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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 2737 | . . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 2 | lsslindf.u | . . . . . . . 8 ⊢ 𝑈 = (LSubSp‘𝑊) | |
| 3 | 1, 2 | lssss 20934 | . . . . . . 7 ⊢ (𝑆 ∈ 𝑈 → 𝑆 ⊆ (Base‘𝑊)) |
| 4 | lsslindf.x | . . . . . . . 8 ⊢ 𝑋 = (𝑊 ↾s 𝑆) | |
| 5 | 4, 1 | ressbas2 17283 | . . . . . . 7 ⊢ (𝑆 ⊆ (Base‘𝑊) → 𝑆 = (Base‘𝑋)) |
| 6 | 3, 5 | syl 17 | . . . . . 6 ⊢ (𝑆 ∈ 𝑈 → 𝑆 = (Base‘𝑋)) |
| 7 | 6 | 3ad2ant2 1135 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → 𝑆 = (Base‘𝑋)) |
| 8 | 7 | sseq2d 4016 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → (𝐹 ⊆ 𝑆 ↔ 𝐹 ⊆ (Base‘𝑋))) |
| 9 | 3 | 3ad2ant2 1135 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → 𝑆 ⊆ (Base‘𝑊)) |
| 10 | sstr2 3990 | . . . . . 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 1194 | . . . . 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 6093 | . . . . 5 ⊢ ran ( I ↾ 𝐹) = 𝐹 | |
| 16 | 15 | sseq1i 4012 | . . . 4 ⊢ (ran ( I ↾ 𝐹) ⊆ 𝑆 ↔ 𝐹 ⊆ 𝑆) |
| 17 | 2, 4 | lsslindf 21850 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran ( I ↾ 𝐹) ⊆ 𝑆) → (( I ↾ 𝐹) LIndF 𝑋 ↔ ( I ↾ 𝐹) LIndF 𝑊)) |
| 18 | 16, 17 | syl3an3br 1410 | . . 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 7465 | . . 3 ⊢ 𝑋 ∈ V |
| 21 | eqid 2737 | . . . 4 ⊢ (Base‘𝑋) = (Base‘𝑋) | |
| 22 | 21 | islinds 21829 | . . 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 21829 | . . 3 ⊢ (𝑊 ∈ LMod → (𝐹 ∈ (LIndS‘𝑊) ↔ (𝐹 ⊆ (Base‘𝑊) ∧ ( I ↾ 𝐹) LIndF 𝑊))) |
| 25 | 24 | 3ad2ant1 1134 | . 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 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 I cid 5577 ran crn 5686 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 ↾s cress 17274 LModclmod 20858 LSubSpclss 20929 LIndF clindf 21824 LIndSclinds 21825 |
| 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-rep 5279 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-int 4947 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-1st 8014 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-3 12330 df-4 12331 df-5 12332 df-6 12333 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-sca 17313 df-vsca 17314 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-mgp 20138 df-ur 20179 df-ring 20232 df-lmod 20860 df-lss 20930 df-lsp 20970 df-lindf 21826 df-linds 21827 |
| This theorem is referenced by: islinds3 21854 lssdimle 33658 dimkerim 33678 fedgmullem2 33681 |
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