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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 20671 | . . . 4 ⊢ (𝑌 ∈ (LIndS‘𝑊) → 𝑌 ⊆ 𝐵) |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑊 ∈ LMod → (𝑌 ∈ (LIndS‘𝑊) → 𝑌 ⊆ 𝐵)) |
4 | eqid 2771 | . . . . . . 7 ⊢ (Base‘𝑋) = (Base‘𝑋) | |
5 | 4 | linds1 20671 | . . . . . 6 ⊢ (𝑌 ∈ (LIndS‘𝑋) → 𝑌 ⊆ (Base‘𝑋)) |
6 | islinds3.x | . . . . . . 7 ⊢ 𝑋 = (𝑊 ↾s (𝐾‘𝑌)) | |
7 | 6, 1 | ressbasss 16410 | . . . . . 6 ⊢ (Base‘𝑋) ⊆ 𝐵 |
8 | 5, 7 | syl6ss 3863 | . . . . 5 ⊢ (𝑌 ∈ (LIndS‘𝑋) → 𝑌 ⊆ 𝐵) |
9 | 8 | adantr 473 | . . . 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 475 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ⊆ 𝐵) → 𝑊 ∈ LMod) | |
12 | eqid 2771 | . . . . . . . . 9 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
13 | islinds3.k | . . . . . . . . 9 ⊢ 𝐾 = (LSpan‘𝑊) | |
14 | 1, 12, 13 | lspcl 19482 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ⊆ 𝐵) → (𝐾‘𝑌) ∈ (LSubSp‘𝑊)) |
15 | 1, 13 | lspssid 19491 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ⊆ 𝐵) → 𝑌 ⊆ (𝐾‘𝑌)) |
16 | eqid 2771 | . . . . . . . . 9 ⊢ (LSpan‘𝑋) = (LSpan‘𝑋) | |
17 | 6, 13, 16, 12 | lsslsp 19521 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝐾‘𝑌) ∈ (LSubSp‘𝑊) ∧ 𝑌 ⊆ (𝐾‘𝑌)) → (𝐾‘𝑌) = ((LSpan‘𝑋)‘𝑌)) |
18 | 11, 14, 15, 17 | syl3anc 1352 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ⊆ 𝐵) → (𝐾‘𝑌) = ((LSpan‘𝑋)‘𝑌)) |
19 | 1, 13 | lspssv 19489 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ⊆ 𝐵) → (𝐾‘𝑌) ⊆ 𝐵) |
20 | 6, 1 | ressbas2 16409 | . . . . . . . 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 2809 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ⊆ 𝐵) → ((LSpan‘𝑋)‘𝑌) = (Base‘𝑋)) |
23 | 22 | biantrud 524 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ⊆ 𝐵) → (𝑌 ∈ (LIndS‘𝑊) ↔ (𝑌 ∈ (LIndS‘𝑊) ∧ ((LSpan‘𝑋)‘𝑌) = (Base‘𝑋)))) |
24 | 12, 6 | lsslinds 20692 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝐾‘𝑌) ∈ (LSubSp‘𝑊) ∧ 𝑌 ⊆ (𝐾‘𝑌)) → (𝑌 ∈ (LIndS‘𝑋) ↔ 𝑌 ∈ (LIndS‘𝑊))) |
25 | 11, 14, 15, 24 | syl3anc 1352 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ⊆ 𝐵) → (𝑌 ∈ (LIndS‘𝑋) ↔ 𝑌 ∈ (LIndS‘𝑊))) |
26 | 25 | bicomd 215 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ⊆ 𝐵) → (𝑌 ∈ (LIndS‘𝑊) ↔ 𝑌 ∈ (LIndS‘𝑋))) |
27 | 26 | anbi1d 621 | . . . . 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 271 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ⊆ 𝐵) → (𝑌 ∈ (LIndS‘𝑊) ↔ (𝑌 ∈ (LIndS‘𝑋) ∧ ((LSpan‘𝑋)‘𝑌) = (Base‘𝑋)))) |
29 | 28 | ex 405 | . . 3 ⊢ (𝑊 ∈ LMod → (𝑌 ⊆ 𝐵 → (𝑌 ∈ (LIndS‘𝑊) ↔ (𝑌 ∈ (LIndS‘𝑋) ∧ ((LSpan‘𝑋)‘𝑌) = (Base‘𝑋))))) |
30 | 3, 10, 29 | pm5.21ndd 372 | . 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 20693 | . 2 ⊢ (𝑌 ∈ 𝐽 ↔ (𝑌 ∈ (LIndS‘𝑋) ∧ ((LSpan‘𝑋)‘𝑌) = (Base‘𝑋))) |
33 | 30, 32 | syl6bbr 281 | 1 ⊢ (𝑊 ∈ LMod → (𝑌 ∈ (LIndS‘𝑊) ↔ 𝑌 ∈ 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ⊆ wss 3822 ‘cfv 6185 (class class class)co 6974 Basecbs 16337 ↾s cress 16338 LModclmod 19368 LSubSpclss 19437 LSpanclspn 19477 LBasisclbs 19580 LIndSclinds 20666 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-sca 16435 df-vsca 16436 df-0g 16569 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-grp 17906 df-minusg 17907 df-sbg 17908 df-subg 18072 df-mgp 18975 df-ur 18987 df-ring 19034 df-lmod 19370 df-lss 19438 df-lsp 19478 df-lbs 19581 df-lindf 20667 df-linds 20668 |
This theorem is referenced by: (None) |
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