar‘𝑀))})(𝑠( ·𝑠 ‘𝑀)𝑆) ≠ (0g‘𝑀) ↔ {𝑆} linIndS 𝑀)) |
22 | 16, 21 | mpbid 235 | 1 ⊢ ((𝑀 ∈ LVec ∧ 𝑆 ∈ (Base‘𝑀) ∧ 𝑆 ≠ (0g‘𝑀)) → {𝑆} linIndS 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∖ cdif 3878 {csn 4525 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 Scalarcsca 16560 ·𝑠 cvsca 16561 0gc0g 16705 LModclmod 19627 LVecclvec 19867 linIndS clininds 44849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-0g 16707 df-gsum 16708 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-mulg 18217 df-cntz 18439 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-drng 19497 df-lmod 19629 df-lvec 19868 df-linc 44815 df-lininds 44851 |
This theorem is referenced by: (None) |
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