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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 20841 | . . 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 20839 | . . . 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 3197 | . . 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 511 | . 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 19082 | . . . 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 2728 | . . . . . 6 β’ (0gβπ) = (0gβπ) | |
| 13 | 12 | subg0cl 19089 | . . . . 5 β’ (π β (SubGrpβπ) β (0gβπ) β π) |
| 14 | 13 | ne0d 4336 | . . . 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 2728 | . . . . . . . . . 10 β’ (+gβπ) = (+gβπ) | |
| 17 | 16 | subgcl 19091 | . . . . . . . . 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 1117 | . . . . . . . 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 481 | . . . . . . 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 3149 | . . . . . 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 3166 | . . . . 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 3166 | . . . 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 454 | . . 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 20818 | . . 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 1341 | . 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 395 = wceq 1534 β wcel 2099 β wne 2937 βwral 3058 β wss 3947 β c0 4323 βcfv 6548 (class class class)co 7420 Basecbs 17180 +gcplusg 17233 Scalarcsca 17236 Β·π cvsca 17237 0gc0g 17421 SubGrpcsubg 19075 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-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 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: lssacs 20851 lmhmima 20932 lmhmpreima 20933 lmhmeql 20940 lsmcl 20968 dsmmlss 21678 issubassa2 21825 mplind 22014 mhplss 22079 fedgmullem2 33328 |
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