MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmatcrng Structured version   Unicode version

Theorem dmatcrng 19525
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.)
Hypotheses
Ref Expression
dmatid.a  |-  A  =  ( N Mat  R )
dmatid.b  |-  B  =  ( Base `  A
)
dmatid.0  |-  .0.  =  ( 0g `  R )
dmatid.d  |-  D  =  ( N DMat  R )
dmatcrng.c  |-  C  =  ( As  D )
Assertion
Ref Expression
dmatcrng  |-  ( ( R  e.  CRing  /\  N  e.  Fin )  ->  C  e.  CRing )

Proof of Theorem dmatcrng
Dummy variables  x  y  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 crngring 17790 . . . 4  |-  ( R  e.  CRing  ->  R  e.  Ring )
2 dmatid.a . . . . 5  |-  A  =  ( N Mat  R )
3 dmatid.b . . . . 5  |-  B  =  ( Base `  A
)
4 dmatid.0 . . . . 5  |-  .0.  =  ( 0g `  R )
5 dmatid.d . . . . 5  |-  D  =  ( N DMat  R )
62, 3, 4, 5dmatsrng 19524 . . . 4  |-  ( ( R  e.  Ring  /\  N  e.  Fin )  ->  D  e.  (SubRing `  A )
)
71, 6sylan 473 . . 3  |-  ( ( R  e.  CRing  /\  N  e.  Fin )  ->  D  e.  (SubRing `  A )
)
8 dmatcrng.c . . . 4  |-  C  =  ( As  D )
98subrgring 18010 . . 3  |-  ( D  e.  (SubRing `  A
)  ->  C  e.  Ring )
107, 9syl 17 . 2  |-  ( ( R  e.  CRing  /\  N  e.  Fin )  ->  C  e.  Ring )
11 simp1lr 1069 . . . . . . . . 9  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing
)  /\  ( x  e.  D  /\  y  e.  D ) )  /\  a  e.  N  /\  b  e.  N )  ->  R  e.  CRing )
12 eqid 2422 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
13 eqid 2422 . . . . . . . . . 10  |-  ( Base `  A )  =  (
Base `  A )
14 simp2 1006 . . . . . . . . . 10  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing
)  /\  ( x  e.  D  /\  y  e.  D ) )  /\  a  e.  N  /\  b  e.  N )  ->  a  e.  N )
15 simp3 1007 . . . . . . . . . 10  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing
)  /\  ( x  e.  D  /\  y  e.  D ) )  /\  a  e.  N  /\  b  e.  N )  ->  b  e.  N )
162, 13, 4, 5dmatmat 19517 . . . . . . . . . . . . 13  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( x  e.  D  ->  x  e.  ( Base `  A ) ) )
1716imp 430 . . . . . . . . . . . 12  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  x  e.  D )  ->  x  e.  ( Base `  A ) )
1817adantrr 721 . . . . . . . . . . 11  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  D  /\  y  e.  D
) )  ->  x  e.  ( Base `  A
) )
19183ad2ant1 1026 . . . . . . . . . 10  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing
)  /\  ( x  e.  D  /\  y  e.  D ) )  /\  a  e.  N  /\  b  e.  N )  ->  x  e.  ( Base `  A ) )
202, 12, 13, 14, 15, 19matecld 19449 . . . . . . . . 9  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing
)  /\  ( x  e.  D  /\  y  e.  D ) )  /\  a  e.  N  /\  b  e.  N )  ->  ( a x b )  e.  ( Base `  R ) )
212, 13, 4, 5dmatmat 19517 . . . . . . . . . . . . 13  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( y  e.  D  ->  y  e.  ( Base `  A ) ) )
2221imp 430 . . . . . . . . . . . 12  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  y  e.  D )  ->  y  e.  ( Base `  A ) )
2322adantrl 720 . . . . . . . . . . 11  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  D  /\  y  e.  D
) )  ->  y  e.  ( Base `  A
) )
24233ad2ant1 1026 . . . . . . . . . 10  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing
)  /\  ( x  e.  D  /\  y  e.  D ) )  /\  a  e.  N  /\  b  e.  N )  ->  y  e.  ( Base `  A ) )
252, 12, 13, 14, 15, 24matecld 19449 . . . . . . . . 9  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing
)  /\  ( x  e.  D  /\  y  e.  D ) )  /\  a  e.  N  /\  b  e.  N )  ->  ( a y b )  e.  ( Base `  R ) )
26 eqid 2422 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
2712, 26crngcom 17794 . . . . . . . . 9  |-  ( ( R  e.  CRing  /\  (
a x b )  e.  ( Base `  R
)  /\  ( a
y b )  e.  ( Base `  R
) )  ->  (
( a x b ) ( .r `  R ) ( a y b ) )  =  ( ( a y b ) ( .r `  R ) ( a x b ) ) )
2811, 20, 25, 27syl3anc 1264 . . . . . . . 8  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing
)  /\  ( x  e.  D  /\  y  e.  D ) )  /\  a  e.  N  /\  b  e.  N )  ->  ( ( a x b ) ( .r
`  R ) ( a y b ) )  =  ( ( a y b ) ( .r `  R
) ( a x b ) ) )
2928ifeq1d 3929 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing
)  /\  ( x  e.  D  /\  y  e.  D ) )  /\  a  e.  N  /\  b  e.  N )  ->  if ( a  =  b ,  ( ( a x b ) ( .r `  R
) ( a y b ) ) ,  .0.  )  =  if ( a  =  b ,  ( ( a y b ) ( .r `  R ) ( a x b ) ) ,  .0.  ) )
3029mpt2eq3dva 6369 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  D  /\  y  e.  D
) )  ->  (
a  e.  N , 
b  e.  N  |->  if ( a  =  b ,  ( ( a x b ) ( .r `  R ) ( a y b ) ) ,  .0.  ) )  =  ( a  e.  N , 
b  e.  N  |->  if ( a  =  b ,  ( ( a y b ) ( .r `  R ) ( a x b ) ) ,  .0.  ) ) )
311anim2i 571 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( N  e.  Fin  /\  R  e.  Ring )
)
322, 3, 4, 5dmatmul 19520 . . . . . . 7  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( x  e.  D  /\  y  e.  D
) )  ->  (
x ( .r `  A ) y )  =  ( a  e.  N ,  b  e.  N  |->  if ( a  =  b ,  ( ( a x b ) ( .r `  R ) ( a y b ) ) ,  .0.  ) ) )
3331, 32sylan 473 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  D  /\  y  e.  D
) )  ->  (
x ( .r `  A ) y )  =  ( a  e.  N ,  b  e.  N  |->  if ( a  =  b ,  ( ( a x b ) ( .r `  R ) ( a y b ) ) ,  .0.  ) ) )
34 pm3.22 450 . . . . . . 7  |-  ( ( x  e.  D  /\  y  e.  D )  ->  ( y  e.  D  /\  x  e.  D
) )
352, 3, 4, 5dmatmul 19520 . . . . . . 7  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  D  /\  x  e.  D
) )  ->  (
y ( .r `  A ) x )  =  ( a  e.  N ,  b  e.  N  |->  if ( a  =  b ,  ( ( a y b ) ( .r `  R ) ( a x b ) ) ,  .0.  ) ) )
3631, 34, 35syl2an 479 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  D  /\  y  e.  D
) )  ->  (
y ( .r `  A ) x )  =  ( a  e.  N ,  b  e.  N  |->  if ( a  =  b ,  ( ( a y b ) ( .r `  R ) ( a x b ) ) ,  .0.  ) ) )
3730, 33, 363eqtr4d 2473 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  D  /\  y  e.  D
) )  ->  (
x ( .r `  A ) y )  =  ( y ( .r `  A ) x ) )
3837ralrimivva 2843 . . . 4  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A. x  e.  D  A. y  e.  D  ( x ( .r
`  A ) y )  =  ( y ( .r `  A
) x ) )
3938ancoms 454 . . 3  |-  ( ( R  e.  CRing  /\  N  e.  Fin )  ->  A. x  e.  D  A. y  e.  D  ( x
( .r `  A
) y )  =  ( y ( .r
`  A ) x ) )
408subrgbas 18016 . . . . . 6  |-  ( D  e.  (SubRing `  A
)  ->  D  =  ( Base `  C )
)
4140eqcomd 2430 . . . . 5  |-  ( D  e.  (SubRing `  A
)  ->  ( Base `  C )  =  D )
42 eqid 2422 . . . . . . . . . 10  |-  ( .r
`  A )  =  ( .r `  A
)
438, 42ressmulr 15249 . . . . . . . . 9  |-  ( D  e.  (SubRing `  A
)  ->  ( .r `  A )  =  ( .r `  C ) )
4443eqcomd 2430 . . . . . . . 8  |-  ( D  e.  (SubRing `  A
)  ->  ( .r `  C )  =  ( .r `  A ) )
4544oveqd 6322 . . . . . . 7  |-  ( D  e.  (SubRing `  A
)  ->  ( x
( .r `  C
) y )  =  ( x ( .r
`  A ) y ) )
4644oveqd 6322 . . . . . . 7  |-  ( D  e.  (SubRing `  A
)  ->  ( y
( .r `  C
) x )  =  ( y ( .r
`  A ) x ) )
4745, 46eqeq12d 2444 . . . . . 6  |-  ( D  e.  (SubRing `  A
)  ->  ( (
x ( .r `  C ) y )  =  ( y ( .r `  C ) x )  <->  ( x
( .r `  A
) y )  =  ( y ( .r
`  A ) x ) ) )
4841, 47raleqbidv 3036 . . . . 5  |-  ( D  e.  (SubRing `  A
)  ->  ( A. y  e.  ( Base `  C ) ( x ( .r `  C
) y )  =  ( y ( .r
`  C ) x )  <->  A. y  e.  D  ( x ( .r
`  A ) y )  =  ( y ( .r `  A
) x ) ) )
4941, 48raleqbidv 3036 . . . 4  |-  ( D  e.  (SubRing `  A
)  ->  ( A. x  e.  ( Base `  C ) A. y  e.  ( Base `  C
) ( x ( .r `  C ) y )  =  ( y ( .r `  C ) x )  <->  A. x  e.  D  A. y  e.  D  ( x ( .r
`  A ) y )  =  ( y ( .r `  A
) x ) ) )
507, 49syl 17 . . 3  |-  ( ( R  e.  CRing  /\  N  e.  Fin )  ->  ( A. x  e.  ( Base `  C ) A. y  e.  ( Base `  C ) ( x ( .r `  C
) y )  =  ( y ( .r
`  C ) x )  <->  A. x  e.  D  A. y  e.  D  ( x ( .r
`  A ) y )  =  ( y ( .r `  A
) x ) ) )
5139, 50mpbird 235 . 2  |-  ( ( R  e.  CRing  /\  N  e.  Fin )  ->  A. x  e.  ( Base `  C
) A. y  e.  ( Base `  C
) ( x ( .r `  C ) y )  =  ( y ( .r `  C ) x ) )
52 eqid 2422 . . 3  |-  ( Base `  C )  =  (
Base `  C )
53 eqid 2422 . . 3  |-  ( .r
`  C )  =  ( .r `  C
)
5452, 53iscrng2 17795 . 2  |-  ( C  e.  CRing 
<->  ( C  e.  Ring  /\ 
A. x  e.  (
Base `  C ) A. y  e.  ( Base `  C ) ( x ( .r `  C ) y )  =  ( y ( .r `  C ) x ) ) )
5510, 51, 54sylanbrc 668 1  |-  ( ( R  e.  CRing  /\  N  e.  Fin )  ->  C  e.  CRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   A.wral 2771   ifcif 3911   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   Fincfn 7580   Basecbs 15120   ↾s cress 15121   .rcmulr 15190   0gc0g 15337   Ringcrg 17779   CRingccrg 17780  SubRingcsubrg 18003   Mat cmat 19430   DMat cdmat 19511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-inf2 8155  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-ot 4007  df-uni 4220  df-int 4256  df-iun 4301  df-iin 4302  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-map 7485  df-ixp 7534  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-fsupp 7893  df-sup 7965  df-oi 8034  df-card 8381  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-fz 11792  df-fzo 11923  df-seq 12220  df-hash 12522  df-struct 15122  df-ndx 15123  df-slot 15124  df-base 15125  df-sets 15126  df-ress 15127  df-plusg 15202  df-mulr 15203  df-sca 15205  df-vsca 15206  df-ip 15207  df-tset 15208  df-ple 15209  df-ds 15211  df-hom 15213  df-cco 15214  df-0g 15339  df-gsum 15340  df-prds 15345  df-pws 15347  df-mre 15491  df-mrc 15492  df-acs 15494  df-mgm 16487  df-sgrp 16526  df-mnd 16536  df-mhm 16581  df-submnd 16582  df-grp 16672  df-minusg 16673  df-sbg 16674  df-mulg 16675  df-subg 16813  df-ghm 16880  df-cntz 16970  df-cmn 17431  df-abl 17432  df-mgp 17723  df-ur 17735  df-ring 17781  df-cring 17782  df-subrg 18005  df-lmod 18092  df-lss 18155  df-sra 18394  df-rgmod 18395  df-dsmm 19293  df-frlm 19308  df-mamu 19407  df-mat 19431  df-dmat 19513
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator