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Mirrors > Home > MPE Home > Th. List > lsslinds | Structured version Visualization version GIF version |
Description: Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
Ref | Expression |
---|---|
lsslindf.u | ⊢ 𝑈 = (LSubSp‘𝑊) |
lsslindf.x | ⊢ 𝑋 = (𝑊 ↾s 𝑆) |
Ref | Expression |
---|---|
lsslinds | ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → (𝐹 ∈ (LIndS‘𝑋) ↔ 𝐹 ∈ (LIndS‘𝑊))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2734 | . . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | lsslindf.u | . . . . . . . 8 ⊢ 𝑈 = (LSubSp‘𝑊) | |
3 | 1, 2 | lssss 20951 | . . . . . . 7 ⊢ (𝑆 ∈ 𝑈 → 𝑆 ⊆ (Base‘𝑊)) |
4 | lsslindf.x | . . . . . . . 8 ⊢ 𝑋 = (𝑊 ↾s 𝑆) | |
5 | 4, 1 | ressbas2 17282 | . . . . . . 7 ⊢ (𝑆 ⊆ (Base‘𝑊) → 𝑆 = (Base‘𝑋)) |
6 | 3, 5 | syl 17 | . . . . . 6 ⊢ (𝑆 ∈ 𝑈 → 𝑆 = (Base‘𝑋)) |
7 | 6 | 3ad2ant2 1133 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → 𝑆 = (Base‘𝑋)) |
8 | 7 | sseq2d 4027 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → (𝐹 ⊆ 𝑆 ↔ 𝐹 ⊆ (Base‘𝑋))) |
9 | 3 | 3ad2ant2 1133 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → 𝑆 ⊆ (Base‘𝑊)) |
10 | sstr2 4001 | . . . . . 6 ⊢ (𝐹 ⊆ 𝑆 → (𝑆 ⊆ (Base‘𝑊) → 𝐹 ⊆ (Base‘𝑊))) | |
11 | 9, 10 | mpan9 506 | . . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) ∧ 𝐹 ⊆ 𝑆) → 𝐹 ⊆ (Base‘𝑊)) |
12 | simpl3 1192 | . . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) ∧ 𝐹 ⊆ (Base‘𝑊)) → 𝐹 ⊆ 𝑆) | |
13 | 11, 12 | impbida 801 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → (𝐹 ⊆ 𝑆 ↔ 𝐹 ⊆ (Base‘𝑊))) |
14 | 8, 13 | bitr3d 281 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → (𝐹 ⊆ (Base‘𝑋) ↔ 𝐹 ⊆ (Base‘𝑊))) |
15 | rnresi 6094 | . . . . 5 ⊢ ran ( I ↾ 𝐹) = 𝐹 | |
16 | 15 | sseq1i 4023 | . . . 4 ⊢ (ran ( I ↾ 𝐹) ⊆ 𝑆 ↔ 𝐹 ⊆ 𝑆) |
17 | 2, 4 | lsslindf 21867 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran ( I ↾ 𝐹) ⊆ 𝑆) → (( I ↾ 𝐹) LIndF 𝑋 ↔ ( I ↾ 𝐹) LIndF 𝑊)) |
18 | 16, 17 | syl3an3br 1407 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → (( I ↾ 𝐹) LIndF 𝑋 ↔ ( I ↾ 𝐹) LIndF 𝑊)) |
19 | 14, 18 | anbi12d 632 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → ((𝐹 ⊆ (Base‘𝑋) ∧ ( I ↾ 𝐹) LIndF 𝑋) ↔ (𝐹 ⊆ (Base‘𝑊) ∧ ( I ↾ 𝐹) LIndF 𝑊))) |
20 | 4 | ovexi 7464 | . . 3 ⊢ 𝑋 ∈ V |
21 | eqid 2734 | . . . 4 ⊢ (Base‘𝑋) = (Base‘𝑋) | |
22 | 21 | islinds 21846 | . . 3 ⊢ (𝑋 ∈ V → (𝐹 ∈ (LIndS‘𝑋) ↔ (𝐹 ⊆ (Base‘𝑋) ∧ ( I ↾ 𝐹) LIndF 𝑋))) |
23 | 20, 22 | mp1i 13 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → (𝐹 ∈ (LIndS‘𝑋) ↔ (𝐹 ⊆ (Base‘𝑋) ∧ ( I ↾ 𝐹) LIndF 𝑋))) |
24 | 1 | islinds 21846 | . . 3 ⊢ (𝑊 ∈ LMod → (𝐹 ∈ (LIndS‘𝑊) ↔ (𝐹 ⊆ (Base‘𝑊) ∧ ( I ↾ 𝐹) LIndF 𝑊))) |
25 | 24 | 3ad2ant1 1132 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → (𝐹 ∈ (LIndS‘𝑊) ↔ (𝐹 ⊆ (Base‘𝑊) ∧ ( I ↾ 𝐹) LIndF 𝑊))) |
26 | 19, 23, 25 | 3bitr4d 311 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → (𝐹 ∈ (LIndS‘𝑋) ↔ 𝐹 ∈ (LIndS‘𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ⊆ wss 3962 class class class wbr 5147 I cid 5581 ran crn 5689 ↾ cres 5690 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 ↾s cress 17273 LModclmod 20874 LSubSpclss 20946 LIndF clindf 21841 LIndSclinds 21842 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-sca 17313 df-vsca 17314 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 df-sbg 18968 df-subg 19153 df-mgp 20152 df-ur 20199 df-ring 20252 df-lmod 20876 df-lss 20947 df-lsp 20987 df-lindf 21843 df-linds 21844 |
This theorem is referenced by: islinds3 21871 lssdimle 33634 dimkerim 33654 fedgmullem2 33657 |
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