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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 | ⊢ 𝐽 = (LBasis‘𝑊) |
lbsind2.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lbsind2.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lbsind2.o | ⊢ 1 = (1r‘𝐹) |
lbsind2.z | ⊢ 0 = (0g‘𝐹) |
Ref | Expression |
---|---|
lbsind2 | ⊢ (((𝑊 ∈ LMod ∧ 1 ≠ 0 ) ∧ 𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) → ¬ 𝐸 ∈ (𝑁‘(𝐵 ∖ {𝐸}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1l 1193 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 1 ≠ 0 ) ∧ 𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) → 𝑊 ∈ LMod) | |
2 | simp2 1133 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 1 ≠ 0 ) ∧ 𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) → 𝐵 ∈ 𝐽) | |
3 | simp3 1134 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 1 ≠ 0 ) ∧ 𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) → 𝐸 ∈ 𝐵) | |
4 | eqid 2823 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
5 | lbsind2.j | . . . . 5 ⊢ 𝐽 = (LBasis‘𝑊) | |
6 | 4, 5 | lbsel 19852 | . . . 4 ⊢ ((𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) → 𝐸 ∈ (Base‘𝑊)) |
7 | 2, 3, 6 | syl2anc 586 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 1 ≠ 0 ) ∧ 𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) → 𝐸 ∈ (Base‘𝑊)) |
8 | lbsind2.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
9 | eqid 2823 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
10 | lbsind2.o | . . . 4 ⊢ 1 = (1r‘𝐹) | |
11 | 4, 8, 9, 10 | lmodvs1 19664 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐸 ∈ (Base‘𝑊)) → ( 1 ( ·𝑠 ‘𝑊)𝐸) = 𝐸) |
12 | 1, 7, 11 | syl2anc 586 | . 2 ⊢ (((𝑊 ∈ LMod ∧ 1 ≠ 0 ) ∧ 𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) → ( 1 ( ·𝑠 ‘𝑊)𝐸) = 𝐸) |
13 | 8 | lmodring 19644 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
14 | eqid 2823 | . . . . 5 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
15 | 14, 10 | ringidcl 19320 | . . . 4 ⊢ (𝐹 ∈ Ring → 1 ∈ (Base‘𝐹)) |
16 | 1, 13, 15 | 3syl 18 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 1 ≠ 0 ) ∧ 𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) → 1 ∈ (Base‘𝐹)) |
17 | simp1r 1194 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 1 ≠ 0 ) ∧ 𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) → 1 ≠ 0 ) | |
18 | lbsind2.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
19 | lbsind2.z | . . . 4 ⊢ 0 = (0g‘𝐹) | |
20 | 4, 5, 18, 8, 9, 14, 19 | lbsind 19854 | . . 3 ⊢ (((𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) ∧ ( 1 ∈ (Base‘𝐹) ∧ 1 ≠ 0 )) → ¬ ( 1 ( ·𝑠 ‘𝑊)𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))) |
21 | 2, 3, 16, 17, 20 | syl22anc 836 | . 2 ⊢ (((𝑊 ∈ LMod ∧ 1 ≠ 0 ) ∧ 𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) → ¬ ( 1 ( ·𝑠 ‘𝑊)𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))) |
22 | 12, 21 | eqneltrrd 2935 | 1 ⊢ (((𝑊 ∈ LMod ∧ 1 ≠ 0 ) ∧ 𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) → ¬ 𝐸 ∈ (𝑁‘(𝐵 ∖ {𝐸}))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∖ cdif 3935 {csn 4569 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 Scalarcsca 16570 ·𝑠 cvsca 16571 0gc0g 16715 1rcur 19253 Ringcrg 19299 LModclmod 19636 LSpanclspn 19745 LBasisclbs 19848 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-plusg 16580 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mgp 19242 df-ur 19254 df-ring 19301 df-lmod 19638 df-lbs 19849 |
This theorem is referenced by: lbspss 19856 islbs2 19928 |
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