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Mirrors > Home > MPE Home > Th. List > dmatcrng | Structured version Visualization version Unicode version |
Description: The subring of diagonal matrices (over a commutative ring) is a commutative ring . (Contributed by AV, 20-Aug-2019.) (Revised by AV, 18-Dec-2019.) |
Ref | Expression |
---|---|
dmatid.a | Mat |
dmatid.b | |
dmatid.0 | |
dmatid.d | DMat |
dmatcrng.c | ↾s |
Ref | Expression |
---|---|
dmatcrng |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngring 18558 | . . . 4 | |
2 | dmatid.a | . . . . 5 Mat | |
3 | dmatid.b | . . . . 5 | |
4 | dmatid.0 | . . . . 5 | |
5 | dmatid.d | . . . . 5 DMat | |
6 | 2, 3, 4, 5 | dmatsrng 20307 | . . . 4 SubRing |
7 | 1, 6 | sylan 488 | . . 3 SubRing |
8 | dmatcrng.c | . . . 4 ↾s | |
9 | 8 | subrgring 18783 | . . 3 SubRing |
10 | 7, 9 | syl 17 | . 2 |
11 | simp1lr 1125 | . . . . . . . . 9 | |
12 | eqid 2622 | . . . . . . . . . 10 | |
13 | eqid 2622 | . . . . . . . . . 10 | |
14 | simp2 1062 | . . . . . . . . . 10 | |
15 | simp3 1063 | . . . . . . . . . 10 | |
16 | 2, 13, 4, 5 | dmatmat 20300 | . . . . . . . . . . . . 13 |
17 | 16 | imp 445 | . . . . . . . . . . . 12 |
18 | 17 | adantrr 753 | . . . . . . . . . . 11 |
19 | 18 | 3ad2ant1 1082 | . . . . . . . . . 10 |
20 | 2, 12, 13, 14, 15, 19 | matecld 20232 | . . . . . . . . 9 |
21 | 2, 13, 4, 5 | dmatmat 20300 | . . . . . . . . . . . . 13 |
22 | 21 | imp 445 | . . . . . . . . . . . 12 |
23 | 22 | adantrl 752 | . . . . . . . . . . 11 |
24 | 23 | 3ad2ant1 1082 | . . . . . . . . . 10 |
25 | 2, 12, 13, 14, 15, 24 | matecld 20232 | . . . . . . . . 9 |
26 | eqid 2622 | . . . . . . . . . 10 | |
27 | 12, 26 | crngcom 18562 | . . . . . . . . 9 |
28 | 11, 20, 25, 27 | syl3anc 1326 | . . . . . . . 8 |
29 | 28 | ifeq1d 4104 | . . . . . . 7 |
30 | 29 | mpt2eq3dva 6719 | . . . . . 6 |
31 | 1 | anim2i 593 | . . . . . . 7 |
32 | 2, 3, 4, 5 | dmatmul 20303 | . . . . . . 7 |
33 | 31, 32 | sylan 488 | . . . . . 6 |
34 | pm3.22 465 | . . . . . . 7 | |
35 | 2, 3, 4, 5 | dmatmul 20303 | . . . . . . 7 |
36 | 31, 34, 35 | syl2an 494 | . . . . . 6 |
37 | 30, 33, 36 | 3eqtr4d 2666 | . . . . 5 |
38 | 37 | ralrimivva 2971 | . . . 4 |
39 | 38 | ancoms 469 | . . 3 |
40 | 8 | subrgbas 18789 | . . . . . 6 SubRing |
41 | 40 | eqcomd 2628 | . . . . 5 SubRing |
42 | eqid 2622 | . . . . . . . . . 10 | |
43 | 8, 42 | ressmulr 16006 | . . . . . . . . 9 SubRing |
44 | 43 | eqcomd 2628 | . . . . . . . 8 SubRing |
45 | 44 | oveqd 6667 | . . . . . . 7 SubRing |
46 | 44 | oveqd 6667 | . . . . . . 7 SubRing |
47 | 45, 46 | eqeq12d 2637 | . . . . . 6 SubRing |
48 | 41, 47 | raleqbidv 3152 | . . . . 5 SubRing |
49 | 41, 48 | raleqbidv 3152 | . . . 4 SubRing |
50 | 7, 49 | syl 17 | . . 3 |
51 | 39, 50 | mpbird 247 | . 2 |
52 | eqid 2622 | . . 3 | |
53 | eqid 2622 | . . 3 | |
54 | 52, 53 | iscrng2 18563 | . 2 |
55 | 10, 51, 54 | sylanbrc 698 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 cif 4086 cfv 5888 (class class class)co 6650 cmpt2 6652 cfn 7955 cbs 15857 ↾s cress 15858 cmulr 15942 c0g 16100 crg 18547 ccrg 18548 SubRingcsubrg 18776 Mat cmat 20213 DMat cdmat 20294 |
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-inf2 8538 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-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 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-se 5074 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-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-oi 8415 df-card 8765 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-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 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-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-0g 16102 df-gsum 16103 df-prds 16108 df-pws 16110 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-subrg 18778 df-lmod 18865 df-lss 18933 df-sra 19172 df-rgmod 19173 df-dsmm 20076 df-frlm 20091 df-mamu 20190 df-mat 20214 df-dmat 20296 |
This theorem is referenced by: (None) |
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