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Mirrors > Home > MPE Home > Th. List > lsslss | Structured version Visualization version GIF version |
Description: The subspaces of a subspace are the smaller subspaces. (Contributed by Stefan O'Rear, 12-Dec-2014.) |
Ref | Expression |
---|---|
lsslss.x | ⊢ 𝑋 = (𝑊 ↾s 𝑈) |
lsslss.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lsslss.t | ⊢ 𝑇 = (LSubSp‘𝑋) |
Ref | Expression |
---|---|
lsslss | ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑉 ∈ 𝑇 ↔ (𝑉 ∈ 𝑆 ∧ 𝑉 ⊆ 𝑈))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsslss.x | . . . 4 ⊢ 𝑋 = (𝑊 ↾s 𝑈) | |
2 | lsslss.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | 1, 2 | lsslmod 20844 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ LMod) |
4 | eqid 2728 | . . . 4 ⊢ (𝑋 ↾s 𝑉) = (𝑋 ↾s 𝑉) | |
5 | eqid 2728 | . . . 4 ⊢ (Base‘𝑋) = (Base‘𝑋) | |
6 | lsslss.t | . . . 4 ⊢ 𝑇 = (LSubSp‘𝑋) | |
7 | 4, 5, 6 | islss3 20843 | . . 3 ⊢ (𝑋 ∈ LMod → (𝑉 ∈ 𝑇 ↔ (𝑉 ⊆ (Base‘𝑋) ∧ (𝑋 ↾s 𝑉) ∈ LMod))) |
8 | 3, 7 | syl 17 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑉 ∈ 𝑇 ↔ (𝑉 ⊆ (Base‘𝑋) ∧ (𝑋 ↾s 𝑉) ∈ LMod))) |
9 | eqid 2728 | . . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
10 | 9, 2 | lssss 20820 | . . . . . 6 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ (Base‘𝑊)) |
11 | 10 | adantl 481 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑈 ⊆ (Base‘𝑊)) |
12 | 1, 9 | ressbas2 17218 | . . . . 5 ⊢ (𝑈 ⊆ (Base‘𝑊) → 𝑈 = (Base‘𝑋)) |
13 | 11, 12 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑈 = (Base‘𝑋)) |
14 | 13 | sseq2d 4012 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑉 ⊆ 𝑈 ↔ 𝑉 ⊆ (Base‘𝑋))) |
15 | 14 | anbi1d 630 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → ((𝑉 ⊆ 𝑈 ∧ (𝑋 ↾s 𝑉) ∈ LMod) ↔ (𝑉 ⊆ (Base‘𝑋) ∧ (𝑋 ↾s 𝑉) ∈ LMod))) |
16 | sstr2 3987 | . . . . . . 7 ⊢ (𝑉 ⊆ 𝑈 → (𝑈 ⊆ (Base‘𝑊) → 𝑉 ⊆ (Base‘𝑊))) | |
17 | 11, 16 | mpan9 506 | . . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ 𝑉 ⊆ 𝑈) → 𝑉 ⊆ (Base‘𝑊)) |
18 | 17 | biantrurd 532 | . . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ 𝑉 ⊆ 𝑈) → ((𝑊 ↾s 𝑉) ∈ LMod ↔ (𝑉 ⊆ (Base‘𝑊) ∧ (𝑊 ↾s 𝑉) ∈ LMod))) |
19 | 1 | oveq1i 7430 | . . . . . . 7 ⊢ (𝑋 ↾s 𝑉) = ((𝑊 ↾s 𝑈) ↾s 𝑉) |
20 | ressabs 17230 | . . . . . . . 8 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑉 ⊆ 𝑈) → ((𝑊 ↾s 𝑈) ↾s 𝑉) = (𝑊 ↾s 𝑉)) | |
21 | 20 | adantll 713 | . . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ 𝑉 ⊆ 𝑈) → ((𝑊 ↾s 𝑈) ↾s 𝑉) = (𝑊 ↾s 𝑉)) |
22 | 19, 21 | eqtrid 2780 | . . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ 𝑉 ⊆ 𝑈) → (𝑋 ↾s 𝑉) = (𝑊 ↾s 𝑉)) |
23 | 22 | eleq1d 2814 | . . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ 𝑉 ⊆ 𝑈) → ((𝑋 ↾s 𝑉) ∈ LMod ↔ (𝑊 ↾s 𝑉) ∈ LMod)) |
24 | eqid 2728 | . . . . . . 7 ⊢ (𝑊 ↾s 𝑉) = (𝑊 ↾s 𝑉) | |
25 | 24, 9, 2 | islss3 20843 | . . . . . 6 ⊢ (𝑊 ∈ LMod → (𝑉 ∈ 𝑆 ↔ (𝑉 ⊆ (Base‘𝑊) ∧ (𝑊 ↾s 𝑉) ∈ LMod))) |
26 | 25 | ad2antrr 725 | . . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ 𝑉 ⊆ 𝑈) → (𝑉 ∈ 𝑆 ↔ (𝑉 ⊆ (Base‘𝑊) ∧ (𝑊 ↾s 𝑉) ∈ LMod))) |
27 | 18, 23, 26 | 3bitr4d 311 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ 𝑉 ⊆ 𝑈) → ((𝑋 ↾s 𝑉) ∈ LMod ↔ 𝑉 ∈ 𝑆)) |
28 | 27 | pm5.32da 578 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → ((𝑉 ⊆ 𝑈 ∧ (𝑋 ↾s 𝑉) ∈ LMod) ↔ (𝑉 ⊆ 𝑈 ∧ 𝑉 ∈ 𝑆))) |
29 | 28 | biancomd 463 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → ((𝑉 ⊆ 𝑈 ∧ (𝑋 ↾s 𝑉) ∈ LMod) ↔ (𝑉 ∈ 𝑆 ∧ 𝑉 ⊆ 𝑈))) |
30 | 8, 15, 29 | 3bitr2d 307 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑉 ∈ 𝑇 ↔ (𝑉 ∈ 𝑆 ∧ 𝑉 ⊆ 𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 ‘cfv 6548 (class class class)co 7420 Basecbs 17180 ↾s cress 17209 LModclmod 20743 LSubSpclss 20815 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-sca 17249 df-vsca 17250 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-minusg 18894 df-sbg 18895 df-subg 19078 df-mgp 20075 df-ur 20122 df-ring 20175 df-lmod 20745 df-lss 20816 |
This theorem is referenced by: lsslsp 20899 lsslspOLD 20900 mplbas2 21980 mplind 22014 lcdlss 41092 lnmlsslnm 42505 lmhmlnmsplit 42511 |
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