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Mirrors > Home > MPE Home > Th. List > islss4 | Structured version Visualization version GIF version |
Description: A linear subspace is a subgroup which respects scalar multiplication. (Contributed by Stefan O'Rear, 11-Dec-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
Ref | Expression |
---|---|
islss4.f | β’ πΉ = (Scalarβπ) |
islss4.b | β’ π΅ = (BaseβπΉ) |
islss4.v | β’ π = (Baseβπ) |
islss4.t | β’ Β· = ( Β·π βπ) |
islss4.s | β’ π = (LSubSpβπ) |
Ref | Expression |
---|---|
islss4 | β’ (π β LMod β (π β π β (π β (SubGrpβπ) β§ βπ β π΅ βπ β π (π Β· π) β π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islss4.s | . . . 4 β’ π = (LSubSpβπ) | |
2 | 1 | lsssubg 20433 | . . 3 β’ ((π β LMod β§ π β π) β π β (SubGrpβπ)) |
3 | islss4.f | . . . . 5 β’ πΉ = (Scalarβπ) | |
4 | islss4.t | . . . . 5 β’ Β· = ( Β·π βπ) | |
5 | islss4.b | . . . . 5 β’ π΅ = (BaseβπΉ) | |
6 | 3, 4, 5, 1 | lssvscl 20431 | . . . 4 β’ (((π β LMod β§ π β π) β§ (π β π΅ β§ π β π)) β (π Β· π) β π) |
7 | 6 | ralrimivva 3194 | . . 3 β’ ((π β LMod β§ π β π) β βπ β π΅ βπ β π (π Β· π) β π) |
8 | 2, 7 | jca 513 | . 2 β’ ((π β LMod β§ π β π) β (π β (SubGrpβπ) β§ βπ β π΅ βπ β π (π Β· π) β π)) |
9 | islss4.v | . . . . 5 β’ π = (Baseβπ) | |
10 | 9 | subgss 18934 | . . . 4 β’ (π β (SubGrpβπ) β π β π) |
11 | 10 | ad2antrl 727 | . . 3 β’ ((π β LMod β§ (π β (SubGrpβπ) β§ βπ β π΅ βπ β π (π Β· π) β π)) β π β π) |
12 | eqid 2733 | . . . . . 6 β’ (0gβπ) = (0gβπ) | |
13 | 12 | subg0cl 18941 | . . . . 5 β’ (π β (SubGrpβπ) β (0gβπ) β π) |
14 | 13 | ne0d 4296 | . . . 4 β’ (π β (SubGrpβπ) β π β β ) |
15 | 14 | ad2antrl 727 | . . 3 β’ ((π β LMod β§ (π β (SubGrpβπ) β§ βπ β π΅ βπ β π (π Β· π) β π)) β π β β ) |
16 | eqid 2733 | . . . . . . . . . 10 β’ (+gβπ) = (+gβπ) | |
17 | 16 | subgcl 18943 | . . . . . . . . 9 β’ ((π β (SubGrpβπ) β§ (π Β· π) β π β§ π β π) β ((π Β· π)(+gβπ)π) β π) |
18 | 17 | 3exp 1120 | . . . . . . . 8 β’ (π β (SubGrpβπ) β ((π Β· π) β π β (π β π β ((π Β· π)(+gβπ)π) β π))) |
19 | 18 | adantl 483 | . . . . . . 7 β’ ((π β LMod β§ π β (SubGrpβπ)) β ((π Β· π) β π β (π β π β ((π Β· π)(+gβπ)π) β π))) |
20 | 19 | ralrimdv 3146 | . . . . . 6 β’ ((π β LMod β§ π β (SubGrpβπ)) β ((π Β· π) β π β βπ β π ((π Β· π)(+gβπ)π) β π)) |
21 | 20 | ralimdv 3163 | . . . . 5 β’ ((π β LMod β§ π β (SubGrpβπ)) β (βπ β π (π Β· π) β π β βπ β π βπ β π ((π Β· π)(+gβπ)π) β π)) |
22 | 21 | ralimdv 3163 | . . . 4 β’ ((π β LMod β§ π β (SubGrpβπ)) β (βπ β π΅ βπ β π (π Β· π) β π β βπ β π΅ βπ β π βπ β π ((π Β· π)(+gβπ)π) β π)) |
23 | 22 | impr 456 | . . 3 β’ ((π β LMod β§ (π β (SubGrpβπ) β§ βπ β π΅ βπ β π (π Β· π) β π)) β βπ β π΅ βπ β π βπ β π ((π Β· π)(+gβπ)π) β π) |
24 | 3, 5, 9, 16, 4, 1 | islss 20410 | . . 3 β’ (π β π β (π β π β§ π β β β§ βπ β π΅ βπ β π βπ β π ((π Β· π)(+gβπ)π) β π)) |
25 | 11, 15, 23, 24 | syl3anbrc 1344 | . 2 β’ ((π β LMod β§ (π β (SubGrpβπ) β§ βπ β π΅ βπ β π (π Β· π) β π)) β π β π) |
26 | 8, 25 | impbida 800 | 1 β’ (π β LMod β (π β π β (π β (SubGrpβπ) β§ βπ β π΅ βπ β π (π Β· π) β π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2940 βwral 3061 β wss 3911 β c0 4283 βcfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 Scalarcsca 17141 Β·π cvsca 17142 0gc0g 17326 SubGrpcsubg 18927 LModclmod 20336 LSubSpclss 20407 |
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-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-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-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 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 |
This theorem is referenced by: lssacs 20443 lmhmima 20523 lmhmpreima 20524 lmhmeql 20531 lsmcl 20559 dsmmlss 21166 issubassa2 21311 mplind 21494 mhplss 21561 fedgmullem2 32382 |
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