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Theorem scmatcrng 19558
Description: The subring of scalar matrices (over a commutative ring) is a commutative ring. (Contributed by AV, 21-Aug-2019.)
Hypotheses
Ref Expression
scmatid.a  |-  A  =  ( N Mat  R )
scmatid.b  |-  B  =  ( Base `  A
)
scmatid.e  |-  E  =  ( Base `  R
)
scmatid.0  |-  .0.  =  ( 0g `  R )
scmatid.s  |-  S  =  ( N ScMat  R )
scmatcrng.c  |-  C  =  ( As  S )
Assertion
Ref Expression
scmatcrng  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  C  e.  CRing )

Proof of Theorem scmatcrng
Dummy variables  x  y  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 crngring 17803 . . . 4  |-  ( R  e.  CRing  ->  R  e.  Ring )
2 scmatid.a . . . . 5  |-  A  =  ( N Mat  R )
3 scmatid.b . . . . 5  |-  B  =  ( Base `  A
)
4 scmatid.e . . . . 5  |-  E  =  ( Base `  R
)
5 scmatid.0 . . . . 5  |-  .0.  =  ( 0g `  R )
6 scmatid.s . . . . 5  |-  S  =  ( N ScMat  R )
72, 3, 4, 5, 6scmatsrng 19557 . . . 4  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  S  e.  (SubRing `  A
) )
81, 7sylan2 477 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  S  e.  (SubRing `  A
) )
9 scmatcrng.c . . . 4  |-  C  =  ( As  S )
109subrgring 18023 . . 3  |-  ( S  e.  (SubRing `  A
)  ->  C  e.  Ring )
118, 10syl 17 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  C  e.  Ring )
12 simp1lr 1073 . . . . . . . 8  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing
)  /\  ( x  e.  S  /\  y  e.  S ) )  /\  a  e.  N  /\  b  e.  N )  ->  R  e.  CRing )
13 eqid 2453 . . . . . . . . 9  |-  ( Base `  A )  =  (
Base `  A )
14 simp2 1010 . . . . . . . . 9  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing
)  /\  ( x  e.  S  /\  y  e.  S ) )  /\  a  e.  N  /\  b  e.  N )  ->  a  e.  N )
15 simp3 1011 . . . . . . . . 9  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing
)  /\  ( x  e.  S  /\  y  e.  S ) )  /\  a  e.  N  /\  b  e.  N )  ->  b  e.  N )
162, 13, 6scmatmat 19546 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( x  e.  S  ->  x  e.  ( Base `  A ) ) )
1716imp 431 . . . . . . . . . . 11  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  x  e.  S )  ->  x  e.  ( Base `  A ) )
1817adantrr 724 . . . . . . . . . 10  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  S  /\  y  e.  S
) )  ->  x  e.  ( Base `  A
) )
19183ad2ant1 1030 . . . . . . . . 9  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing
)  /\  ( x  e.  S  /\  y  e.  S ) )  /\  a  e.  N  /\  b  e.  N )  ->  x  e.  ( Base `  A ) )
202, 4, 13, 14, 15, 19matecld 19463 . . . . . . . 8  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing
)  /\  ( x  e.  S  /\  y  e.  S ) )  /\  a  e.  N  /\  b  e.  N )  ->  ( a x b )  e.  E )
212, 13, 6scmatmat 19546 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( y  e.  S  ->  y  e.  ( Base `  A ) ) )
2221imp 431 . . . . . . . . . . 11  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  y  e.  S )  ->  y  e.  ( Base `  A ) )
2322adantrl 723 . . . . . . . . . 10  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  S  /\  y  e.  S
) )  ->  y  e.  ( Base `  A
) )
24233ad2ant1 1030 . . . . . . . . 9  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing
)  /\  ( x  e.  S  /\  y  e.  S ) )  /\  a  e.  N  /\  b  e.  N )  ->  y  e.  ( Base `  A ) )
252, 4, 13, 14, 15, 24matecld 19463 . . . . . . . 8  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing
)  /\  ( x  e.  S  /\  y  e.  S ) )  /\  a  e.  N  /\  b  e.  N )  ->  ( a y b )  e.  E )
26 eqid 2453 . . . . . . . . 9  |-  ( .r
`  R )  =  ( .r `  R
)
274, 26crngcom 17807 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  (
a x b )  e.  E  /\  (
a y b )  e.  E )  -> 
( ( a x b ) ( .r
`  R ) ( a y b ) )  =  ( ( a y b ) ( .r `  R
) ( a x b ) ) )
2812, 20, 25, 27syl3anc 1269 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing
)  /\  ( x  e.  S  /\  y  e.  S ) )  /\  a  e.  N  /\  b  e.  N )  ->  ( ( a x b ) ( .r
`  R ) ( a y b ) )  =  ( ( a y b ) ( .r `  R
) ( a x b ) ) )
2928ifeq1d 3901 . . . . . 6  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing
)  /\  ( x  e.  S  /\  y  e.  S ) )  /\  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 6360 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  S  /\  y  e.  S
) )  ->  (
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 573 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( N  e.  Fin  /\  R  e.  Ring )
)
3231adantr 467 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  S  /\  y  e.  S
) )  ->  ( N  e.  Fin  /\  R  e.  Ring ) )
33 eqid 2453 . . . . . . . . . 10  |-  ( N DMat 
R )  =  ( N DMat  R )
342, 3, 4, 5, 6, 33scmatdmat 19552 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( x  e.  S  ->  x  e.  ( N DMat 
R ) ) )
351, 34sylan2 477 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( x  e.  S  ->  x  e.  ( N DMat 
R ) ) )
362, 3, 4, 5, 6, 33scmatdmat 19552 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( y  e.  S  ->  y  e.  ( N DMat 
R ) ) )
371, 36sylan2 477 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( y  e.  S  ->  y  e.  ( N DMat 
R ) ) )
3835, 37anim12d 566 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( ( x  e.  S  /\  y  e.  S )  ->  (
x  e.  ( N DMat 
R )  /\  y  e.  ( N DMat  R ) ) ) )
3938imp 431 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  S  /\  y  e.  S
) )  ->  (
x  e.  ( N DMat 
R )  /\  y  e.  ( N DMat  R ) ) )
402, 3, 5, 33dmatmul 19534 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( x  e.  ( N DMat  R )  /\  y  e.  ( N DMat  R ) ) )  -> 
( x ( .r
`  A ) y )  =  ( a  e.  N ,  b  e.  N  |->  if ( a  =  b ,  ( ( a x b ) ( .r
`  R ) ( a y b ) ) ,  .0.  )
) )
4132, 39, 40syl2anc 667 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  S  /\  y  e.  S
) )  ->  (
x ( .r `  A ) y )  =  ( a  e.  N ,  b  e.  N  |->  if ( a  =  b ,  ( ( a x b ) ( .r `  R ) ( a y b ) ) ,  .0.  ) ) )
4239ancomd 453 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  S  /\  y  e.  S
) )  ->  (
y  e.  ( N DMat 
R )  /\  x  e.  ( N DMat  R ) ) )
432, 3, 5, 33dmatmul 19534 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  ( N DMat  R )  /\  x  e.  ( N DMat  R ) ) )  -> 
( y ( .r
`  A ) x )  =  ( a  e.  N ,  b  e.  N  |->  if ( a  =  b ,  ( ( a y b ) ( .r
`  R ) ( a x b ) ) ,  .0.  )
) )
4432, 42, 43syl2anc 667 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  S  /\  y  e.  S
) )  ->  (
y ( .r `  A ) x )  =  ( a  e.  N ,  b  e.  N  |->  if ( a  =  b ,  ( ( a y b ) ( .r `  R ) ( a x b ) ) ,  .0.  ) ) )
4530, 41, 443eqtr4d 2497 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  S  /\  y  e.  S
) )  ->  (
x ( .r `  A ) y )  =  ( y ( .r `  A ) x ) )
4645ralrimivva 2811 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A. x  e.  S  A. y  e.  S  ( x ( .r
`  A ) y )  =  ( y ( .r `  A
) x ) )
479subrgbas 18029 . . . . . 6  |-  ( S  e.  (SubRing `  A
)  ->  S  =  ( Base `  C )
)
4847eqcomd 2459 . . . . 5  |-  ( S  e.  (SubRing `  A
)  ->  ( Base `  C )  =  S )
49 eqid 2453 . . . . . . . . . 10  |-  ( .r
`  A )  =  ( .r `  A
)
509, 49ressmulr 15262 . . . . . . . . 9  |-  ( S  e.  (SubRing `  A
)  ->  ( .r `  A )  =  ( .r `  C ) )
5150eqcomd 2459 . . . . . . . 8  |-  ( S  e.  (SubRing `  A
)  ->  ( .r `  C )  =  ( .r `  A ) )
5251oveqd 6312 . . . . . . 7  |-  ( S  e.  (SubRing `  A
)  ->  ( x
( .r `  C
) y )  =  ( x ( .r
`  A ) y ) )
5351oveqd 6312 . . . . . . 7  |-  ( S  e.  (SubRing `  A
)  ->  ( y
( .r `  C
) x )  =  ( y ( .r
`  A ) x ) )
5452, 53eqeq12d 2468 . . . . . 6  |-  ( S  e.  (SubRing `  A
)  ->  ( (
x ( .r `  C ) y )  =  ( y ( .r `  C ) x )  <->  ( x
( .r `  A
) y )  =  ( y ( .r
`  A ) x ) ) )
5548, 54raleqbidv 3003 . . . . 5  |-  ( S  e.  (SubRing `  A
)  ->  ( A. y  e.  ( Base `  C ) ( x ( .r `  C
) y )  =  ( y ( .r
`  C ) x )  <->  A. y  e.  S  ( x ( .r
`  A ) y )  =  ( y ( .r `  A
) x ) ) )
5648, 55raleqbidv 3003 . . . 4  |-  ( S  e.  (SubRing `  A
)  ->  ( A. x  e.  ( Base `  C ) A. y  e.  ( Base `  C
) ( x ( .r `  C ) y )  =  ( y ( .r `  C ) x )  <->  A. x  e.  S  A. y  e.  S  ( x ( .r
`  A ) y )  =  ( y ( .r `  A
) x ) ) )
578, 56syl 17 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( A. x  e.  ( Base `  C
) A. y  e.  ( Base `  C
) ( x ( .r `  C ) y )  =  ( y ( .r `  C ) x )  <->  A. x  e.  S  A. y  e.  S  ( x ( .r
`  A ) y )  =  ( y ( .r `  A
) x ) ) )
5846, 57mpbird 236 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A. x  e.  ( Base `  C ) A. y  e.  ( Base `  C ) ( x ( .r `  C
) y )  =  ( y ( .r
`  C ) x ) )
59 eqid 2453 . . 3  |-  ( Base `  C )  =  (
Base `  C )
60 eqid 2453 . . 3  |-  ( .r
`  C )  =  ( .r `  C
)
6159, 60iscrng2 17808 . 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 ) ) )
6211, 58, 61sylanbrc 671 1  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  C  e.  CRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889   A.wral 2739   ifcif 3883   ` cfv 5585  (class class class)co 6295    |-> cmpt2 6297   Fincfn 7574   Basecbs 15133   ↾s cress 15134   .rcmulr 15203   0gc0g 15350   Ringcrg 17792   CRingccrg 17793  SubRingcsubrg 18016   Mat cmat 19444   DMat cdmat 19525   ScMat cscmat 19526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-ot 3979  df-uni 4202  df-int 4238  df-iun 4283  df-iin 4284  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6920  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7889  df-sup 7961  df-oi 8030  df-card 8378  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  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 12221  df-hash 12523  df-struct 15135  df-ndx 15136  df-slot 15137  df-base 15138  df-sets 15139  df-ress 15140  df-plusg 15215  df-mulr 15216  df-sca 15218  df-vsca 15219  df-ip 15220  df-tset 15221  df-ple 15222  df-ds 15224  df-hom 15226  df-cco 15227  df-0g 15352  df-gsum 15353  df-prds 15358  df-pws 15360  df-mre 15504  df-mrc 15505  df-acs 15507  df-mgm 16500  df-sgrp 16539  df-mnd 16549  df-mhm 16594  df-submnd 16595  df-grp 16685  df-minusg 16686  df-sbg 16687  df-mulg 16688  df-subg 16826  df-ghm 16893  df-cntz 16983  df-cmn 17444  df-abl 17445  df-mgp 17736  df-ur 17748  df-ring 17794  df-cring 17795  df-subrg 18018  df-lmod 18105  df-lss 18168  df-sra 18407  df-rgmod 18408  df-dsmm 19307  df-frlm 19322  df-mamu 19421  df-mat 19445  df-dmat 19527  df-scmat 19528
This theorem is referenced by: (None)
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