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Mirrors > Home > MPE Home > Th. List > scmatsgrp1 | Structured version Visualization version GIF version |
Description: The set of scalar matrices is a subgroup of the group/ring of diagonal matrices. (Contributed by AV, 21-Aug-2019.) |
Ref | Expression |
---|---|
scmatid.a | β’ π΄ = (π Mat π ) |
scmatid.b | β’ π΅ = (Baseβπ΄) |
scmatid.e | β’ πΈ = (Baseβπ ) |
scmatid.0 | β’ 0 = (0gβπ ) |
scmatid.s | β’ π = (π ScMat π ) |
scmatsgrp1.d | β’ π· = (π DMat π ) |
scmatsgrp1.c | β’ πΆ = (π΄ βΎs π·) |
Ref | Expression |
---|---|
scmatsgrp1 | β’ ((π β Fin β§ π β Ring) β π β (SubGrpβπΆ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | scmatid.a | . . . . 5 β’ π΄ = (π Mat π ) | |
2 | scmatid.b | . . . . 5 β’ π΅ = (Baseβπ΄) | |
3 | scmatid.e | . . . . 5 β’ πΈ = (Baseβπ ) | |
4 | scmatid.0 | . . . . 5 β’ 0 = (0gβπ ) | |
5 | scmatid.s | . . . . 5 β’ π = (π ScMat π ) | |
6 | scmatsgrp1.d | . . . . 5 β’ π· = (π DMat π ) | |
7 | 1, 2, 3, 4, 5, 6 | scmatdmat 22433 | . . . 4 β’ ((π β Fin β§ π β Ring) β (π₯ β π β π₯ β π·)) |
8 | 7 | ssrdv 3978 | . . 3 β’ ((π β Fin β§ π β Ring) β π β π·) |
9 | 1, 2, 4, 6 | dmatsgrp 22417 | . . . . 5 β’ ((π β Ring β§ π β Fin) β π· β (SubGrpβπ΄)) |
10 | 9 | ancoms 457 | . . . 4 β’ ((π β Fin β§ π β Ring) β π· β (SubGrpβπ΄)) |
11 | scmatsgrp1.c | . . . . . 6 β’ πΆ = (π΄ βΎs π·) | |
12 | 11 | subgbas 19087 | . . . . 5 β’ (π· β (SubGrpβπ΄) β π· = (BaseβπΆ)) |
13 | 12 | eqcomd 2731 | . . . 4 β’ (π· β (SubGrpβπ΄) β (BaseβπΆ) = π·) |
14 | 10, 13 | syl 17 | . . 3 β’ ((π β Fin β§ π β Ring) β (BaseβπΆ) = π·) |
15 | 8, 14 | sseqtrrd 4014 | . 2 β’ ((π β Fin β§ π β Ring) β π β (BaseβπΆ)) |
16 | 1, 2, 3, 4, 5 | scmatid 22432 | . . 3 β’ ((π β Fin β§ π β Ring) β (1rβπ΄) β π) |
17 | 16 | ne0d 4331 | . 2 β’ ((π β Fin β§ π β Ring) β π β β ) |
18 | 10 | adantr 479 | . . . . 5 β’ (((π β Fin β§ π β Ring) β§ (π₯ β π β§ π¦ β π)) β π· β (SubGrpβπ΄)) |
19 | 7 | com12 32 | . . . . . . 7 β’ (π₯ β π β ((π β Fin β§ π β Ring) β π₯ β π·)) |
20 | 19 | adantr 479 | . . . . . 6 β’ ((π₯ β π β§ π¦ β π) β ((π β Fin β§ π β Ring) β π₯ β π·)) |
21 | 20 | impcom 406 | . . . . 5 β’ (((π β Fin β§ π β Ring) β§ (π₯ β π β§ π¦ β π)) β π₯ β π·) |
22 | 1, 2, 3, 4, 5, 6 | scmatdmat 22433 | . . . . . . 7 β’ ((π β Fin β§ π β Ring) β (π¦ β π β π¦ β π·)) |
23 | 22 | a1d 25 | . . . . . 6 β’ ((π β Fin β§ π β Ring) β (π₯ β π β (π¦ β π β π¦ β π·))) |
24 | 23 | imp32 417 | . . . . 5 β’ (((π β Fin β§ π β Ring) β§ (π₯ β π β§ π¦ β π)) β π¦ β π·) |
25 | eqid 2725 | . . . . . . 7 β’ (-gβπ΄) = (-gβπ΄) | |
26 | eqid 2725 | . . . . . . 7 β’ (-gβπΆ) = (-gβπΆ) | |
27 | 25, 11, 26 | subgsub 19095 | . . . . . 6 β’ ((π· β (SubGrpβπ΄) β§ π₯ β π· β§ π¦ β π·) β (π₯(-gβπ΄)π¦) = (π₯(-gβπΆ)π¦)) |
28 | 27 | eqcomd 2731 | . . . . 5 β’ ((π· β (SubGrpβπ΄) β§ π₯ β π· β§ π¦ β π·) β (π₯(-gβπΆ)π¦) = (π₯(-gβπ΄)π¦)) |
29 | 18, 21, 24, 28 | syl3anc 1368 | . . . 4 β’ (((π β Fin β§ π β Ring) β§ (π₯ β π β§ π¦ β π)) β (π₯(-gβπΆ)π¦) = (π₯(-gβπ΄)π¦)) |
30 | 1, 2, 3, 4, 5 | scmatsubcl 22435 | . . . 4 β’ (((π β Fin β§ π β Ring) β§ (π₯ β π β§ π¦ β π)) β (π₯(-gβπ΄)π¦) β π) |
31 | 29, 30 | eqeltrd 2825 | . . 3 β’ (((π β Fin β§ π β Ring) β§ (π₯ β π β§ π¦ β π)) β (π₯(-gβπΆ)π¦) β π) |
32 | 31 | ralrimivva 3191 | . 2 β’ ((π β Fin β§ π β Ring) β βπ₯ β π βπ¦ β π (π₯(-gβπΆ)π¦) β π) |
33 | 1, 2, 4, 6 | dmatsrng 22419 | . . . . 5 β’ ((π β Ring β§ π β Fin) β π· β (SubRingβπ΄)) |
34 | 33 | ancoms 457 | . . . 4 β’ ((π β Fin β§ π β Ring) β π· β (SubRingβπ΄)) |
35 | 11 | subrgring 20515 | . . . 4 β’ (π· β (SubRingβπ΄) β πΆ β Ring) |
36 | 34, 35 | syl 17 | . . 3 β’ ((π β Fin β§ π β Ring) β πΆ β Ring) |
37 | ringgrp 20180 | . . 3 β’ (πΆ β Ring β πΆ β Grp) | |
38 | eqid 2725 | . . . 4 β’ (BaseβπΆ) = (BaseβπΆ) | |
39 | 38, 26 | issubg4 19102 | . . 3 β’ (πΆ β Grp β (π β (SubGrpβπΆ) β (π β (BaseβπΆ) β§ π β β β§ βπ₯ β π βπ¦ β π (π₯(-gβπΆ)π¦) β π))) |
40 | 36, 37, 39 | 3syl 18 | . 2 β’ ((π β Fin β§ π β Ring) β (π β (SubGrpβπΆ) β (π β (BaseβπΆ) β§ π β β β§ βπ₯ β π βπ¦ β π (π₯(-gβπΆ)π¦) β π))) |
41 | 15, 17, 32, 40 | mpbir3and 1339 | 1 β’ ((π β Fin β§ π β Ring) β π β (SubGrpβπΆ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2930 βwral 3051 β wss 3940 β c0 4318 βcfv 6542 (class class class)co 7415 Fincfn 8960 Basecbs 17177 βΎs cress 17206 0gc0g 17418 Grpcgrp 18892 -gcsg 18894 SubGrpcsubg 19077 1rcur 20123 Ringcrg 20175 SubRingcsubrg 20508 Mat cmat 22323 DMat cdmat 22406 ScMat cscmat 22407 |
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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-om 7868 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-sup 9463 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-fz 13515 df-fzo 13658 df-seq 13997 df-hash 14320 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-hom 17254 df-cco 17255 df-0g 17420 df-gsum 17421 df-prds 17426 df-pws 17428 df-mre 17563 df-mrc 17564 df-acs 17566 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18737 df-submnd 18738 df-grp 18895 df-minusg 18896 df-sbg 18897 df-mulg 19026 df-subg 19080 df-ghm 19170 df-cntz 19270 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-subrng 20485 df-subrg 20510 df-lmod 20747 df-lss 20818 df-sra 21060 df-rgmod 21061 df-dsmm 21668 df-frlm 21683 df-mamu 22307 df-mat 22324 df-dmat 22408 df-scmat 22409 |
This theorem is referenced by: scmatsrng1 22441 |
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