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Mirrors > Home > MPE Home > Th. List > islinds4 | Structured version Visualization version GIF version |
Description: A set is independent in a vector space iff it is a subset of some basis. (AC equivalent) (Contributed by Stefan O'Rear, 24-Feb-2015.) |
Ref | Expression |
---|---|
islinds4.j | ⊢ 𝐽 = (LBasis‘𝑊) |
Ref | Expression |
---|---|
islinds4 | ⊢ (𝑊 ∈ LVec → (𝑌 ∈ (LIndS‘𝑊) ↔ ∃𝑏 ∈ 𝐽 𝑌 ⊆ 𝑏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 485 | . . . 4 ⊢ ((𝑊 ∈ LVec ∧ 𝑌 ∈ (LIndS‘𝑊)) → 𝑊 ∈ LVec) | |
2 | eqid 2823 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
3 | 2 | linds1 20956 | . . . . 5 ⊢ (𝑌 ∈ (LIndS‘𝑊) → 𝑌 ⊆ (Base‘𝑊)) |
4 | 3 | adantl 484 | . . . 4 ⊢ ((𝑊 ∈ LVec ∧ 𝑌 ∈ (LIndS‘𝑊)) → 𝑌 ⊆ (Base‘𝑊)) |
5 | lveclmod 19880 | . . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
6 | 5 | ad2antrr 724 | . . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ 𝑌 ∈ (LIndS‘𝑊)) ∧ 𝑥 ∈ 𝑌) → 𝑊 ∈ LMod) |
7 | eqid 2823 | . . . . . . . . 9 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
8 | 7 | lvecdrng 19879 | . . . . . . . 8 ⊢ (𝑊 ∈ LVec → (Scalar‘𝑊) ∈ DivRing) |
9 | drngnzr 20037 | . . . . . . . 8 ⊢ ((Scalar‘𝑊) ∈ DivRing → (Scalar‘𝑊) ∈ NzRing) | |
10 | 8, 9 | syl 17 | . . . . . . 7 ⊢ (𝑊 ∈ LVec → (Scalar‘𝑊) ∈ NzRing) |
11 | 10 | ad2antrr 724 | . . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ 𝑌 ∈ (LIndS‘𝑊)) ∧ 𝑥 ∈ 𝑌) → (Scalar‘𝑊) ∈ NzRing) |
12 | simplr 767 | . . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ 𝑌 ∈ (LIndS‘𝑊)) ∧ 𝑥 ∈ 𝑌) → 𝑌 ∈ (LIndS‘𝑊)) | |
13 | simpr 487 | . . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ 𝑌 ∈ (LIndS‘𝑊)) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑌) | |
14 | eqid 2823 | . . . . . . 7 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
15 | 14, 7 | lindsind2 20965 | . . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ NzRing) ∧ 𝑌 ∈ (LIndS‘𝑊) ∧ 𝑥 ∈ 𝑌) → ¬ 𝑥 ∈ ((LSpan‘𝑊)‘(𝑌 ∖ {𝑥}))) |
16 | 6, 11, 12, 13, 15 | syl211anc 1372 | . . . . 5 ⊢ (((𝑊 ∈ LVec ∧ 𝑌 ∈ (LIndS‘𝑊)) ∧ 𝑥 ∈ 𝑌) → ¬ 𝑥 ∈ ((LSpan‘𝑊)‘(𝑌 ∖ {𝑥}))) |
17 | 16 | ralrimiva 3184 | . . . 4 ⊢ ((𝑊 ∈ LVec ∧ 𝑌 ∈ (LIndS‘𝑊)) → ∀𝑥 ∈ 𝑌 ¬ 𝑥 ∈ ((LSpan‘𝑊)‘(𝑌 ∖ {𝑥}))) |
18 | islinds4.j | . . . . 5 ⊢ 𝐽 = (LBasis‘𝑊) | |
19 | 18, 2, 14 | lbsext 19937 | . . . 4 ⊢ ((𝑊 ∈ LVec ∧ 𝑌 ⊆ (Base‘𝑊) ∧ ∀𝑥 ∈ 𝑌 ¬ 𝑥 ∈ ((LSpan‘𝑊)‘(𝑌 ∖ {𝑥}))) → ∃𝑏 ∈ 𝐽 𝑌 ⊆ 𝑏) |
20 | 1, 4, 17, 19 | syl3anc 1367 | . . 3 ⊢ ((𝑊 ∈ LVec ∧ 𝑌 ∈ (LIndS‘𝑊)) → ∃𝑏 ∈ 𝐽 𝑌 ⊆ 𝑏) |
21 | 20 | ex 415 | . 2 ⊢ (𝑊 ∈ LVec → (𝑌 ∈ (LIndS‘𝑊) → ∃𝑏 ∈ 𝐽 𝑌 ⊆ 𝑏)) |
22 | 5 | ad2antrr 724 | . . . 4 ⊢ (((𝑊 ∈ LVec ∧ 𝑏 ∈ 𝐽) ∧ 𝑌 ⊆ 𝑏) → 𝑊 ∈ LMod) |
23 | 18 | lbslinds 20979 | . . . . . 6 ⊢ 𝐽 ⊆ (LIndS‘𝑊) |
24 | 23 | sseli 3965 | . . . . 5 ⊢ (𝑏 ∈ 𝐽 → 𝑏 ∈ (LIndS‘𝑊)) |
25 | 24 | ad2antlr 725 | . . . 4 ⊢ (((𝑊 ∈ LVec ∧ 𝑏 ∈ 𝐽) ∧ 𝑌 ⊆ 𝑏) → 𝑏 ∈ (LIndS‘𝑊)) |
26 | simpr 487 | . . . 4 ⊢ (((𝑊 ∈ LVec ∧ 𝑏 ∈ 𝐽) ∧ 𝑌 ⊆ 𝑏) → 𝑌 ⊆ 𝑏) | |
27 | lindsss 20970 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑏 ∈ (LIndS‘𝑊) ∧ 𝑌 ⊆ 𝑏) → 𝑌 ∈ (LIndS‘𝑊)) | |
28 | 22, 25, 26, 27 | syl3anc 1367 | . . 3 ⊢ (((𝑊 ∈ LVec ∧ 𝑏 ∈ 𝐽) ∧ 𝑌 ⊆ 𝑏) → 𝑌 ∈ (LIndS‘𝑊)) |
29 | 28 | rexlimdva2 3289 | . 2 ⊢ (𝑊 ∈ LVec → (∃𝑏 ∈ 𝐽 𝑌 ⊆ 𝑏 → 𝑌 ∈ (LIndS‘𝑊))) |
30 | 21, 29 | impbid 214 | 1 ⊢ (𝑊 ∈ LVec → (𝑌 ∈ (LIndS‘𝑊) ↔ ∃𝑏 ∈ 𝐽 𝑌 ⊆ 𝑏)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 ∖ cdif 3935 ⊆ wss 3938 {csn 4569 ‘cfv 6357 Basecbs 16485 Scalarcsca 16570 DivRingcdr 19504 LModclmod 19636 LSpanclspn 19745 LBasisclbs 19848 LVecclvec 19876 NzRingcnzr 20032 LIndSclinds 20951 |
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-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-ac2 9887 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-int 4879 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-se 5517 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-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-rpss 7451 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-dju 9332 df-card 9370 df-ac 9544 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-3 11704 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-sbg 18110 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-drng 19506 df-lmod 19638 df-lss 19706 df-lsp 19746 df-lbs 19849 df-lvec 19877 df-nzr 20033 df-lindf 20952 df-linds 20953 |
This theorem is referenced by: lssdimle 31008 dimkerim 31025 |
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