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Theorem dmatcrng 18766
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 crngrng 16991 . . . 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 18765 . . . 4  |-  ( ( R  e.  Ring  /\  N  e.  Fin )  ->  D  e.  (SubRing `  A )
)
71, 6sylan 471 . . 3  |-  ( ( R  e.  CRing  /\  N  e.  Fin )  ->  D  e.  (SubRing `  A )
)
8 dmatcrng.c . . . 4  |-  C  =  ( As  D )
98subrgrng 17210 . . 3  |-  ( D  e.  (SubRing `  A
)  ->  C  e.  Ring )
107, 9syl 16 . 2  |-  ( ( R  e.  CRing  /\  N  e.  Fin )  ->  C  e.  Ring )
11 simp1lr 1055 . . . . . . . . 9  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing
)  /\  ( x  e.  D  /\  y  e.  D ) )  /\  a  e.  N  /\  b  e.  N )  ->  R  e.  CRing )
12 eqid 2462 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
13 eqid 2462 . . . . . . . . . 10  |-  ( Base `  A )  =  (
Base `  A )
14 simp2 992 . . . . . . . . . 10  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing
)  /\  ( x  e.  D  /\  y  e.  D ) )  /\  a  e.  N  /\  b  e.  N )  ->  a  e.  N )
15 simp3 993 . . . . . . . . . 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 18758 . . . . . . . . . . . . 13  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( x  e.  D  ->  x  e.  ( Base `  A ) ) )
1716imp 429 . . . . . . . . . . . 12  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  x  e.  D )  ->  x  e.  ( Base `  A ) )
1817adantrr 716 . . . . . . . . . . 11  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  D  /\  y  e.  D
) )  ->  x  e.  ( Base `  A
) )
19183ad2ant1 1012 . . . . . . . . . 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 18690 . . . . . . . . 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 18758 . . . . . . . . . . . . 13  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( y  e.  D  ->  y  e.  ( Base `  A ) ) )
2221imp 429 . . . . . . . . . . . 12  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  y  e.  D )  ->  y  e.  ( Base `  A ) )
2322adantrl 715 . . . . . . . . . . 11  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  D  /\  y  e.  D
) )  ->  y  e.  ( Base `  A
) )
24233ad2ant1 1012 . . . . . . . . . 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 18690 . . . . . . . . 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 2462 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
2712, 26crngcom 16995 . . . . . . . . 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 1223 . . . . . . . 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 3952 . . . . . . 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 6338 . . . . . 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 569 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( N  e.  Fin  /\  R  e.  Ring )
)
322, 3, 4, 5dmatmul 18761 . . . . . . 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 471 . . . . . 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 pm3.22 449 . . . . . . 7  |-  ( ( x  e.  D  /\  y  e.  D )  ->  ( y  e.  D  /\  x  e.  D
) )
352, 3, 4, 5dmatmul 18761 . . . . . . 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 477 . . . . . 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 2513 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  D  /\  y  e.  D
) )  ->  (
x ( .r `  A ) y )  =  ( y ( .r `  A ) x ) )
3837ralrimivva 2880 . . . 4  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A. x  e.  D  A. y  e.  D  ( x ( .r
`  A ) y )  =  ( y ( .r `  A
) x ) )
3938ancoms 453 . . 3  |-  ( ( R  e.  CRing  /\  N  e.  Fin )  ->  A. x  e.  D  A. y  e.  D  ( x
( .r `  A
) y )  =  ( y ( .r
`  A ) x ) )
408subrgbas 17216 . . . . . 6  |-  ( D  e.  (SubRing `  A
)  ->  D  =  ( Base `  C )
)
4140eqcomd 2470 . . . . 5  |-  ( D  e.  (SubRing `  A
)  ->  ( Base `  C )  =  D )
42 eqid 2462 . . . . . . . . . 10  |-  ( .r
`  A )  =  ( .r `  A
)
438, 42ressmulr 14599 . . . . . . . . 9  |-  ( D  e.  (SubRing `  A
)  ->  ( .r `  A )  =  ( .r `  C ) )
4443eqcomd 2470 . . . . . . . 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 2484 . . . . . 6  |-  ( D  e.  (SubRing `  A
)  ->  ( (
x ( .r `  C ) y )  =  ( y ( .r `  C ) x )  <->  ( x
( .r `  A
) y )  =  ( y ( .r
`  A ) x ) ) )
4841, 47raleqbidv 3067 . . . . 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 3067 . . . 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 16 . . 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 2462 . . 3  |-  ( Base `  C )  =  (
Base `  C )
53 eqid 2462 . . 3  |-  ( .r
`  C )  =  ( .r `  C
)
5452, 53iscrng2 16996 . 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 664 1  |-  ( ( R  e.  CRing  /\  N  e.  Fin )  ->  C  e.  CRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2809   ifcif 3934   ` cfv 5581  (class class class)co 6277    |-> cmpt2 6279   Fincfn 7508   Basecbs 14481   ↾s cress 14482   .rcmulr 14547   0gc0g 14686   Ringcrg 16981   CRingccrg 16982  SubRingcsubrg 17203   Mat cmat 18671   DMat cdmat 18752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-ot 4031  df-uni 4241  df-int 4278  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-om 6674  df-1st 6776  df-2nd 6777  df-supp 6894  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7462  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-fsupp 7821  df-sup 7892  df-oi 7926  df-card 8311  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-fz 11664  df-fzo 11784  df-seq 12066  df-hash 12363  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-mulr 14560  df-sca 14562  df-vsca 14563  df-ip 14564  df-tset 14565  df-ple 14566  df-ds 14568  df-hom 14570  df-cco 14571  df-0g 14688  df-gsum 14689  df-prds 14694  df-pws 14696  df-mre 14832  df-mrc 14833  df-acs 14835  df-mnd 15723  df-mhm 15772  df-submnd 15773  df-grp 15853  df-minusg 15854  df-sbg 15855  df-mulg 15856  df-subg 15988  df-ghm 16055  df-cntz 16145  df-cmn 16591  df-abl 16592  df-mgp 16927  df-ur 16939  df-rng 16983  df-cring 16984  df-subrg 17205  df-lmod 17292  df-lss 17357  df-sra 17596  df-rgmod 17597  df-dsmm 18525  df-frlm 18540  df-mamu 18648  df-mat 18672  df-dmat 18754
This theorem is referenced by: (None)
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