Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2622 | . . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | lsslindf.u | . . . . . . . 8 ⊢ 𝑈 = (LSubSp‘𝑊) | |
3 | 1, 2 | lssss 18937 | . . . . . . 7 ⊢ (𝑆 ∈ 𝑈 → 𝑆 ⊆ (Base‘𝑊)) |
4 | lsslindf.x | . . . . . . . 8 ⊢ 𝑋 = (𝑊 ↾s 𝑆) | |
5 | 4, 1 | ressbas2 15931 | . . . . . . 7 ⊢ (𝑆 ⊆ (Base‘𝑊) → 𝑆 = (Base‘𝑋)) |
6 | 3, 5 | syl 17 | . . . . . 6 ⊢ (𝑆 ∈ 𝑈 → 𝑆 = (Base‘𝑋)) |
7 | 6 | 3ad2ant2 1083 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → 𝑆 = (Base‘𝑋)) |
8 | 7 | sseq2d 3633 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → (𝐹 ⊆ 𝑆 ↔ 𝐹 ⊆ (Base‘𝑋))) |
9 | 3 | 3ad2ant2 1083 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → 𝑆 ⊆ (Base‘𝑊)) |
10 | sstr2 3610 | . . . . . 6 ⊢ (𝐹 ⊆ 𝑆 → (𝑆 ⊆ (Base‘𝑊) → 𝐹 ⊆ (Base‘𝑊))) | |
11 | 9, 10 | mpan9 486 | . . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) ∧ 𝐹 ⊆ 𝑆) → 𝐹 ⊆ (Base‘𝑊)) |
12 | simpl3 1066 | . . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) ∧ 𝐹 ⊆ (Base‘𝑊)) → 𝐹 ⊆ 𝑆) | |
13 | 11, 12 | impbida 877 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → (𝐹 ⊆ 𝑆 ↔ 𝐹 ⊆ (Base‘𝑊))) |
14 | 8, 13 | bitr3d 270 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → (𝐹 ⊆ (Base‘𝑋) ↔ 𝐹 ⊆ (Base‘𝑊))) |
15 | rnresi 5479 | . . . . 5 ⊢ ran ( I ↾ 𝐹) = 𝐹 | |
16 | 15 | sseq1i 3629 | . . . 4 ⊢ (ran ( I ↾ 𝐹) ⊆ 𝑆 ↔ 𝐹 ⊆ 𝑆) |
17 | 2, 4 | lsslindf 20169 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran ( I ↾ 𝐹) ⊆ 𝑆) → (( I ↾ 𝐹) LIndF 𝑋 ↔ ( I ↾ 𝐹) LIndF 𝑊)) |
18 | 16, 17 | syl3an3br 1367 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → (( I ↾ 𝐹) LIndF 𝑋 ↔ ( I ↾ 𝐹) LIndF 𝑊)) |
19 | 14, 18 | anbi12d 747 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → ((𝐹 ⊆ (Base‘𝑋) ∧ ( I ↾ 𝐹) LIndF 𝑋) ↔ (𝐹 ⊆ (Base‘𝑊) ∧ ( I ↾ 𝐹) LIndF 𝑊))) |
20 | ovex 6678 | . . . 4 ⊢ (𝑊 ↾s 𝑆) ∈ V | |
21 | 4, 20 | eqeltri 2697 | . . 3 ⊢ 𝑋 ∈ V |
22 | eqid 2622 | . . . 4 ⊢ (Base‘𝑋) = (Base‘𝑋) | |
23 | 22 | islinds 20148 | . . 3 ⊢ (𝑋 ∈ V → (𝐹 ∈ (LIndS‘𝑋) ↔ (𝐹 ⊆ (Base‘𝑋) ∧ ( I ↾ 𝐹) LIndF 𝑋))) |
24 | 21, 23 | mp1i 13 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → (𝐹 ∈ (LIndS‘𝑋) ↔ (𝐹 ⊆ (Base‘𝑋) ∧ ( I ↾ 𝐹) LIndF 𝑋))) |
25 | 1 | islinds 20148 | . . 3 ⊢ (𝑊 ∈ LMod → (𝐹 ∈ (LIndS‘𝑊) ↔ (𝐹 ⊆ (Base‘𝑊) ∧ ( I ↾ 𝐹) LIndF 𝑊))) |
26 | 25 | 3ad2ant1 1082 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → (𝐹 ∈ (LIndS‘𝑊) ↔ (𝐹 ⊆ (Base‘𝑊) ∧ ( I ↾ 𝐹) LIndF 𝑊))) |
27 | 19, 24, 26 | 3bitr4d 300 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → (𝐹 ∈ (LIndS‘𝑋) ↔ 𝐹 ∈ (LIndS‘𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 class class class wbr 4653 I cid 5023 ran crn 5115 ↾ cres 5116 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 ↾s cress 15858 LModclmod 18863 LSubSpclss 18932 LIndF clindf 20143 LIndSclinds 20144 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-sca 15957 df-vsca 15958 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-mgp 18490 df-ur 18502 df-ring 18549 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lindf 20145 df-linds 20146 |
This theorem is referenced by: islinds3 20173 |
Copyright terms: Public domain | W3C validator |