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Mirrors > Home > MPE Home > Th. List > issubassa | Structured version Visualization version GIF version |
Description: The subalgebras of an associative algebra are exactly the subrings (under the ring multiplication) that are simultaneously subspaces (under the scalar multiplication from the vector space). (Contributed by Mario Carneiro, 7-Jan-2015.) |
Ref | Expression |
---|---|
issubassa.s | β’ π = (π βΎs π΄) |
issubassa.l | β’ πΏ = (LSubSpβπ) |
issubassa.v | β’ π = (Baseβπ) |
issubassa.o | β’ 1 = (1rβπ) |
Ref | Expression |
---|---|
issubassa | β’ ((π β AssAlg β§ 1 β π΄ β§ π΄ β π) β (π β AssAlg β (π΄ β (SubRingβπ) β§ π΄ β πΏ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1192 | . . . . 5 β’ (((π β AssAlg β§ 1 β π΄ β§ π΄ β π) β§ π β AssAlg) β π β AssAlg) | |
2 | assaring 21283 | . . . . 5 β’ (π β AssAlg β π β Ring) | |
3 | 1, 2 | syl 17 | . . . 4 β’ (((π β AssAlg β§ 1 β π΄ β§ π΄ β π) β§ π β AssAlg) β π β Ring) |
4 | issubassa.s | . . . . 5 β’ π = (π βΎs π΄) | |
5 | assaring 21283 | . . . . . 6 β’ (π β AssAlg β π β Ring) | |
6 | 5 | adantl 483 | . . . . 5 β’ (((π β AssAlg β§ 1 β π΄ β§ π΄ β π) β§ π β AssAlg) β π β Ring) |
7 | 4, 6 | eqeltrrid 2839 | . . . 4 β’ (((π β AssAlg β§ 1 β π΄ β§ π΄ β π) β§ π β AssAlg) β (π βΎs π΄) β Ring) |
8 | simpl3 1194 | . . . . 5 β’ (((π β AssAlg β§ 1 β π΄ β§ π΄ β π) β§ π β AssAlg) β π΄ β π) | |
9 | simpl2 1193 | . . . . 5 β’ (((π β AssAlg β§ 1 β π΄ β§ π΄ β π) β§ π β AssAlg) β 1 β π΄) | |
10 | 8, 9 | jca 513 | . . . 4 β’ (((π β AssAlg β§ 1 β π΄ β§ π΄ β π) β§ π β AssAlg) β (π΄ β π β§ 1 β π΄)) |
11 | issubassa.v | . . . . 5 β’ π = (Baseβπ) | |
12 | issubassa.o | . . . . 5 β’ 1 = (1rβπ) | |
13 | 11, 12 | issubrg 20236 | . . . 4 β’ (π΄ β (SubRingβπ) β ((π β Ring β§ (π βΎs π΄) β Ring) β§ (π΄ β π β§ 1 β π΄))) |
14 | 3, 7, 10, 13 | syl21anbrc 1345 | . . 3 β’ (((π β AssAlg β§ 1 β π΄ β§ π΄ β π) β§ π β AssAlg) β π΄ β (SubRingβπ)) |
15 | assalmod 21282 | . . . . 5 β’ (π β AssAlg β π β LMod) | |
16 | 15 | adantl 483 | . . . 4 β’ (((π β AssAlg β§ 1 β π΄ β§ π΄ β π) β§ π β AssAlg) β π β LMod) |
17 | assalmod 21282 | . . . . 5 β’ (π β AssAlg β π β LMod) | |
18 | issubassa.l | . . . . . 6 β’ πΏ = (LSubSpβπ) | |
19 | 4, 11, 18 | islss3 20435 | . . . . 5 β’ (π β LMod β (π΄ β πΏ β (π΄ β π β§ π β LMod))) |
20 | 1, 17, 19 | 3syl 18 | . . . 4 β’ (((π β AssAlg β§ 1 β π΄ β§ π΄ β π) β§ π β AssAlg) β (π΄ β πΏ β (π΄ β π β§ π β LMod))) |
21 | 8, 16, 20 | mpbir2and 712 | . . 3 β’ (((π β AssAlg β§ 1 β π΄ β§ π΄ β π) β§ π β AssAlg) β π΄ β πΏ) |
22 | 14, 21 | jca 513 | . 2 β’ (((π β AssAlg β§ 1 β π΄ β§ π΄ β π) β§ π β AssAlg) β (π΄ β (SubRingβπ) β§ π΄ β πΏ)) |
23 | 4, 18 | issubassa3 21287 | . . 3 β’ ((π β AssAlg β§ (π΄ β (SubRingβπ) β§ π΄ β πΏ)) β π β AssAlg) |
24 | 23 | 3ad2antl1 1186 | . 2 β’ (((π β AssAlg β§ 1 β π΄ β§ π΄ β π) β§ (π΄ β (SubRingβπ) β§ π΄ β πΏ)) β π β AssAlg) |
25 | 22, 24 | impbida 800 | 1 β’ ((π β AssAlg β§ 1 β π΄ β§ π΄ β π) β (π β AssAlg β (π΄ β (SubRingβπ) β§ π΄ β πΏ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wss 3911 βcfv 6497 (class class class)co 7358 Basecbs 17088 βΎs cress 17117 1rcur 19918 Ringcrg 19969 SubRingcsubrg 20232 LModclmod 20336 LSubSpclss 20407 AssAlgcasa 21272 |
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-3 12222 df-4 12223 df-5 12224 df-6 12225 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-sca 17154 df-vsca 17155 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-subrg 20234 df-lmod 20338 df-lss 20408 df-assa 21275 |
This theorem is referenced by: mplassa 21443 ply1assa 21586 |
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