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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 20800 | . . 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 20798 | . . . 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 3192 | . . 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 19050 | . . . 4 β’ (π β (SubGrpβπ) β π β π) |
| 11 | 10 | ad2antrl 725 | . . 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 2724 | . . . . . 6 β’ (0gβπ) = (0gβπ) | |
| 13 | 12 | subg0cl 19057 | . . . . 5 β’ (π β (SubGrpβπ) β (0gβπ) β π) |
| 14 | 13 | ne0d 4328 | . . . 4 β’ (π β (SubGrpβπ) β π β β ) |
| 15 | 14 | ad2antrl 725 | . . 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 2724 | . . . . . . . . . 10 β’ (+gβπ) = (+gβπ) | |
| 17 | 16 | subgcl 19059 | . . . . . . . . 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 1116 | . . . . . . . 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 3144 | . . . . . 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 3161 | . . . . 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 3161 | . . . 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 20777 | . . 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 1340 | . 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 798 | 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 1533 β wcel 2098 β wne 2932 βwral 3053 β wss 3941 β c0 4315 βcfv 6534 (class class class)co 7402 Basecbs 17149 +gcplusg 17202 Scalarcsca 17205 Β·π cvsca 17206 0gc0g 17390 SubGrpcsubg 19043 LModclmod 20702 LSubSpclss 20774 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-0g 17392 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-grp 18862 df-minusg 18863 df-sbg 18864 df-subg 19046 df-mgp 20036 df-ur 20083 df-ring 20136 df-lmod 20704 df-lss 20775 |
| This theorem is referenced by: lssacs 20810 lmhmima 20891 lmhmpreima 20892 lmhmeql 20899 lsmcl 20927 dsmmlss 21628 issubassa2 21775 mplind 21962 mhplss 22027 fedgmullem2 33223 |
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