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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 20568 | . . 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 20566 | . . . 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 3201 | . . 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 19007 | . . . 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 19014 | . . . . 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 2733 | . . . . . . . . . 10 β’ (+gβπ) = (+gβπ) | |
| 17 | 16 | subgcl 19016 | . . . . . . . . 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 3153 | . . . . . 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 3170 | . . . . 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 3170 | . . . 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 20545 | . . 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 2941 βwral 3062 β wss 3949 β c0 4323 βcfv 6544 (class class class)co 7409 Basecbs 17144 +gcplusg 17197 Scalarcsca 17200 Β·π cvsca 17201 0gc0g 17385 SubGrpcsubg 19000 LModclmod 20471 LSubSpclss 20542 |
| 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 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
| 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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-0g 17387 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-grp 18822 df-minusg 18823 df-sbg 18824 df-subg 19003 df-mgp 19988 df-ur 20005 df-ring 20058 df-lmod 20473 df-lss 20543 |
| This theorem is referenced by: lssacs 20578 lmhmima 20658 lmhmpreima 20659 lmhmeql 20666 lsmcl 20694 dsmmlss 21299 issubassa2 21446 mplind 21631 mhplss 21698 fedgmullem2 32715 |
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