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Theorem scmatcrng 19150
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 17336 . . . 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 19149 . . . 4  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  S  e.  (SubRing `  A
) )
81, 7sylan2 474 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  S  e.  (SubRing `  A
) )
9 scmatcrng.c . . . 4  |-  C  =  ( As  S )
109subrgring 17559 . . 3  |-  ( S  e.  (SubRing `  A
)  ->  C  e.  Ring )
118, 10syl 16 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  C  e.  Ring )
12 simp1lr 1060 . . . . . . . 8  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing
)  /\  ( x  e.  S  /\  y  e.  S ) )  /\  a  e.  N  /\  b  e.  N )  ->  R  e.  CRing )
13 eqid 2457 . . . . . . . . 9  |-  ( Base `  A )  =  (
Base `  A )
14 simp2 997 . . . . . . . . 9  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing
)  /\  ( x  e.  S  /\  y  e.  S ) )  /\  a  e.  N  /\  b  e.  N )  ->  a  e.  N )
15 simp3 998 . . . . . . . . 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 19138 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( x  e.  S  ->  x  e.  ( Base `  A ) ) )
1716imp 429 . . . . . . . . . . 11  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  x  e.  S )  ->  x  e.  ( Base `  A ) )
1817adantrr 716 . . . . . . . . . 10  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  S  /\  y  e.  S
) )  ->  x  e.  ( Base `  A
) )
19183ad2ant1 1017 . . . . . . . . 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 19055 . . . . . . . 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 19138 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( y  e.  S  ->  y  e.  ( Base `  A ) ) )
2221imp 429 . . . . . . . . . . 11  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  y  e.  S )  ->  y  e.  ( Base `  A ) )
2322adantrl 715 . . . . . . . . . 10  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  S  /\  y  e.  S
) )  ->  y  e.  ( Base `  A
) )
24233ad2ant1 1017 . . . . . . . . 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 19055 . . . . . . . 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 2457 . . . . . . . . 9  |-  ( .r
`  R )  =  ( .r `  R
)
274, 26crngcom 17340 . . . . . . . 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 1228 . . . . . . 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 3962 . . . . . 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 569 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( N  e.  Fin  /\  R  e.  Ring )
)
3231adantr 465 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  S  /\  y  e.  S
) )  ->  ( N  e.  Fin  /\  R  e.  Ring ) )
33 eqid 2457 . . . . . . . . . 10  |-  ( N DMat 
R )  =  ( N DMat  R )
342, 3, 4, 5, 6, 33scmatdmat 19144 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( x  e.  S  ->  x  e.  ( N DMat 
R ) ) )
351, 34sylan2 474 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( x  e.  S  ->  x  e.  ( N DMat 
R ) ) )
362, 3, 4, 5, 6, 33scmatdmat 19144 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( y  e.  S  ->  y  e.  ( N DMat 
R ) ) )
371, 36sylan2 474 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( y  e.  S  ->  y  e.  ( N DMat 
R ) ) )
3835, 37anim12d 563 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( ( x  e.  S  /\  y  e.  S )  ->  (
x  e.  ( N DMat 
R )  /\  y  e.  ( N DMat  R ) ) ) )
3938imp 429 . . . . . 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 19126 . . . . . 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 661 . . . . 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 451 . . . . . 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 19126 . . . . . 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 661 . . . . 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 2508 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  S  /\  y  e.  S
) )  ->  (
x ( .r `  A ) y )  =  ( y ( .r `  A ) x ) )
4645ralrimivva 2878 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A. x  e.  S  A. y  e.  S  ( x ( .r
`  A ) y )  =  ( y ( .r `  A
) x ) )
479subrgbas 17565 . . . . . 6  |-  ( S  e.  (SubRing `  A
)  ->  S  =  ( Base `  C )
)
4847eqcomd 2465 . . . . 5  |-  ( S  e.  (SubRing `  A
)  ->  ( Base `  C )  =  S )
49 eqid 2457 . . . . . . . . . 10  |-  ( .r
`  A )  =  ( .r `  A
)
509, 49ressmulr 14769 . . . . . . . . 9  |-  ( S  e.  (SubRing `  A
)  ->  ( .r `  A )  =  ( .r `  C ) )
5150eqcomd 2465 . . . . . . . 8  |-  ( S  e.  (SubRing `  A
)  ->  ( .r `  C )  =  ( .r `  A ) )
5251oveqd 6313 . . . . . . 7  |-  ( S  e.  (SubRing `  A
)  ->  ( x
( .r `  C
) y )  =  ( x ( .r
`  A ) y ) )
5351oveqd 6313 . . . . . . 7  |-  ( S  e.  (SubRing `  A
)  ->  ( y
( .r `  C
) x )  =  ( y ( .r
`  A ) x ) )
5452, 53eqeq12d 2479 . . . . . 6  |-  ( S  e.  (SubRing `  A
)  ->  ( (
x ( .r `  C ) y )  =  ( y ( .r `  C ) x )  <->  ( x
( .r `  A
) y )  =  ( y ( .r
`  A ) x ) ) )
5548, 54raleqbidv 3068 . . . . 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 3068 . . . 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 16 . . 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 232 . 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 2457 . . 3  |-  ( Base `  C )  =  (
Base `  C )
60 eqid 2457 . . 3  |-  ( .r
`  C )  =  ( .r `  C
)
6159, 60iscrng2 17341 . 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 664 1  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  C  e.  CRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   ifcif 3944   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   Fincfn 7535   Basecbs 14644   ↾s cress 14645   .rcmulr 14713   0gc0g 14857   Ringcrg 17325   CRingccrg 17326  SubRingcsubrg 17552   Mat cmat 19036   DMat cdmat 19117   ScMat cscmat 19118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-ot 4041  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-fzo 11822  df-seq 12111  df-hash 12409  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-hom 14736  df-cco 14737  df-0g 14859  df-gsum 14860  df-prds 14865  df-pws 14867  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-mhm 16093  df-submnd 16094  df-grp 16184  df-minusg 16185  df-sbg 16186  df-mulg 16187  df-subg 16325  df-ghm 16392  df-cntz 16482  df-cmn 16927  df-abl 16928  df-mgp 17269  df-ur 17281  df-ring 17327  df-cring 17328  df-subrg 17554  df-lmod 17641  df-lss 17706  df-sra 17945  df-rgmod 17946  df-dsmm 18890  df-frlm 18905  df-mamu 19013  df-mat 19037  df-dmat 19119  df-scmat 19120
This theorem is referenced by: (None)
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