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Theorem dmatcrng 19294
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 17527 . . . 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 19293 . . . 4  |-  ( ( R  e.  Ring  /\  N  e.  Fin )  ->  D  e.  (SubRing `  A )
)
71, 6sylan 469 . . 3  |-  ( ( R  e.  CRing  /\  N  e.  Fin )  ->  D  e.  (SubRing `  A )
)
8 dmatcrng.c . . . 4  |-  C  =  ( As  D )
98subrgring 17750 . . 3  |-  ( D  e.  (SubRing `  A
)  ->  C  e.  Ring )
107, 9syl 17 . 2  |-  ( ( R  e.  CRing  /\  N  e.  Fin )  ->  C  e.  Ring )
11 simp1lr 1061 . . . . . . . . 9  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing
)  /\  ( x  e.  D  /\  y  e.  D ) )  /\  a  e.  N  /\  b  e.  N )  ->  R  e.  CRing )
12 eqid 2402 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
13 eqid 2402 . . . . . . . . . 10  |-  ( Base `  A )  =  (
Base `  A )
14 simp2 998 . . . . . . . . . 10  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing
)  /\  ( x  e.  D  /\  y  e.  D ) )  /\  a  e.  N  /\  b  e.  N )  ->  a  e.  N )
15 simp3 999 . . . . . . . . . 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 19286 . . . . . . . . . . . . 13  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( x  e.  D  ->  x  e.  ( Base `  A ) ) )
1716imp 427 . . . . . . . . . . . 12  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  x  e.  D )  ->  x  e.  ( Base `  A ) )
1817adantrr 715 . . . . . . . . . . 11  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  D  /\  y  e.  D
) )  ->  x  e.  ( Base `  A
) )
19183ad2ant1 1018 . . . . . . . . . 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 19218 . . . . . . . . 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 19286 . . . . . . . . . . . . 13  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( y  e.  D  ->  y  e.  ( Base `  A ) ) )
2221imp 427 . . . . . . . . . . . 12  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  y  e.  D )  ->  y  e.  ( Base `  A ) )
2322adantrl 714 . . . . . . . . . . 11  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  D  /\  y  e.  D
) )  ->  y  e.  ( Base `  A
) )
24233ad2ant1 1018 . . . . . . . . . 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 19218 . . . . . . . . 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 2402 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
2712, 26crngcom 17531 . . . . . . . . 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 1230 . . . . . . . 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 3902 . . . . . . 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 6341 . . . . . 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 567 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( N  e.  Fin  /\  R  e.  Ring )
)
322, 3, 4, 5dmatmul 19289 . . . . . . 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 469 . . . . . 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 pm3.22 447 . . . . . . 7  |-  ( ( x  e.  D  /\  y  e.  D )  ->  ( y  e.  D  /\  x  e.  D
) )
352, 3, 4, 5dmatmul 19289 . . . . . . 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 475 . . . . . 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 2453 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  D  /\  y  e.  D
) )  ->  (
x ( .r `  A ) y )  =  ( y ( .r `  A ) x ) )
3837ralrimivva 2824 . . . 4  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A. x  e.  D  A. y  e.  D  ( x ( .r
`  A ) y )  =  ( y ( .r `  A
) x ) )
3938ancoms 451 . . 3  |-  ( ( R  e.  CRing  /\  N  e.  Fin )  ->  A. x  e.  D  A. y  e.  D  ( x
( .r `  A
) y )  =  ( y ( .r
`  A ) x ) )
408subrgbas 17756 . . . . . 6  |-  ( D  e.  (SubRing `  A
)  ->  D  =  ( Base `  C )
)
4140eqcomd 2410 . . . . 5  |-  ( D  e.  (SubRing `  A
)  ->  ( Base `  C )  =  D )
42 eqid 2402 . . . . . . . . . 10  |-  ( .r
`  A )  =  ( .r `  A
)
438, 42ressmulr 14964 . . . . . . . . 9  |-  ( D  e.  (SubRing `  A
)  ->  ( .r `  A )  =  ( .r `  C ) )
4443eqcomd 2410 . . . . . . . 8  |-  ( D  e.  (SubRing `  A
)  ->  ( .r `  C )  =  ( .r `  A ) )
4544oveqd 6294 . . . . . . 7  |-  ( D  e.  (SubRing `  A
)  ->  ( x
( .r `  C
) y )  =  ( x ( .r
`  A ) y ) )
4644oveqd 6294 . . . . . . 7  |-  ( D  e.  (SubRing `  A
)  ->  ( y
( .r `  C
) x )  =  ( y ( .r
`  A ) x ) )
4745, 46eqeq12d 2424 . . . . . 6  |-  ( D  e.  (SubRing `  A
)  ->  ( (
x ( .r `  C ) y )  =  ( y ( .r `  C ) x )  <->  ( x
( .r `  A
) y )  =  ( y ( .r
`  A ) x ) ) )
4841, 47raleqbidv 3017 . . . . 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 3017 . . . 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 232 . 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 2402 . . 3  |-  ( Base `  C )  =  (
Base `  C )
53 eqid 2402 . . 3  |-  ( .r
`  C )  =  ( .r `  C
)
5452, 53iscrng2 17532 . 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 662 1  |-  ( ( R  e.  CRing  /\  N  e.  Fin )  ->  C  e.  CRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2753   ifcif 3884   ` cfv 5568  (class class class)co 6277    |-> cmpt2 6279   Fincfn 7553   Basecbs 14839   ↾s cress 14840   .rcmulr 14908   0gc0g 15052   Ringcrg 17516   CRingccrg 17517  SubRingcsubrg 17743   Mat cmat 19199   DMat cdmat 19280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-ot 3980  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6520  df-om 6683  df-1st 6783  df-2nd 6784  df-supp 6902  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-ixp 7507  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-fsupp 7863  df-sup 7934  df-oi 7968  df-card 8351  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-7 10639  df-8 10640  df-9 10641  df-10 10642  df-n0 10836  df-z 10905  df-dec 11019  df-uz 11127  df-fz 11725  df-fzo 11853  df-seq 12150  df-hash 12451  df-struct 14841  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-mulr 14921  df-sca 14923  df-vsca 14924  df-ip 14925  df-tset 14926  df-ple 14927  df-ds 14929  df-hom 14931  df-cco 14932  df-0g 15054  df-gsum 15055  df-prds 15060  df-pws 15062  df-mre 15198  df-mrc 15199  df-acs 15201  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-mhm 16288  df-submnd 16289  df-grp 16379  df-minusg 16380  df-sbg 16381  df-mulg 16382  df-subg 16520  df-ghm 16587  df-cntz 16677  df-cmn 17122  df-abl 17123  df-mgp 17460  df-ur 17472  df-ring 17518  df-cring 17519  df-subrg 17745  df-lmod 17832  df-lss 17897  df-sra 18136  df-rgmod 18137  df-dsmm 19059  df-frlm 19074  df-mamu 19176  df-mat 19200  df-dmat 19282
This theorem is referenced by: (None)
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