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Mirrors > Home > MPE Home > Th. List > islinds3 | Structured version Visualization version GIF version |
Description: A subset is linearly independent iff it is a basis of its span. (Contributed by Stefan O'Rear, 25-Feb-2015.) |
Ref | Expression |
---|---|
islinds3.b | ⊢ 𝐵 = (Base‘𝑊) |
islinds3.k | ⊢ 𝐾 = (LSpan‘𝑊) |
islinds3.x | ⊢ 𝑋 = (𝑊 ↾s (𝐾‘𝑌)) |
islinds3.j | ⊢ 𝐽 = (LBasis‘𝑋) |
Ref | Expression |
---|---|
islinds3 | ⊢ (𝑊 ∈ LMod → (𝑌 ∈ (LIndS‘𝑊) ↔ 𝑌 ∈ 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islinds3.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑊) | |
2 | 1 | linds1 20948 | . . . 4 ⊢ (𝑌 ∈ (LIndS‘𝑊) → 𝑌 ⊆ 𝐵) |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑊 ∈ LMod → (𝑌 ∈ (LIndS‘𝑊) → 𝑌 ⊆ 𝐵)) |
4 | eqid 2821 | . . . . . . 7 ⊢ (Base‘𝑋) = (Base‘𝑋) | |
5 | 4 | linds1 20948 | . . . . . 6 ⊢ (𝑌 ∈ (LIndS‘𝑋) → 𝑌 ⊆ (Base‘𝑋)) |
6 | islinds3.x | . . . . . . 7 ⊢ 𝑋 = (𝑊 ↾s (𝐾‘𝑌)) | |
7 | 6, 1 | ressbasss 16550 | . . . . . 6 ⊢ (Base‘𝑋) ⊆ 𝐵 |
8 | 5, 7 | sstrdi 3978 | . . . . 5 ⊢ (𝑌 ∈ (LIndS‘𝑋) → 𝑌 ⊆ 𝐵) |
9 | 8 | adantr 483 | . . . 4 ⊢ ((𝑌 ∈ (LIndS‘𝑋) ∧ ((LSpan‘𝑋)‘𝑌) = (Base‘𝑋)) → 𝑌 ⊆ 𝐵) |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑊 ∈ LMod → ((𝑌 ∈ (LIndS‘𝑋) ∧ ((LSpan‘𝑋)‘𝑌) = (Base‘𝑋)) → 𝑌 ⊆ 𝐵)) |
11 | simpl 485 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ⊆ 𝐵) → 𝑊 ∈ LMod) | |
12 | eqid 2821 | . . . . . . . . 9 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
13 | islinds3.k | . . . . . . . . 9 ⊢ 𝐾 = (LSpan‘𝑊) | |
14 | 1, 12, 13 | lspcl 19742 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ⊆ 𝐵) → (𝐾‘𝑌) ∈ (LSubSp‘𝑊)) |
15 | 1, 13 | lspssid 19751 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ⊆ 𝐵) → 𝑌 ⊆ (𝐾‘𝑌)) |
16 | eqid 2821 | . . . . . . . . 9 ⊢ (LSpan‘𝑋) = (LSpan‘𝑋) | |
17 | 6, 13, 16, 12 | lsslsp 19781 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝐾‘𝑌) ∈ (LSubSp‘𝑊) ∧ 𝑌 ⊆ (𝐾‘𝑌)) → (𝐾‘𝑌) = ((LSpan‘𝑋)‘𝑌)) |
18 | 11, 14, 15, 17 | syl3anc 1367 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ⊆ 𝐵) → (𝐾‘𝑌) = ((LSpan‘𝑋)‘𝑌)) |
19 | 1, 13 | lspssv 19749 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ⊆ 𝐵) → (𝐾‘𝑌) ⊆ 𝐵) |
20 | 6, 1 | ressbas2 16549 | . . . . . . . 8 ⊢ ((𝐾‘𝑌) ⊆ 𝐵 → (𝐾‘𝑌) = (Base‘𝑋)) |
21 | 19, 20 | syl 17 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ⊆ 𝐵) → (𝐾‘𝑌) = (Base‘𝑋)) |
22 | 18, 21 | eqtr3d 2858 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ⊆ 𝐵) → ((LSpan‘𝑋)‘𝑌) = (Base‘𝑋)) |
23 | 22 | biantrud 534 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ⊆ 𝐵) → (𝑌 ∈ (LIndS‘𝑊) ↔ (𝑌 ∈ (LIndS‘𝑊) ∧ ((LSpan‘𝑋)‘𝑌) = (Base‘𝑋)))) |
24 | 12, 6 | lsslinds 20969 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝐾‘𝑌) ∈ (LSubSp‘𝑊) ∧ 𝑌 ⊆ (𝐾‘𝑌)) → (𝑌 ∈ (LIndS‘𝑋) ↔ 𝑌 ∈ (LIndS‘𝑊))) |
25 | 11, 14, 15, 24 | syl3anc 1367 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ⊆ 𝐵) → (𝑌 ∈ (LIndS‘𝑋) ↔ 𝑌 ∈ (LIndS‘𝑊))) |
26 | 25 | bicomd 225 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ⊆ 𝐵) → (𝑌 ∈ (LIndS‘𝑊) ↔ 𝑌 ∈ (LIndS‘𝑋))) |
27 | 26 | anbi1d 631 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ⊆ 𝐵) → ((𝑌 ∈ (LIndS‘𝑊) ∧ ((LSpan‘𝑋)‘𝑌) = (Base‘𝑋)) ↔ (𝑌 ∈ (LIndS‘𝑋) ∧ ((LSpan‘𝑋)‘𝑌) = (Base‘𝑋)))) |
28 | 23, 27 | bitrd 281 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ⊆ 𝐵) → (𝑌 ∈ (LIndS‘𝑊) ↔ (𝑌 ∈ (LIndS‘𝑋) ∧ ((LSpan‘𝑋)‘𝑌) = (Base‘𝑋)))) |
29 | 28 | ex 415 | . . 3 ⊢ (𝑊 ∈ LMod → (𝑌 ⊆ 𝐵 → (𝑌 ∈ (LIndS‘𝑊) ↔ (𝑌 ∈ (LIndS‘𝑋) ∧ ((LSpan‘𝑋)‘𝑌) = (Base‘𝑋))))) |
30 | 3, 10, 29 | pm5.21ndd 383 | . 2 ⊢ (𝑊 ∈ LMod → (𝑌 ∈ (LIndS‘𝑊) ↔ (𝑌 ∈ (LIndS‘𝑋) ∧ ((LSpan‘𝑋)‘𝑌) = (Base‘𝑋)))) |
31 | islinds3.j | . . 3 ⊢ 𝐽 = (LBasis‘𝑋) | |
32 | 4, 31, 16 | islbs4 20970 | . 2 ⊢ (𝑌 ∈ 𝐽 ↔ (𝑌 ∈ (LIndS‘𝑋) ∧ ((LSpan‘𝑋)‘𝑌) = (Base‘𝑋))) |
33 | 30, 32 | syl6bbr 291 | 1 ⊢ (𝑊 ∈ LMod → (𝑌 ∈ (LIndS‘𝑊) ↔ 𝑌 ∈ 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 ↾s cress 16478 LModclmod 19628 LSubSpclss 19697 LSpanclspn 19737 LBasisclbs 19840 LIndSclinds 20943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-sca 16575 df-vsca 16576 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-mgp 19234 df-ur 19246 df-ring 19293 df-lmod 19630 df-lss 19698 df-lsp 19738 df-lbs 19841 df-lindf 20944 df-linds 20945 |
This theorem is referenced by: (None) |
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