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Mirrors > Home > MPE Home > Th. List > issubassa2 | Structured version Visualization version Unicode version |
Description: A subring of a unital algebra is a subspace and thus a subalgebra iff it contains all scalar multiples of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.) |
Ref | Expression |
---|---|
issubassa2.a | algSc |
issubassa2.l |
Ref | Expression |
---|---|
issubassa2 | AssAlg SubRing |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issubassa2.a | . . . . 5 algSc | |
2 | eqid 2622 | . . . . 5 | |
3 | eqid 2622 | . . . . 5 | |
4 | 1, 2, 3 | rnascl 19343 | . . . 4 AssAlg |
5 | 4 | ad2antrr 762 | . . 3 AssAlg SubRing |
6 | issubassa2.l | . . . 4 | |
7 | assalmod 19319 | . . . . 5 AssAlg | |
8 | 7 | ad2antrr 762 | . . . 4 AssAlg SubRing |
9 | simpr 477 | . . . 4 AssAlg SubRing | |
10 | 2 | subrg1cl 18788 | . . . . 5 SubRing |
11 | 10 | ad2antlr 763 | . . . 4 AssAlg SubRing |
12 | 6, 3, 8, 9, 11 | lspsnel5a 18996 | . . 3 AssAlg SubRing |
13 | 5, 12 | eqsstrd 3639 | . 2 AssAlg SubRing |
14 | subrgsubg 18786 | . . . 4 SubRing SubGrp | |
15 | 14 | ad2antlr 763 | . . 3 AssAlg SubRing SubGrp |
16 | simplll 798 | . . . . . 6 AssAlg SubRing Scalar AssAlg | |
17 | simprl 794 | . . . . . 6 AssAlg SubRing Scalar Scalar | |
18 | eqid 2622 | . . . . . . . . . 10 | |
19 | 18 | subrgss 18781 | . . . . . . . . 9 SubRing |
20 | 19 | ad2antlr 763 | . . . . . . . 8 AssAlg SubRing |
21 | 20 | sselda 3603 | . . . . . . 7 AssAlg SubRing |
22 | 21 | adantrl 752 | . . . . . 6 AssAlg SubRing Scalar |
23 | eqid 2622 | . . . . . . 7 Scalar Scalar | |
24 | eqid 2622 | . . . . . . 7 Scalar Scalar | |
25 | eqid 2622 | . . . . . . 7 | |
26 | eqid 2622 | . . . . . . 7 | |
27 | 1, 23, 24, 18, 25, 26 | asclmul1 19339 | . . . . . 6 AssAlg Scalar |
28 | 16, 17, 22, 27 | syl3anc 1326 | . . . . 5 AssAlg SubRing Scalar |
29 | simpllr 799 | . . . . . 6 AssAlg SubRing Scalar SubRing | |
30 | simplr 792 | . . . . . . . 8 AssAlg SubRing Scalar | |
31 | 1, 23, 24 | asclfn 19336 | . . . . . . . . . 10 Scalar |
32 | 31 | a1i 11 | . . . . . . . . 9 AssAlg SubRing Scalar |
33 | fnfvelrn 6356 | . . . . . . . . 9 Scalar Scalar | |
34 | 32, 33 | sylan 488 | . . . . . . . 8 AssAlg SubRing Scalar |
35 | 30, 34 | sseldd 3604 | . . . . . . 7 AssAlg SubRing Scalar |
36 | 35 | adantrr 753 | . . . . . 6 AssAlg SubRing Scalar |
37 | simprr 796 | . . . . . 6 AssAlg SubRing Scalar | |
38 | 25 | subrgmcl 18792 | . . . . . 6 SubRing |
39 | 29, 36, 37, 38 | syl3anc 1326 | . . . . 5 AssAlg SubRing Scalar |
40 | 28, 39 | eqeltrrd 2702 | . . . 4 AssAlg SubRing Scalar |
41 | 40 | ralrimivva 2971 | . . 3 AssAlg SubRing Scalar |
42 | 23, 24, 18, 26, 6 | islss4 18962 | . . . . 5 SubGrp Scalar |
43 | 7, 42 | syl 17 | . . . 4 AssAlg SubGrp Scalar |
44 | 43 | ad2antrr 762 | . . 3 AssAlg SubRing SubGrp Scalar |
45 | 15, 41, 44 | mpbir2and 957 | . 2 AssAlg SubRing |
46 | 13, 45 | impbida 877 | 1 AssAlg SubRing |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 wss 3574 csn 4177 crn 5115 wfn 5883 cfv 5888 (class class class)co 6650 cbs 15857 cmulr 15942 Scalarcsca 15944 cvsca 15945 SubGrpcsubg 17588 cur 18501 SubRingcsubrg 18776 clmod 18863 clss 18932 clspn 18971 AssAlgcasa 19309 algSccascl 19311 |
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-int 4476 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-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 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-lsp 18972 df-assa 19312 df-ascl 19314 |
This theorem is referenced by: aspval2 19347 |
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