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Mirrors > Home > MPE Home > Th. List > issubassa | Structured version Visualization version Unicode 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 | ↾s |
issubassa.l | |
issubassa.v | |
issubassa.o |
Ref | Expression |
---|---|
issubassa | AssAlg AssAlg SubRing |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1064 | . . . . . 6 AssAlg AssAlg AssAlg | |
2 | assaring 19320 | . . . . . 6 AssAlg | |
3 | 1, 2 | syl 17 | . . . . 5 AssAlg AssAlg |
4 | issubassa.s | . . . . . 6 ↾s | |
5 | assaring 19320 | . . . . . . 7 AssAlg | |
6 | 5 | adantl 482 | . . . . . 6 AssAlg AssAlg |
7 | 4, 6 | syl5eqelr 2706 | . . . . 5 AssAlg AssAlg ↾s |
8 | 3, 7 | jca 554 | . . . 4 AssAlg AssAlg ↾s |
9 | simpl3 1066 | . . . . 5 AssAlg AssAlg | |
10 | simpl2 1065 | . . . . 5 AssAlg AssAlg | |
11 | 9, 10 | jca 554 | . . . 4 AssAlg AssAlg |
12 | issubassa.v | . . . . 5 | |
13 | issubassa.o | . . . . 5 | |
14 | 12, 13 | issubrg 18780 | . . . 4 SubRing ↾s |
15 | 8, 11, 14 | sylanbrc 698 | . . 3 AssAlg AssAlg SubRing |
16 | assalmod 19319 | . . . . 5 AssAlg | |
17 | 16 | adantl 482 | . . . 4 AssAlg AssAlg |
18 | assalmod 19319 | . . . . 5 AssAlg | |
19 | issubassa.l | . . . . . 6 | |
20 | 4, 12, 19 | islss3 18959 | . . . . 5 |
21 | 1, 18, 20 | 3syl 18 | . . . 4 AssAlg AssAlg |
22 | 9, 17, 21 | mpbir2and 957 | . . 3 AssAlg AssAlg |
23 | 15, 22 | jca 554 | . 2 AssAlg AssAlg SubRing |
24 | 12 | subrgss 18781 | . . . . . 6 SubRing |
25 | 24 | ad2antrl 764 | . . . . 5 AssAlg SubRing |
26 | 4, 12 | ressbas2 15931 | . . . . 5 |
27 | 25, 26 | syl 17 | . . . 4 AssAlg SubRing |
28 | eqid 2622 | . . . . . 6 Scalar Scalar | |
29 | 4, 28 | resssca 16031 | . . . . 5 SubRing Scalar Scalar |
30 | 29 | ad2antrl 764 | . . . 4 AssAlg SubRing Scalar Scalar |
31 | eqidd 2623 | . . . 4 AssAlg SubRing Scalar Scalar | |
32 | eqid 2622 | . . . . . 6 | |
33 | 4, 32 | ressvsca 16032 | . . . . 5 SubRing |
34 | 33 | ad2antrl 764 | . . . 4 AssAlg SubRing |
35 | eqid 2622 | . . . . . 6 | |
36 | 4, 35 | ressmulr 16006 | . . . . 5 SubRing |
37 | 36 | ad2antrl 764 | . . . 4 AssAlg SubRing |
38 | simpr 477 | . . . . 5 SubRing | |
39 | 4, 19 | lsslmod 18960 | . . . . 5 |
40 | 18, 38, 39 | syl2an 494 | . . . 4 AssAlg SubRing |
41 | 4 | subrgring 18783 | . . . . 5 SubRing |
42 | 41 | ad2antrl 764 | . . . 4 AssAlg SubRing |
43 | 28 | assasca 19321 | . . . . 5 AssAlg Scalar |
44 | 43 | adantr 481 | . . . 4 AssAlg SubRing Scalar |
45 | simpll 790 | . . . . 5 AssAlg SubRing Scalar AssAlg | |
46 | simpr1 1067 | . . . . 5 AssAlg SubRing Scalar Scalar | |
47 | 25 | adantr 481 | . . . . . 6 AssAlg SubRing Scalar |
48 | simpr2 1068 | . . . . . 6 AssAlg SubRing Scalar | |
49 | 47, 48 | sseldd 3604 | . . . . 5 AssAlg SubRing Scalar |
50 | simpr3 1069 | . . . . . 6 AssAlg SubRing Scalar | |
51 | 47, 50 | sseldd 3604 | . . . . 5 AssAlg SubRing Scalar |
52 | eqid 2622 | . . . . . 6 Scalar Scalar | |
53 | 12, 28, 52, 32, 35 | assaass 19317 | . . . . 5 AssAlg Scalar |
54 | 45, 46, 49, 51, 53 | syl13anc 1328 | . . . 4 AssAlg SubRing Scalar |
55 | 12, 28, 52, 32, 35 | assaassr 19318 | . . . . 5 AssAlg Scalar |
56 | 45, 46, 49, 51, 55 | syl13anc 1328 | . . . 4 AssAlg SubRing Scalar |
57 | 27, 30, 31, 34, 37, 40, 42, 44, 54, 56 | isassad 19323 | . . 3 AssAlg SubRing AssAlg |
58 | 57 | 3ad2antl1 1223 | . 2 AssAlg SubRing AssAlg |
59 | 23, 58 | impbida 877 | 1 AssAlg AssAlg SubRing |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wss 3574 cfv 5888 (class class class)co 6650 cbs 15857 ↾s cress 15858 cmulr 15942 Scalarcsca 15944 cvsca 15945 cur 18501 crg 18547 ccrg 18548 SubRingcsubrg 18776 clmod 18863 clss 18932 AssAlgcasa 19309 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-mgp 18490 df-ur 18502 df-ring 18549 df-subrg 18778 df-lmod 18865 df-lss 18933 df-assa 19312 |
This theorem is referenced by: mplassa 19454 ply1assa 19569 |
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