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Theorem mat1dimcrng 19439
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 7599 . . 3  |-  { E }  e.  Fin
2 crngring 17729 . . . 4  |-  ( R  e.  CRing  ->  R  e.  Ring )
32adantr 466 . . 3  |-  ( ( R  e.  CRing  /\  E  e.  V )  ->  R  e.  Ring )
4 mat1dim.a . . . 4  |-  A  =  ( { E } Mat  R )
54matring 19405 . . 3  |-  ( ( { E }  e.  Fin  /\  R  e.  Ring )  ->  A  e.  Ring )
61, 3, 5sylancr 667 . 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 19433 . . . . . 6  |-  ( ( R  e.  Ring  /\  E  e.  V )  ->  (
x  e.  ( Base `  A )  <->  E. a  e.  B  x  =  { <. O ,  a
>. } ) )
104, 7, 8mat1dimelbas 19433 . . . . . 6  |-  ( ( R  e.  Ring  /\  E  e.  V )  ->  (
y  e.  ( Base `  A )  <->  E. b  e.  B  y  =  { <. O ,  b
>. } ) )
119, 10anbi12d 715 . . . . 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 473 . . . 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 758 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  CRing  /\  E  e.  V )  /\  ( a  e.  B  /\  b  e.  B ) )  ->  R  e.  CRing )
14 simprl 762 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  CRing  /\  E  e.  V )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
a  e.  B )
15 simprr 764 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  CRing  /\  E  e.  V )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
b  e.  B )
16 eqid 2423 . . . . . . . . . . . . . . . . . . 19  |-  ( .r
`  R )  =  ( .r `  R
)
177, 16crngcom 17733 . . . . . . . . . . . . . . . . . 18  |-  ( ( R  e.  CRing  /\  a  e.  B  /\  b  e.  B )  ->  (
a ( .r `  R ) b )  =  ( b ( .r `  R ) a ) )
1813, 14, 15, 17syl3anc 1264 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  CRing  /\  E  e.  V )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a ( .r
`  R ) b )  =  ( b ( .r `  R
) a ) )
1918opeq2d 4132 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  CRing  /\  E  e.  V )  /\  ( a  e.  B  /\  b  e.  B ) )  ->  <. O ,  ( a ( .r `  R
) b ) >.  =  <. O ,  ( b ( .r `  R ) a )
>. )
2019sneqd 3948 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  CRing  /\  E  e.  V )  /\  ( a  e.  B  /\  b  e.  B ) )  ->  { <. O ,  ( a ( .r `  R ) b )
>. }  =  { <. O ,  ( b ( .r `  R ) a ) >. } )
212anim1i 570 . . . . . . . . . . . . . . . 16  |-  ( ( R  e.  CRing  /\  E  e.  V )  ->  ( R  e.  Ring  /\  E  e.  V ) )
224, 7, 8mat1dimmul 19438 . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . 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 pm3.22 450 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  B  /\  b  e.  B )  ->  ( b  e.  B  /\  a  e.  B
) )
254, 7, 8mat1dimmul 19438 . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . 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 618 . . . . . . . . . . . . 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 466 . . . . . . . . . . . 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 430 . . . . . . . . . . 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 466 . . . . . . . . . 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 6253 . . . . . . . . . . . . 13  |-  ( ( x  =  { <. O ,  a >. }  /\  y  =  { <. O , 
b >. } )  -> 
( x ( .r
`  A ) y )  =  ( {
<. O ,  a >. }  ( .r `  A ) { <. O ,  b >. } ) )
3332ex 435 . . . . . . . . . . . 12  |-  ( x  =  { <. O , 
a >. }  ->  (
y  =  { <. O ,  b >. }  ->  ( x ( .r `  A ) y )  =  ( { <. O ,  a >. }  ( .r `  A ) {
<. O ,  b >. } ) ) )
3433ad2antlr 731 . . . . . . . . . . 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 430 . . . . . . . . . 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 6253 . . . . . . . . . . . . 13  |-  ( ( y  =  { <. O ,  b >. }  /\  x  =  { <. O , 
a >. } )  -> 
( y ( .r
`  A ) x )  =  ( {
<. O ,  b >. }  ( .r `  A ) { <. O ,  a >. } ) )
3736expcom 436 . . . . . . . . . . . 12  |-  ( x  =  { <. O , 
a >. }  ->  (
y  =  { <. O ,  b >. }  ->  ( y ( .r `  A ) x )  =  ( { <. O ,  b >. }  ( .r `  A ) {
<. O ,  a >. } ) ) )
3837ad2antlr 731 . . . . . . . . . . 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 430 . . . . . . . . . 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 2467 . . . . . . . . 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 435 . . . . . . . 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 2851 . . . . . . 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 435 . . . . . 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 2851 . . . . 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 432 . . . 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 218 . . 3  |-  ( ( R  e.  CRing  /\  E  e.  V )  ->  (
( x  e.  (
Base `  A )  /\  y  e.  ( Base `  A ) )  ->  ( x ( .r `  A ) y )  =  ( y ( .r `  A ) x ) ) )
4746ralrimivv 2780 . 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 2423 . . 3  |-  ( Base `  A )  =  (
Base `  A )
49 eqid 2423 . . 3  |-  ( .r
`  A )  =  ( .r `  A
)
5048, 49iscrng2 17734 . 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 668 1  |-  ( ( R  e.  CRing  /\  E  e.  V )  ->  A  e.  CRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   A.wral 2709   E.wrex 2710   {csn 3936   <.cop 3942   ` cfv 5539  (class class class)co 6244   Fincfn 7519   Basecbs 15059   .rcmulr 15129   Ringcrg 17718   CRingccrg 17719   Mat cmat 19369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-rep 4474  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536  ax-inf2 8094  ax-cnex 9541  ax-resscn 9542  ax-1cn 9543  ax-icn 9544  ax-addcl 9545  ax-addrcl 9546  ax-mulcl 9547  ax-mulrcl 9548  ax-mulcom 9549  ax-addass 9550  ax-mulass 9551  ax-distr 9552  ax-i2m1 9553  ax-1ne0 9554  ax-1rid 9555  ax-rnegex 9556  ax-rrecex 9557  ax-cnre 9558  ax-pre-lttri 9559  ax-pre-lttrn 9560  ax-pre-ltadd 9561  ax-pre-mulgt0 9562
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-nel 2597  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-pss 3390  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-ot 3945  df-uni 4158  df-int 4194  df-iun 4239  df-iin 4240  df-br 4362  df-opab 4421  df-mpt 4422  df-tr 4457  df-eprel 4702  df-id 4706  df-po 4712  df-so 4713  df-fr 4750  df-se 4751  df-we 4752  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-pred 5337  df-ord 5383  df-on 5384  df-lim 5385  df-suc 5386  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-isom 5548  df-riota 6206  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-of 6484  df-om 6646  df-1st 6746  df-2nd 6747  df-supp 6865  df-wrecs 6978  df-recs 7040  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-ixp 7473  df-en 7520  df-dom 7521  df-sdom 7522  df-fin 7523  df-fsupp 7832  df-sup 7904  df-oi 7973  df-card 8320  df-pnf 9623  df-mnf 9624  df-xr 9625  df-ltxr 9626  df-le 9627  df-sub 9808  df-neg 9809  df-nn 10556  df-2 10614  df-3 10615  df-4 10616  df-5 10617  df-6 10618  df-7 10619  df-8 10620  df-9 10621  df-10 10622  df-n0 10816  df-z 10884  df-dec 10998  df-uz 11106  df-fz 11731  df-fzo 11862  df-seq 12159  df-hash 12461  df-struct 15061  df-ndx 15062  df-slot 15063  df-base 15064  df-sets 15065  df-ress 15066  df-plusg 15141  df-mulr 15142  df-sca 15144  df-vsca 15145  df-ip 15146  df-tset 15147  df-ple 15148  df-ds 15150  df-hom 15152  df-cco 15153  df-0g 15278  df-gsum 15279  df-prds 15284  df-pws 15286  df-mre 15430  df-mrc 15431  df-acs 15433  df-mgm 16426  df-sgrp 16465  df-mnd 16475  df-mhm 16520  df-submnd 16521  df-grp 16611  df-minusg 16612  df-sbg 16613  df-mulg 16614  df-subg 16752  df-ghm 16819  df-cntz 16909  df-cmn 17370  df-abl 17371  df-mgp 17662  df-ur 17674  df-ring 17720  df-cring 17721  df-subrg 17944  df-lmod 18031  df-lss 18094  df-sra 18333  df-rgmod 18334  df-dsmm 19232  df-frlm 19247  df-mamu 19346  df-mat 19370
This theorem is referenced by: (None)
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