|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 1198 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 1 ≠ 0 ) ∧ 𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) → 𝑊 ∈ LMod) | |
| 2 | simp2 1138 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 1 ≠ 0 ) ∧ 𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) → 𝐵 ∈ 𝐽) | |
| 3 | simp3 1139 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 1 ≠ 0 ) ∧ 𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) → 𝐸 ∈ 𝐵) | |
| 4 | eqid 2737 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 5 | lbsind2.j | . . . . 5 ⊢ 𝐽 = (LBasis‘𝑊) | |
| 6 | 4, 5 | lbsel 21077 | . . . 4 ⊢ ((𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) → 𝐸 ∈ (Base‘𝑊)) | 
| 7 | 2, 3, 6 | syl2anc 584 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 1 ≠ 0 ) ∧ 𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) → 𝐸 ∈ (Base‘𝑊)) | 
| 8 | lbsind2.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
| 9 | eqid 2737 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
| 10 | lbsind2.o | . . . 4 ⊢ 1 = (1r‘𝐹) | |
| 11 | 4, 8, 9, 10 | lmodvs1 20888 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐸 ∈ (Base‘𝑊)) → ( 1 ( ·𝑠 ‘𝑊)𝐸) = 𝐸) | 
| 12 | 1, 7, 11 | syl2anc 584 | . 2 ⊢ (((𝑊 ∈ LMod ∧ 1 ≠ 0 ) ∧ 𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) → ( 1 ( ·𝑠 ‘𝑊)𝐸) = 𝐸) | 
| 13 | 8 | lmodring 20866 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) | 
| 14 | eqid 2737 | . . . . 5 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
| 15 | 14, 10 | ringidcl 20262 | . . . 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 1199 | . . 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 21079 | . . 3 ⊢ (((𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) ∧ ( 1 ∈ (Base‘𝐹) ∧ 1 ≠ 0 )) → ¬ ( 1 ( ·𝑠 ‘𝑊)𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))) | 
| 21 | 2, 3, 16, 17, 20 | syl22anc 839 | . 2 ⊢ (((𝑊 ∈ LMod ∧ 1 ≠ 0 ) ∧ 𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) → ¬ ( 1 ( ·𝑠 ‘𝑊)𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))) | 
| 22 | 12, 21 | eqneltrrd 2862 | 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 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 {csn 4626 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 1rcur 20178 Ringcrg 20230 LModclmod 20858 LSpanclspn 20969 LBasisclbs 21073 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mgp 20138 df-ur 20179 df-ring 20232 df-lmod 20860 df-lbs 21074 | 
| This theorem is referenced by: lbspss 21081 islbs2 21156 dimlssid 33683 | 
| Copyright terms: Public domain | W3C validator |