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Theorem matassa 27450
Description: Existence of the matrix algebra. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypothesis
Ref Expression
matassa.a  |-  A  =  ( N Mat  R )
Assertion
Ref Expression
matassa  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A  e. AssAlg )

Proof of Theorem matassa
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 matassa.a . . 3  |-  A  =  ( N Mat  R )
2 eqid 2436 . . 3  |-  ( Base `  R )  =  (
Base `  R )
31, 2matbas2 27444 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( ( Base `  R
)  ^m  ( N  X.  N ) )  =  ( Base `  A
) )
41matsca2 27443 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  R  =  (Scalar `  A
) )
5 eqidd 2437 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( Base `  R )  =  ( Base `  R
) )
6 eqidd 2437 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( .s `  A
)  =  ( .s
`  A ) )
7 eqid 2436 . . 3  |-  ( R maMul  <. N ,  N ,  N >. )  =  ( R maMul  <. N ,  N ,  N >. )
81, 7matmulr 27436 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( R maMul  <. N ,  N ,  N >. )  =  ( .r `  A ) )
9 crngrng 15667 . . 3  |-  ( R  e.  CRing  ->  R  e.  Ring )
101matlmod 27448 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  LMod )
119, 10sylan2 461 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A  e.  LMod )
121matrng 27449 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  Ring )
139, 12sylan2 461 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A  e.  Ring )
14 simpr 448 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  R  e.  CRing )
159ad2antlr 708 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  R  e.  Ring )
16 simpll 731 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  N  e.  Fin )
17 eqid 2436 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
18 simpr1 963 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  x  e.  ( Base `  R )
)
19 simpr2 964 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  y  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
20 simpr3 965 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
212, 15, 7, 16, 16, 16, 17, 18, 19, 20mamuvs1 27432 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  ( (
( ( N  X.  N )  X.  {
x } )  o F ( .r `  R ) y ) ( R maMul  <. N ,  N ,  N >. ) z )  =  ( ( ( N  X.  N )  X.  {
x } )  o F ( .r `  R ) ( y ( R maMul  <. N ,  N ,  N >. ) z ) ) )
223adantr 452 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  ( ( Base `  R )  ^m  ( N  X.  N
) )  =  (
Base `  A )
)
2319, 22eleqtrd 2512 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  y  e.  ( Base `  A )
)
24 eqid 2436 . . . . . 6  |-  ( Base `  A )  =  (
Base `  A )
25 eqid 2436 . . . . . 6  |-  ( .s
`  A )  =  ( .s `  A
)
26 eqid 2436 . . . . . 6  |-  ( N  X.  N )  =  ( N  X.  N
)
271, 24, 2, 25, 17, 26matvsca2 27447 . . . . 5  |-  ( ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  A
) )  ->  (
x ( .s `  A ) y )  =  ( ( ( N  X.  N )  X.  { x }
)  o F ( .r `  R ) y ) )
2818, 23, 27syl2anc 643 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  ( x
( .s `  A
) y )  =  ( ( ( N  X.  N )  X. 
{ x } )  o F ( .r
`  R ) y ) )
2928oveq1d 6089 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  ( (
x ( .s `  A ) y ) ( R maMul  <. N ,  N ,  N >. ) z )  =  ( ( ( ( N  X.  N )  X. 
{ x } )  o F ( .r
`  R ) y ) ( R maMul  <. N ,  N ,  N >. ) z ) )
302, 15, 7, 16, 16, 16, 19, 20mamucl 27425 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  ( y
( R maMul  <. N ,  N ,  N >. ) z )  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
3130, 22eleqtrd 2512 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  ( y
( R maMul  <. N ,  N ,  N >. ) z )  e.  (
Base `  A )
)
321, 24, 2, 25, 17, 26matvsca2 27447 . . . 4  |-  ( ( x  e.  ( Base `  R )  /\  (
y ( R maMul  <. N ,  N ,  N >. ) z )  e.  (
Base `  A )
)  ->  ( x
( .s `  A
) ( y ( R maMul  <. N ,  N ,  N >. ) z ) )  =  ( ( ( N  X.  N
)  X.  { x } )  o F ( .r `  R
) ( y ( R maMul  <. N ,  N ,  N >. ) z ) ) )
3318, 31, 32syl2anc 643 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  ( x
( .s `  A
) ( y ( R maMul  <. N ,  N ,  N >. ) z ) )  =  ( ( ( N  X.  N
)  X.  { x } )  o F ( .r `  R
) ( y ( R maMul  <. N ,  N ,  N >. ) z ) ) )
3421, 29, 333eqtr4d 2478 . 2  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  ( (
x ( .s `  A ) y ) ( R maMul  <. N ,  N ,  N >. ) z )  =  ( x ( .s `  A ) ( y ( R maMul  <. N ,  N ,  N >. ) z ) ) )
35 simplr 732 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  R  e.  CRing
)
3635, 2, 17, 7, 16, 16, 16, 19, 18, 20mamuvs2 27433 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  ( y
( R maMul  <. N ,  N ,  N >. ) ( ( ( N  X.  N )  X. 
{ x } )  o F ( .r
`  R ) z ) )  =  ( ( ( N  X.  N )  X.  {
x } )  o F ( .r `  R ) ( y ( R maMul  <. N ,  N ,  N >. ) z ) ) )
3720, 22eleqtrd 2512 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  z  e.  ( Base `  A )
)
381, 24, 2, 25, 17, 26matvsca2 27447 . . . . 5  |-  ( ( x  e.  ( Base `  R )  /\  z  e.  ( Base `  A
) )  ->  (
x ( .s `  A ) z )  =  ( ( ( N  X.  N )  X.  { x }
)  o F ( .r `  R ) z ) )
3918, 37, 38syl2anc 643 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  ( x
( .s `  A
) z )  =  ( ( ( N  X.  N )  X. 
{ x } )  o F ( .r
`  R ) z ) )
4039oveq2d 6090 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  ( y
( R maMul  <. N ,  N ,  N >. ) ( x ( .s
`  A ) z ) )  =  ( y ( R maMul  <. N ,  N ,  N >. ) ( ( ( N  X.  N )  X. 
{ x } )  o F ( .r
`  R ) z ) ) )
4136, 40, 333eqtr4d 2478 . 2  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  ( y
( R maMul  <. N ,  N ,  N >. ) ( x ( .s
`  A ) z ) )  =  ( x ( .s `  A ) ( y ( R maMul  <. N ,  N ,  N >. ) z ) ) )
423, 4, 5, 6, 8, 11, 13, 14, 34, 41isassad 16375 1  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A  e. AssAlg )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {csn 3807   <.cotp 3811    X. cxp 4869   ` cfv 5447  (class class class)co 6074    o Fcof 6296    ^m cmap 7011   Fincfn 7102   Basecbs 13462   .rcmulr 13523   .scvsca 13526   Ringcrg 15653   CRingccrg 15654   LModclmod 15943  AssAlgcasa 16362   maMul cmmul 27408   Mat cmat 27409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-inf2 7589  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-ot 3817  df-uni 4009  df-int 4044  df-iun 4088  df-iin 4089  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-se 4535  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-isom 5456  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-of 6298  df-1st 6342  df-2nd 6343  df-riota 6542  df-recs 6626  df-rdg 6661  df-1o 6717  df-oadd 6721  df-er 6898  df-map 7013  df-ixp 7057  df-en 7103  df-dom 7104  df-sdom 7105  df-fin 7106  df-sup 7439  df-oi 7472  df-card 7819  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-nn 9994  df-2 10051  df-3 10052  df-4 10053  df-5 10054  df-6 10055  df-7 10056  df-8 10057  df-9 10058  df-10 10059  df-n0 10215  df-z 10276  df-dec 10376  df-uz 10482  df-fz 11037  df-fzo 11129  df-seq 11317  df-hash 11612  df-struct 13464  df-ndx 13465  df-slot 13466  df-base 13467  df-sets 13468  df-ress 13469  df-plusg 13535  df-mulr 13536  df-sca 13538  df-vsca 13539  df-tset 13541  df-ple 13542  df-ds 13544  df-hom 13546  df-cco 13547  df-prds 13664  df-pws 13666  df-0g 13720  df-gsum 13721  df-mre 13804  df-mrc 13805  df-acs 13807  df-mnd 14683  df-mhm 14731  df-submnd 14732  df-grp 14805  df-minusg 14806  df-sbg 14807  df-mulg 14808  df-subg 14934  df-ghm 14997  df-cntz 15109  df-cmn 15407  df-abl 15408  df-mgp 15642  df-rng 15656  df-cring 15657  df-ur 15658  df-subrg 15859  df-lmod 15945  df-lss 16002  df-sra 16237  df-rgmod 16238  df-assa 16365  df-dsmm 27167  df-frlm 27183  df-mamu 27410  df-mat 27411
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