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Mirrors > Home > MPE Home > Th. List > lindsss | Structured version Visualization version GIF version |
Description: Any subset of an independent set is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
Ref | Expression |
---|---|
lindsss | β’ ((π β LMod β§ πΉ β (LIndSβπ) β§ πΊ β πΉ) β πΊ β (LIndSβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . . . 6 β’ (Baseβπ) = (Baseβπ) | |
2 | 1 | linds1 21232 | . . . . 5 β’ (πΉ β (LIndSβπ) β πΉ β (Baseβπ)) |
3 | 2 | adantl 483 | . . . 4 β’ ((π β LMod β§ πΉ β (LIndSβπ)) β πΉ β (Baseβπ)) |
4 | sstr2 3952 | . . . 4 β’ (πΊ β πΉ β (πΉ β (Baseβπ) β πΊ β (Baseβπ))) | |
5 | 3, 4 | syl5com 31 | . . 3 β’ ((π β LMod β§ πΉ β (LIndSβπ)) β (πΊ β πΉ β πΊ β (Baseβπ))) |
6 | 5 | 3impia 1118 | . 2 β’ ((π β LMod β§ πΉ β (LIndSβπ) β§ πΊ β πΉ) β πΊ β (Baseβπ)) |
7 | simp1 1137 | . . . 4 β’ ((π β LMod β§ πΉ β (LIndSβπ) β§ πΊ β πΉ) β π β LMod) | |
8 | linds2 21233 | . . . . 5 β’ (πΉ β (LIndSβπ) β ( I βΎ πΉ) LIndF π) | |
9 | 8 | 3ad2ant2 1135 | . . . 4 β’ ((π β LMod β§ πΉ β (LIndSβπ) β§ πΊ β πΉ) β ( I βΎ πΉ) LIndF π) |
10 | lindfres 21245 | . . . 4 β’ ((π β LMod β§ ( I βΎ πΉ) LIndF π) β (( I βΎ πΉ) βΎ πΊ) LIndF π) | |
11 | 7, 9, 10 | syl2anc 585 | . . 3 β’ ((π β LMod β§ πΉ β (LIndSβπ) β§ πΊ β πΉ) β (( I βΎ πΉ) βΎ πΊ) LIndF π) |
12 | resabs1 5968 | . . . . 5 β’ (πΊ β πΉ β (( I βΎ πΉ) βΎ πΊ) = ( I βΎ πΊ)) | |
13 | 12 | breq1d 5116 | . . . 4 β’ (πΊ β πΉ β ((( I βΎ πΉ) βΎ πΊ) LIndF π β ( I βΎ πΊ) LIndF π)) |
14 | 13 | 3ad2ant3 1136 | . . 3 β’ ((π β LMod β§ πΉ β (LIndSβπ) β§ πΊ β πΉ) β ((( I βΎ πΉ) βΎ πΊ) LIndF π β ( I βΎ πΊ) LIndF π)) |
15 | 11, 14 | mpbid 231 | . 2 β’ ((π β LMod β§ πΉ β (LIndSβπ) β§ πΊ β πΉ) β ( I βΎ πΊ) LIndF π) |
16 | 1 | islinds 21231 | . . 3 β’ (π β LMod β (πΊ β (LIndSβπ) β (πΊ β (Baseβπ) β§ ( I βΎ πΊ) LIndF π))) |
17 | 16 | 3ad2ant1 1134 | . 2 β’ ((π β LMod β§ πΉ β (LIndSβπ) β§ πΊ β πΉ) β (πΊ β (LIndSβπ) β (πΊ β (Baseβπ) β§ ( I βΎ πΊ) LIndF π))) |
18 | 6, 15, 17 | mpbir2and 712 | 1 β’ ((π β LMod β§ πΉ β (LIndSβπ) β§ πΊ β πΉ) β πΊ β (LIndSβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 β wcel 2107 β wss 3911 class class class wbr 5106 I cid 5531 βΎ cres 5636 βcfv 6497 Basecbs 17088 LModclmod 20336 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-1cn 11114 ax-addcl 11116 |
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-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-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-nn 12159 df-slot 17059 df-ndx 17071 df-base 17089 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-lmod 20338 df-lss 20408 df-lsp 20448 df-lindf 21228 df-linds 21229 |
This theorem is referenced by: islinds4 21257 linds2eq 32216 dimkerim 32379 |
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