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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 4722 | . . . . 5 ⊢ (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ∖ {(0g‘(Scalar‘𝑀))}) → 𝑠 ≠ (0g‘(Scalar‘𝑀))) | |
2 | 1 | adantl 484 | . . . 4 ⊢ (((𝑀 ∈ LVec ∧ 𝑆 ∈ (Base‘𝑀) ∧ 𝑆 ≠ (0g‘𝑀)) ∧ 𝑠 ∈ ((Base‘(Scalar‘𝑀)) ∖ {(0g‘(Scalar‘𝑀))})) → 𝑠 ≠ (0g‘(Scalar‘𝑀))) |
3 | simpl3 1189 | . . . 4 ⊢ (((𝑀 ∈ LVec ∧ 𝑆 ∈ (Base‘𝑀) ∧ 𝑆 ≠ (0g‘𝑀)) ∧ 𝑠 ∈ ((Base‘(Scalar‘𝑀)) ∖ {(0g‘(Scalar‘𝑀))})) → 𝑆 ≠ (0g‘𝑀)) | |
4 | eqid 2821 | . . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
5 | eqid 2821 | . . . . 5 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀) | |
6 | eqid 2821 | . . . . 5 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀) | |
7 | eqid 2821 | . . . . 5 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀)) | |
8 | eqid 2821 | . . . . 5 ⊢ (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀)) | |
9 | eqid 2821 | . . . . 5 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
10 | simpl1 1187 | . . . . 5 ⊢ (((𝑀 ∈ LVec ∧ 𝑆 ∈ (Base‘𝑀) ∧ 𝑆 ≠ (0g‘𝑀)) ∧ 𝑠 ∈ ((Base‘(Scalar‘𝑀)) ∖ {(0g‘(Scalar‘𝑀))})) → 𝑀 ∈ LVec) | |
11 | eldifi 4103 | . . . . . 6 ⊢ (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ∖ {(0g‘(Scalar‘𝑀))}) → 𝑠 ∈ (Base‘(Scalar‘𝑀))) | |
12 | 11 | adantl 484 | . . . . 5 ⊢ (((𝑀 ∈ LVec ∧ 𝑆 ∈ (Base‘𝑀) ∧ 𝑆 ≠ (0g‘𝑀)) ∧ 𝑠 ∈ ((Base‘(Scalar‘𝑀)) ∖ {(0g‘(Scalar‘𝑀))})) → 𝑠 ∈ (Base‘(Scalar‘𝑀))) |
13 | simpl2 1188 | . . . . 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 19881 | . . . 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 711 | . . 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 3182 | . 2 ⊢ ((𝑀 ∈ LVec ∧ 𝑆 ∈ (Base‘𝑀) ∧ 𝑆 ≠ (0g‘𝑀)) → ∀𝑠 ∈ ((Base‘(Scalar‘𝑀)) ∖ {(0g‘(Scalar‘𝑀))})(𝑠( ·𝑠 ‘𝑀)𝑆) ≠ (0g‘𝑀)) |
17 | lveclmod 19878 | . . . . 5 ⊢ (𝑀 ∈ LVec → 𝑀 ∈ LMod) | |
18 | 17 | anim1i 616 | . . . 4 ⊢ ((𝑀 ∈ LVec ∧ 𝑆 ∈ (Base‘𝑀)) → (𝑀 ∈ LMod ∧ 𝑆 ∈ (Base‘𝑀))) |
19 | 18 | 3adant3 1128 | . . 3 ⊢ ((𝑀 ∈ LVec ∧ 𝑆 ∈ (Base‘𝑀) ∧ 𝑆 ≠ (0g‘𝑀)) → (𝑀 ∈ LMod ∧ 𝑆 ∈ (Base‘𝑀))) |
20 | 4, 6, 7, 8, 9, 5 | snlindsntor 44546 | . . 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 234 | 1 ⊢ ((𝑀 ∈ LVec ∧ 𝑆 ∈ (Base‘𝑀) ∧ 𝑆 ≠ (0g‘𝑀)) → {𝑆} linIndS 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∖ cdif 3933 {csn 4567 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 Scalarcsca 16568 ·𝑠 cvsca 16569 0gc0g 16713 LModclmod 19634 LVecclvec 19874 linIndS clininds 44515 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-0g 16715 df-gsum 16716 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-mulg 18225 df-cntz 18447 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-drng 19504 df-lmod 19636 df-lvec 19875 df-linc 44481 df-lininds 44517 |
This theorem is referenced by: (None) |
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