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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 | β’ π = (LSubSpβπ) |
lsslindf.x | β’ π = (π βΎs π) |
Ref | Expression |
---|---|
lsslinds | β’ ((π β LMod β§ π β π β§ πΉ β π) β (πΉ β (LIndSβπ) β πΉ β (LIndSβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . . . . . 8 β’ (Baseβπ) = (Baseβπ) | |
2 | lsslindf.u | . . . . . . . 8 β’ π = (LSubSpβπ) | |
3 | 1, 2 | lssss 20412 | . . . . . . 7 β’ (π β π β π β (Baseβπ)) |
4 | lsslindf.x | . . . . . . . 8 β’ π = (π βΎs π) | |
5 | 4, 1 | ressbas2 17125 | . . . . . . 7 β’ (π β (Baseβπ) β π = (Baseβπ)) |
6 | 3, 5 | syl 17 | . . . . . 6 β’ (π β π β π = (Baseβπ)) |
7 | 6 | 3ad2ant2 1135 | . . . . 5 β’ ((π β LMod β§ π β π β§ πΉ β π) β π = (Baseβπ)) |
8 | 7 | sseq2d 3977 | . . . 4 β’ ((π β LMod β§ π β π β§ πΉ β π) β (πΉ β π β πΉ β (Baseβπ))) |
9 | 3 | 3ad2ant2 1135 | . . . . . 6 β’ ((π β LMod β§ π β π β§ πΉ β π) β π β (Baseβπ)) |
10 | sstr2 3952 | . . . . . 6 β’ (πΉ β π β (π β (Baseβπ) β πΉ β (Baseβπ))) | |
11 | 9, 10 | mpan9 508 | . . . . 5 β’ (((π β LMod β§ π β π β§ πΉ β π) β§ πΉ β π) β πΉ β (Baseβπ)) |
12 | simpl3 1194 | . . . . 5 β’ (((π β LMod β§ π β π β§ πΉ β π) β§ πΉ β (Baseβπ)) β πΉ β π) | |
13 | 11, 12 | impbida 800 | . . . 4 β’ ((π β LMod β§ π β π β§ πΉ β π) β (πΉ β π β πΉ β (Baseβπ))) |
14 | 8, 13 | bitr3d 281 | . . 3 β’ ((π β LMod β§ π β π β§ πΉ β π) β (πΉ β (Baseβπ) β πΉ β (Baseβπ))) |
15 | rnresi 6028 | . . . . 5 β’ ran ( I βΎ πΉ) = πΉ | |
16 | 15 | sseq1i 3973 | . . . 4 β’ (ran ( I βΎ πΉ) β π β πΉ β π) |
17 | 2, 4 | lsslindf 21252 | . . . 4 β’ ((π β LMod β§ π β π β§ ran ( I βΎ πΉ) β π) β (( I βΎ πΉ) LIndF π β ( I βΎ πΉ) LIndF π)) |
18 | 16, 17 | syl3an3br 1409 | . . 3 β’ ((π β LMod β§ π β π β§ πΉ β π) β (( I βΎ πΉ) LIndF π β ( I βΎ πΉ) LIndF π)) |
19 | 14, 18 | anbi12d 632 | . 2 β’ ((π β LMod β§ π β π β§ πΉ β π) β ((πΉ β (Baseβπ) β§ ( I βΎ πΉ) LIndF π) β (πΉ β (Baseβπ) β§ ( I βΎ πΉ) LIndF π))) |
20 | 4 | ovexi 7392 | . . 3 β’ π β V |
21 | eqid 2733 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
22 | 21 | islinds 21231 | . . 3 β’ (π β V β (πΉ β (LIndSβπ) β (πΉ β (Baseβπ) β§ ( I βΎ πΉ) LIndF π))) |
23 | 20, 22 | mp1i 13 | . 2 β’ ((π β LMod β§ π β π β§ πΉ β π) β (πΉ β (LIndSβπ) β (πΉ β (Baseβπ) β§ ( I βΎ πΉ) LIndF π))) |
24 | 1 | islinds 21231 | . . 3 β’ (π β LMod β (πΉ β (LIndSβπ) β (πΉ β (Baseβπ) β§ ( I βΎ πΉ) LIndF π))) |
25 | 24 | 3ad2ant1 1134 | . 2 β’ ((π β LMod β§ π β π β§ πΉ β π) β (πΉ β (LIndSβπ) β (πΉ β (Baseβπ) β§ ( I βΎ πΉ) LIndF π))) |
26 | 19, 23, 25 | 3bitr4d 311 | 1 β’ ((π β LMod β§ π β π β§ πΉ β π) β (πΉ β (LIndSβπ) β πΉ β (LIndSβπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 Vcvv 3444 β wss 3911 class class class wbr 5106 I cid 5531 ran crn 5635 βΎ cres 5636 βcfv 6497 (class class class)co 7358 Basecbs 17088 βΎs cress 17117 LModclmod 20336 LSubSpclss 20407 LIndF clindf 21226 LIndSclinds 21227 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-sca 17154 df-vsca 17155 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-minusg 18757 df-sbg 18758 df-subg 18930 df-mgp 19902 df-ur 19919 df-ring 19971 df-lmod 20338 df-lss 20408 df-lsp 20448 df-lindf 21228 df-linds 21229 |
This theorem is referenced by: islinds3 21256 lssdimle 32360 dimkerim 32379 fedgmullem2 32382 |
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