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Theorem mat1dimcrng 19271
Description: The algebra of matrices with dimension 1 over a commutative ring is a commutative ring. (Contributed by AV, 16-Aug-2019.)
Hypotheses
Ref Expression
mat1dim.a  |-  A  =  ( { E } Mat  R )
mat1dim.b  |-  B  =  ( Base `  R
)
mat1dim.o  |-  O  = 
<. E ,  E >.
Assertion
Ref Expression
mat1dimcrng  |-  ( ( R  e.  CRing  /\  E  e.  V )  ->  A  e.  CRing )

Proof of Theorem mat1dimcrng
Dummy variables  x  y  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snfi 7634 . . 3  |-  { E }  e.  Fin
2 crngring 17529 . . . 4  |-  ( R  e.  CRing  ->  R  e.  Ring )
32adantr 463 . . 3  |-  ( ( R  e.  CRing  /\  E  e.  V )  ->  R  e.  Ring )
4 mat1dim.a . . . 4  |-  A  =  ( { E } Mat  R )
54matring 19237 . . 3  |-  ( ( { E }  e.  Fin  /\  R  e.  Ring )  ->  A  e.  Ring )
61, 3, 5sylancr 661 . 2  |-  ( ( R  e.  CRing  /\  E  e.  V )  ->  A  e.  Ring )
7 mat1dim.b . . . . . . 7  |-  B  =  ( Base `  R
)
8 mat1dim.o . . . . . . 7  |-  O  = 
<. E ,  E >.
94, 7, 8mat1dimelbas 19265 . . . . . 6  |-  ( ( R  e.  Ring  /\  E  e.  V )  ->  (
x  e.  ( Base `  A )  <->  E. a  e.  B  x  =  { <. O ,  a
>. } ) )
104, 7, 8mat1dimelbas 19265 . . . . . 6  |-  ( ( R  e.  Ring  /\  E  e.  V )  ->  (
y  e.  ( Base `  A )  <->  E. b  e.  B  y  =  { <. O ,  b
>. } ) )
119, 10anbi12d 709 . . . . 5  |-  ( ( R  e.  Ring  /\  E  e.  V )  ->  (
( x  e.  (
Base `  A )  /\  y  e.  ( Base `  A ) )  <-> 
( E. a  e.  B  x  =  { <. O ,  a >. }  /\  E. b  e.  B  y  =  { <. O ,  b >. } ) ) )
122, 11sylan 469 . . . 4  |-  ( ( R  e.  CRing  /\  E  e.  V )  ->  (
( x  e.  (
Base `  A )  /\  y  e.  ( Base `  A ) )  <-> 
( E. a  e.  B  x  =  { <. O ,  a >. }  /\  E. b  e.  B  y  =  { <. O ,  b >. } ) ) )
13 simpll 752 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  CRing  /\  E  e.  V )  /\  ( a  e.  B  /\  b  e.  B ) )  ->  R  e.  CRing )
14 simprl 756 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  CRing  /\  E  e.  V )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
a  e.  B )
15 simprr 758 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  CRing  /\  E  e.  V )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
b  e.  B )
16 eqid 2402 . . . . . . . . . . . . . . . . . . 19  |-  ( .r
`  R )  =  ( .r `  R
)
177, 16crngcom 17533 . . . . . . . . . . . . . . . . . 18  |-  ( ( R  e.  CRing  /\  a  e.  B  /\  b  e.  B )  ->  (
a ( .r `  R ) b )  =  ( b ( .r `  R ) a ) )
1813, 14, 15, 17syl3anc 1230 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  CRing  /\  E  e.  V )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a ( .r
`  R ) b )  =  ( b ( .r `  R
) a ) )
1918opeq2d 4166 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  CRing  /\  E  e.  V )  /\  ( a  e.  B  /\  b  e.  B ) )  ->  <. O ,  ( a ( .r `  R
) b ) >.  =  <. O ,  ( b ( .r `  R ) a )
>. )
2019sneqd 3984 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  CRing  /\  E  e.  V )  /\  ( a  e.  B  /\  b  e.  B ) )  ->  { <. O ,  ( a ( .r `  R ) b )
>. }  =  { <. O ,  ( b ( .r `  R ) a ) >. } )
212anim1i 566 . . . . . . . . . . . . . . . 16  |-  ( ( R  e.  CRing  /\  E  e.  V )  ->  ( R  e.  Ring  /\  E  e.  V ) )
224, 7, 8mat1dimmul 19270 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  Ring  /\  E  e.  V )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( { <. O , 
a >. }  ( .r
`  A ) {
<. O ,  b >. } )  =  { <. O ,  ( a ( .r `  R
) b ) >. } )
2321, 22sylan 469 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  CRing  /\  E  e.  V )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( { <. O , 
a >. }  ( .r
`  A ) {
<. O ,  b >. } )  =  { <. O ,  ( a ( .r `  R
) b ) >. } )
24 pm3.22 447 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  B  /\  b  e.  B )  ->  ( b  e.  B  /\  a  e.  B
) )
254, 7, 8mat1dimmul 19270 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  Ring  /\  E  e.  V )  /\  ( b  e.  B  /\  a  e.  B ) )  -> 
( { <. O , 
b >. }  ( .r
`  A ) {
<. O ,  a >. } )  =  { <. O ,  ( b ( .r `  R
) a ) >. } )
2621, 24, 25syl2an 475 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  CRing  /\  E  e.  V )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( { <. O , 
b >. }  ( .r
`  A ) {
<. O ,  a >. } )  =  { <. O ,  ( b ( .r `  R
) a ) >. } )
2720, 23, 263eqtr4d 2453 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  CRing  /\  E  e.  V )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( { <. O , 
a >. }  ( .r
`  A ) {
<. O ,  b >. } )  =  ( { <. O ,  b
>. }  ( .r `  A ) { <. O ,  a >. } ) )
2827expr 613 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  CRing  /\  E  e.  V )  /\  a  e.  B
)  ->  ( b  e.  B  ->  ( {
<. O ,  a >. }  ( .r `  A ) { <. O ,  b >. } )  =  ( { <. O ,  b >. }  ( .r `  A ) {
<. O ,  a >. } ) ) )
2928adantr 463 . . . . . . . . . . . 12  |-  ( ( ( ( R  e. 
CRing  /\  E  e.  V
)  /\  a  e.  B )  /\  x  =  { <. O ,  a
>. } )  ->  (
b  e.  B  -> 
( { <. O , 
a >. }  ( .r
`  A ) {
<. O ,  b >. } )  =  ( { <. O ,  b
>. }  ( .r `  A ) { <. O ,  a >. } ) ) )
3029imp 427 . . . . . . . . . . 11  |-  ( ( ( ( ( R  e.  CRing  /\  E  e.  V )  /\  a  e.  B )  /\  x  =  { <. O ,  a
>. } )  /\  b  e.  B )  ->  ( { <. O ,  a
>. }  ( .r `  A ) { <. O ,  b >. } )  =  ( { <. O ,  b >. }  ( .r `  A ) {
<. O ,  a >. } ) )
3130adantr 463 . . . . . . . . . 10  |-  ( ( ( ( ( ( R  e.  CRing  /\  E  e.  V )  /\  a  e.  B )  /\  x  =  { <. O ,  a
>. } )  /\  b  e.  B )  /\  y  =  { <. O ,  b
>. } )  ->  ( { <. O ,  a
>. }  ( .r `  A ) { <. O ,  b >. } )  =  ( { <. O ,  b >. }  ( .r `  A ) {
<. O ,  a >. } ) )
32 oveq12 6287 . . . . . . . . . . . . 13  |-  ( ( x  =  { <. O ,  a >. }  /\  y  =  { <. O , 
b >. } )  -> 
( x ( .r
`  A ) y )  =  ( {
<. O ,  a >. }  ( .r `  A ) { <. O ,  b >. } ) )
3332ex 432 . . . . . . . . . . . 12  |-  ( x  =  { <. O , 
a >. }  ->  (
y  =  { <. O ,  b >. }  ->  ( x ( .r `  A ) y )  =  ( { <. O ,  a >. }  ( .r `  A ) {
<. O ,  b >. } ) ) )
3433ad2antlr 725 . . . . . . . . . . 11  |-  ( ( ( ( ( R  e.  CRing  /\  E  e.  V )  /\  a  e.  B )  /\  x  =  { <. O ,  a
>. } )  /\  b  e.  B )  ->  (
y  =  { <. O ,  b >. }  ->  ( x ( .r `  A ) y )  =  ( { <. O ,  a >. }  ( .r `  A ) {
<. O ,  b >. } ) ) )
3534imp 427 . . . . . . . . . 10  |-  ( ( ( ( ( ( R  e.  CRing  /\  E  e.  V )  /\  a  e.  B )  /\  x  =  { <. O ,  a
>. } )  /\  b  e.  B )  /\  y  =  { <. O ,  b
>. } )  ->  (
x ( .r `  A ) y )  =  ( { <. O ,  a >. }  ( .r `  A ) {
<. O ,  b >. } ) )
36 oveq12 6287 . . . . . . . . . . . . 13  |-  ( ( y  =  { <. O ,  b >. }  /\  x  =  { <. O , 
a >. } )  -> 
( y ( .r
`  A ) x )  =  ( {
<. O ,  b >. }  ( .r `  A ) { <. O ,  a >. } ) )
3736expcom 433 . . . . . . . . . . . 12  |-  ( x  =  { <. O , 
a >. }  ->  (
y  =  { <. O ,  b >. }  ->  ( y ( .r `  A ) x )  =  ( { <. O ,  b >. }  ( .r `  A ) {
<. O ,  a >. } ) ) )
3837ad2antlr 725 . . . . . . . . . . 11  |-  ( ( ( ( ( R  e.  CRing  /\  E  e.  V )  /\  a  e.  B )  /\  x  =  { <. O ,  a
>. } )  /\  b  e.  B )  ->  (
y  =  { <. O ,  b >. }  ->  ( y ( .r `  A ) x )  =  ( { <. O ,  b >. }  ( .r `  A ) {
<. O ,  a >. } ) ) )
3938imp 427 . . . . . . . . . 10  |-  ( ( ( ( ( ( R  e.  CRing  /\  E  e.  V )  /\  a  e.  B )  /\  x  =  { <. O ,  a
>. } )  /\  b  e.  B )  /\  y  =  { <. O ,  b
>. } )  ->  (
y ( .r `  A ) x )  =  ( { <. O ,  b >. }  ( .r `  A ) {
<. O ,  a >. } ) )
4031, 35, 393eqtr4d 2453 . . . . . . . . 9  |-  ( ( ( ( ( ( R  e.  CRing  /\  E  e.  V )  /\  a  e.  B )  /\  x  =  { <. O ,  a
>. } )  /\  b  e.  B )  /\  y  =  { <. O ,  b
>. } )  ->  (
x ( .r `  A ) y )  =  ( y ( .r `  A ) x ) )
4140ex 432 . . . . . . . 8  |-  ( ( ( ( ( R  e.  CRing  /\  E  e.  V )  /\  a  e.  B )  /\  x  =  { <. O ,  a
>. } )  /\  b  e.  B )  ->  (
y  =  { <. O ,  b >. }  ->  ( x ( .r `  A ) y )  =  ( y ( .r `  A ) x ) ) )
4241rexlimdva 2896 . . . . . . 7  |-  ( ( ( ( R  e. 
CRing  /\  E  e.  V
)  /\  a  e.  B )  /\  x  =  { <. O ,  a
>. } )  ->  ( E. b  e.  B  y  =  { <. O , 
b >. }  ->  (
x ( .r `  A ) y )  =  ( y ( .r `  A ) x ) ) )
4342ex 432 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  E  e.  V )  /\  a  e.  B
)  ->  ( x  =  { <. O ,  a
>. }  ->  ( E. b  e.  B  y  =  { <. O ,  b
>. }  ->  ( x
( .r `  A
) y )  =  ( y ( .r
`  A ) x ) ) ) )
4443rexlimdva 2896 . . . . 5  |-  ( ( R  e.  CRing  /\  E  e.  V )  ->  ( E. a  e.  B  x  =  { <. O , 
a >. }  ->  ( E. b  e.  B  y  =  { <. O , 
b >. }  ->  (
x ( .r `  A ) y )  =  ( y ( .r `  A ) x ) ) ) )
4544impd 429 . . . 4  |-  ( ( R  e.  CRing  /\  E  e.  V )  ->  (
( E. a  e.  B  x  =  { <. O ,  a >. }  /\  E. b  e.  B  y  =  { <. O ,  b >. } )  ->  (
x ( .r `  A ) y )  =  ( y ( .r `  A ) x ) ) )
4612, 45sylbid 215 . . 3  |-  ( ( R  e.  CRing  /\  E  e.  V )  ->  (
( x  e.  (
Base `  A )  /\  y  e.  ( Base `  A ) )  ->  ( x ( .r `  A ) y )  =  ( y ( .r `  A ) x ) ) )
4746ralrimivv 2824 . 2  |-  ( ( R  e.  CRing  /\  E  e.  V )  ->  A. x  e.  ( Base `  A
) A. y  e.  ( Base `  A
) ( x ( .r `  A ) y )  =  ( y ( .r `  A ) x ) )
48 eqid 2402 . . 3  |-  ( Base `  A )  =  (
Base `  A )
49 eqid 2402 . . 3  |-  ( .r
`  A )  =  ( .r `  A
)
5048, 49iscrng2 17534 . 2  |-  ( A  e.  CRing 
<->  ( A  e.  Ring  /\ 
A. x  e.  (
Base `  A ) A. y  e.  ( Base `  A ) ( x ( .r `  A ) y )  =  ( y ( .r `  A ) x ) ) )
516, 47, 50sylanbrc 662 1  |-  ( ( R  e.  CRing  /\  E  e.  V )  ->  A  e.  CRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2754   E.wrex 2755   {csn 3972   <.cop 3978   ` cfv 5569  (class class class)co 6278   Fincfn 7554   Basecbs 14841   .rcmulr 14910   Ringcrg 17518   CRingccrg 17519   Mat cmat 19201
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 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
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 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-ot 3981  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-sup 7935  df-oi 7969  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-fz 11727  df-fzo 11855  df-seq 12152  df-hash 12453  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-sca 14925  df-vsca 14926  df-ip 14927  df-tset 14928  df-ple 14929  df-ds 14931  df-hom 14933  df-cco 14934  df-0g 15056  df-gsum 15057  df-prds 15062  df-pws 15064  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-mhm 16290  df-submnd 16291  df-grp 16381  df-minusg 16382  df-sbg 16383  df-mulg 16384  df-subg 16522  df-ghm 16589  df-cntz 16679  df-cmn 17124  df-abl 17125  df-mgp 17462  df-ur 17474  df-ring 17520  df-cring 17521  df-subrg 17747  df-lmod 17834  df-lss 17899  df-sra 18138  df-rgmod 18139  df-dsmm 19061  df-frlm 19076  df-mamu 19178  df-mat 19202
This theorem is referenced by: (None)
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